![Page 1: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/1.jpg)
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Introduction Equivalence Relations Equivalence Classes Partitions
Equivalence Relations
Bernd Schroder
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 2: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/2.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Sometimes, We Want to Classify Objects ViaCertain Characteristics
1. We have not defined negative differences yet, but 3−5 and2−4 should be “the same”, even though they involvedifferent symbols. (But here we’ll actually think ofequivalence as equality, and we will see shortly that thatcan be problematic.)
2. Classification helps organize different entities. “food”,“clothing”, “pharmaceuticals”, etc. are in separate sectionsof a store. (Apples and steaks are both foods, so they areequivalent. They are certainly not equal.)
3. The elephant in the room: race. Take your presenter as anexample.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 3: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/3.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Sometimes, We Want to Classify Objects ViaCertain Characteristics
1. We have not defined negative differences yet, but 3−5 and2−4 should be “the same”, even though they involvedifferent symbols.
(But here we’ll actually think ofequivalence as equality, and we will see shortly that thatcan be problematic.)
2. Classification helps organize different entities. “food”,“clothing”, “pharmaceuticals”, etc. are in separate sectionsof a store. (Apples and steaks are both foods, so they areequivalent. They are certainly not equal.)
3. The elephant in the room: race. Take your presenter as anexample.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 4: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/4.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Sometimes, We Want to Classify Objects ViaCertain Characteristics
1. We have not defined negative differences yet, but 3−5 and2−4 should be “the same”, even though they involvedifferent symbols. (But here we’ll actually think ofequivalence as equality, and we will see shortly that thatcan be problematic.)
2. Classification helps organize different entities. “food”,“clothing”, “pharmaceuticals”, etc. are in separate sectionsof a store. (Apples and steaks are both foods, so they areequivalent. They are certainly not equal.)
3. The elephant in the room: race. Take your presenter as anexample.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 5: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/5.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Sometimes, We Want to Classify Objects ViaCertain Characteristics
1. We have not defined negative differences yet, but 3−5 and2−4 should be “the same”, even though they involvedifferent symbols. (But here we’ll actually think ofequivalence as equality, and we will see shortly that thatcan be problematic.)
2. Classification helps organize different entities. “food”,“clothing”, “pharmaceuticals”, etc. are in separate sectionsof a store.
(Apples and steaks are both foods, so they areequivalent. They are certainly not equal.)
3. The elephant in the room: race. Take your presenter as anexample.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 6: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/6.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Sometimes, We Want to Classify Objects ViaCertain Characteristics
1. We have not defined negative differences yet, but 3−5 and2−4 should be “the same”, even though they involvedifferent symbols. (But here we’ll actually think ofequivalence as equality, and we will see shortly that thatcan be problematic.)
2. Classification helps organize different entities. “food”,“clothing”, “pharmaceuticals”, etc. are in separate sectionsof a store. (Apples and steaks are both foods, so they areequivalent. They are certainly not equal.)
3. The elephant in the room: race. Take your presenter as anexample.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 7: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/7.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Sometimes, We Want to Classify Objects ViaCertain Characteristics
1. We have not defined negative differences yet, but 3−5 and2−4 should be “the same”, even though they involvedifferent symbols. (But here we’ll actually think ofequivalence as equality, and we will see shortly that thatcan be problematic.)
2. Classification helps organize different entities. “food”,“clothing”, “pharmaceuticals”, etc. are in separate sectionsof a store. (Apples and steaks are both foods, so they areequivalent. They are certainly not equal.)
3. The elephant in the room: race.
Take your presenter as anexample.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 8: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/8.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Sometimes, We Want to Classify Objects ViaCertain Characteristics
1. We have not defined negative differences yet, but 3−5 and2−4 should be “the same”, even though they involvedifferent symbols. (But here we’ll actually think ofequivalence as equality, and we will see shortly that thatcan be problematic.)
2. Classification helps organize different entities. “food”,“clothing”, “pharmaceuticals”, etc. are in separate sectionsof a store. (Apples and steaks are both foods, so they areequivalent. They are certainly not equal.)
3. The elephant in the room: race. Take your presenter as anexample.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 9: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/9.jpg)
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Introduction Equivalence Relations Equivalence Classes Partitions
Relations Should Be The Right Tool
Typically we compare objects pairwise.1. When representing numbers, 3−5 is equivalent to 2−4,
because both represent the same number.2. When classifying items in a store, an apple is equivalent to
a steak in the sense that they are both foods.3. In terms of race, your presenter is equivalent to his wife.
The type of relation that we are looking for should retain keyproperties of equality without necessarily being equality.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 10: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/10.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Relations Should Be The Right ToolTypically we compare objects pairwise.
1. When representing numbers, 3−5 is equivalent to 2−4,because both represent the same number.
2. When classifying items in a store, an apple is equivalent toa steak in the sense that they are both foods.
3. In terms of race, your presenter is equivalent to his wife.The type of relation that we are looking for should retain keyproperties of equality without necessarily being equality.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 11: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/11.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Relations Should Be The Right ToolTypically we compare objects pairwise.
1. When representing numbers, 3−5 is equivalent to 2−4,because both represent the same number.
2. When classifying items in a store, an apple is equivalent toa steak in the sense that they are both foods.
3. In terms of race, your presenter is equivalent to his wife.The type of relation that we are looking for should retain keyproperties of equality without necessarily being equality.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 12: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/12.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Relations Should Be The Right ToolTypically we compare objects pairwise.
1. When representing numbers, 3−5 is equivalent to 2−4,because both represent the same number.
2. When classifying items in a store, an apple is equivalent toa steak in the sense that they are both foods.
3. In terms of race, your presenter is equivalent to his wife.The type of relation that we are looking for should retain keyproperties of equality without necessarily being equality.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 13: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/13.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Relations Should Be The Right ToolTypically we compare objects pairwise.
1. When representing numbers, 3−5 is equivalent to 2−4,because both represent the same number.
2. When classifying items in a store, an apple is equivalent toa steak in the sense that they are both foods.
3. In terms of race, your presenter is equivalent to his wife.
The type of relation that we are looking for should retain keyproperties of equality without necessarily being equality.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 14: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/14.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Relations Should Be The Right ToolTypically we compare objects pairwise.
1. When representing numbers, 3−5 is equivalent to 2−4,because both represent the same number.
2. When classifying items in a store, an apple is equivalent toa steak in the sense that they are both foods.
3. In terms of race, your presenter is equivalent to his wife.The type of relation that we are looking for should retain keyproperties of equality without necessarily being equality.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 15: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/15.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Definition.
Let X be a set. A relation ∼⊆ X×X is called anequivalence relation iff
1. ∼ is reflexive. That is, for all x ∈ X we have x∼ x.2. ∼ is symmetric. That is, for all x,y ∈ X we have x∼ y iff
y∼ x.3. ∼ is transitive. That is, for all x,y,z ∈ X we have that
x∼ y and y∼ z implies x∼ z.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 16: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/16.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Definition. Let X be a set.
A relation ∼⊆ X×X is called anequivalence relation iff
1. ∼ is reflexive. That is, for all x ∈ X we have x∼ x.2. ∼ is symmetric. That is, for all x,y ∈ X we have x∼ y iff
y∼ x.3. ∼ is transitive. That is, for all x,y,z ∈ X we have that
x∼ y and y∼ z implies x∼ z.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 17: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/17.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Definition. Let X be a set. A relation ∼⊆ X×X is called anequivalence relation iff
1. ∼ is reflexive. That is, for all x ∈ X we have x∼ x.2. ∼ is symmetric. That is, for all x,y ∈ X we have x∼ y iff
y∼ x.3. ∼ is transitive. That is, for all x,y,z ∈ X we have that
x∼ y and y∼ z implies x∼ z.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 18: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/18.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Definition. Let X be a set. A relation ∼⊆ X×X is called anequivalence relation iff
1. ∼ is reflexive. That is, for all x ∈ X we have x∼ x.
2. ∼ is symmetric. That is, for all x,y ∈ X we have x∼ y iffy∼ x.
3. ∼ is transitive. That is, for all x,y,z ∈ X we have thatx∼ y and y∼ z implies x∼ z.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 19: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/19.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Definition. Let X be a set. A relation ∼⊆ X×X is called anequivalence relation iff
1. ∼ is reflexive. That is, for all x ∈ X we have x∼ x.2. ∼ is symmetric. That is, for all x,y ∈ X we have x∼ y iff
y∼ x.
3. ∼ is transitive. That is, for all x,y,z ∈ X we have thatx∼ y and y∼ z implies x∼ z.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 20: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/20.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Definition. Let X be a set. A relation ∼⊆ X×X is called anequivalence relation iff
1. ∼ is reflexive. That is, for all x ∈ X we have x∼ x.2. ∼ is symmetric. That is, for all x,y ∈ X we have x∼ y iff
y∼ x.3. ∼ is transitive. That is, for all x,y,z ∈ X we have that
x∼ y and y∼ z implies x∼ z.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 21: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/21.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Example.
The relation ∼⊆ N×N defined by m∼ n iff everypower of 2 that divides m also divides n and vice versa is anequivalence relation.
Proof. Reflexivity. Let n ∈ N. Every power of 2 that divides n,divides n. So n∼ n.Symmetry. Let m,n ∈ N be so that m∼ n. Then every power of2 that divides m divides n and vice versa. Hence every power of2 that divides n divides m and vice versa. Therefore n∼ m.Transitivity. Let k,m,n ∈ N be so that k ∼ m and m∼ n. Thenevery power of 2 that divides k divides m and every power of 2that divides m divides n. Hence every power of 2 that divides kdivides n. The “vice versa” part is proved similarly, so k∼ n.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 22: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/22.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Example. The relation ∼⊆ N×N defined by m∼ n
iff everypower of 2 that divides m also divides n and vice versa is anequivalence relation.
Proof. Reflexivity. Let n ∈ N. Every power of 2 that divides n,divides n. So n∼ n.Symmetry. Let m,n ∈ N be so that m∼ n. Then every power of2 that divides m divides n and vice versa. Hence every power of2 that divides n divides m and vice versa. Therefore n∼ m.Transitivity. Let k,m,n ∈ N be so that k ∼ m and m∼ n. Thenevery power of 2 that divides k divides m and every power of 2that divides m divides n. Hence every power of 2 that divides kdivides n. The “vice versa” part is proved similarly, so k∼ n.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 23: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/23.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Example. The relation ∼⊆ N×N defined by m∼ n iff everypower of 2 that divides m also divides n and vice versa is anequivalence relation.
Proof. Reflexivity. Let n ∈ N. Every power of 2 that divides n,divides n. So n∼ n.Symmetry. Let m,n ∈ N be so that m∼ n. Then every power of2 that divides m divides n and vice versa. Hence every power of2 that divides n divides m and vice versa. Therefore n∼ m.Transitivity. Let k,m,n ∈ N be so that k ∼ m and m∼ n. Thenevery power of 2 that divides k divides m and every power of 2that divides m divides n. Hence every power of 2 that divides kdivides n. The “vice versa” part is proved similarly, so k∼ n.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 24: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/24.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Example. The relation ∼⊆ N×N defined by m∼ n iff everypower of 2 that divides m also divides n and vice versa is anequivalence relation.
Proof.
Reflexivity. Let n ∈ N. Every power of 2 that divides n,divides n. So n∼ n.Symmetry. Let m,n ∈ N be so that m∼ n. Then every power of2 that divides m divides n and vice versa. Hence every power of2 that divides n divides m and vice versa. Therefore n∼ m.Transitivity. Let k,m,n ∈ N be so that k ∼ m and m∼ n. Thenevery power of 2 that divides k divides m and every power of 2that divides m divides n. Hence every power of 2 that divides kdivides n. The “vice versa” part is proved similarly, so k∼ n.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 25: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/25.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Example. The relation ∼⊆ N×N defined by m∼ n iff everypower of 2 that divides m also divides n and vice versa is anequivalence relation.
Proof. Reflexivity.
Let n ∈ N. Every power of 2 that divides n,divides n. So n∼ n.Symmetry. Let m,n ∈ N be so that m∼ n. Then every power of2 that divides m divides n and vice versa. Hence every power of2 that divides n divides m and vice versa. Therefore n∼ m.Transitivity. Let k,m,n ∈ N be so that k ∼ m and m∼ n. Thenevery power of 2 that divides k divides m and every power of 2that divides m divides n. Hence every power of 2 that divides kdivides n. The “vice versa” part is proved similarly, so k∼ n.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 26: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/26.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Example. The relation ∼⊆ N×N defined by m∼ n iff everypower of 2 that divides m also divides n and vice versa is anequivalence relation.
Proof. Reflexivity. Let n ∈ N.
Every power of 2 that divides n,divides n. So n∼ n.Symmetry. Let m,n ∈ N be so that m∼ n. Then every power of2 that divides m divides n and vice versa. Hence every power of2 that divides n divides m and vice versa. Therefore n∼ m.Transitivity. Let k,m,n ∈ N be so that k ∼ m and m∼ n. Thenevery power of 2 that divides k divides m and every power of 2that divides m divides n. Hence every power of 2 that divides kdivides n. The “vice versa” part is proved similarly, so k∼ n.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 27: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/27.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Example. The relation ∼⊆ N×N defined by m∼ n iff everypower of 2 that divides m also divides n and vice versa is anequivalence relation.
Proof. Reflexivity. Let n ∈ N. Every power of 2 that divides n
,divides n. So n∼ n.Symmetry. Let m,n ∈ N be so that m∼ n. Then every power of2 that divides m divides n and vice versa. Hence every power of2 that divides n divides m and vice versa. Therefore n∼ m.Transitivity. Let k,m,n ∈ N be so that k ∼ m and m∼ n. Thenevery power of 2 that divides k divides m and every power of 2that divides m divides n. Hence every power of 2 that divides kdivides n. The “vice versa” part is proved similarly, so k∼ n.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 28: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/28.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Example. The relation ∼⊆ N×N defined by m∼ n iff everypower of 2 that divides m also divides n and vice versa is anequivalence relation.
Proof. Reflexivity. Let n ∈ N. Every power of 2 that divides n,divides n.
So n∼ n.Symmetry. Let m,n ∈ N be so that m∼ n. Then every power of2 that divides m divides n and vice versa. Hence every power of2 that divides n divides m and vice versa. Therefore n∼ m.Transitivity. Let k,m,n ∈ N be so that k ∼ m and m∼ n. Thenevery power of 2 that divides k divides m and every power of 2that divides m divides n. Hence every power of 2 that divides kdivides n. The “vice versa” part is proved similarly, so k∼ n.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 29: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/29.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Example. The relation ∼⊆ N×N defined by m∼ n iff everypower of 2 that divides m also divides n and vice versa is anequivalence relation.
Proof. Reflexivity. Let n ∈ N. Every power of 2 that divides n,divides n. So n∼ n.
Symmetry. Let m,n ∈ N be so that m∼ n. Then every power of2 that divides m divides n and vice versa. Hence every power of2 that divides n divides m and vice versa. Therefore n∼ m.Transitivity. Let k,m,n ∈ N be so that k ∼ m and m∼ n. Thenevery power of 2 that divides k divides m and every power of 2that divides m divides n. Hence every power of 2 that divides kdivides n. The “vice versa” part is proved similarly, so k∼ n.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 30: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/30.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Example. The relation ∼⊆ N×N defined by m∼ n iff everypower of 2 that divides m also divides n and vice versa is anequivalence relation.
Proof. Reflexivity. Let n ∈ N. Every power of 2 that divides n,divides n. So n∼ n.Symmetry.
Let m,n ∈ N be so that m∼ n. Then every power of2 that divides m divides n and vice versa. Hence every power of2 that divides n divides m and vice versa. Therefore n∼ m.Transitivity. Let k,m,n ∈ N be so that k ∼ m and m∼ n. Thenevery power of 2 that divides k divides m and every power of 2that divides m divides n. Hence every power of 2 that divides kdivides n. The “vice versa” part is proved similarly, so k∼ n.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 31: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/31.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Example. The relation ∼⊆ N×N defined by m∼ n iff everypower of 2 that divides m also divides n and vice versa is anequivalence relation.
Proof. Reflexivity. Let n ∈ N. Every power of 2 that divides n,divides n. So n∼ n.Symmetry. Let m,n ∈ N be so that m∼ n.
Then every power of2 that divides m divides n and vice versa. Hence every power of2 that divides n divides m and vice versa. Therefore n∼ m.Transitivity. Let k,m,n ∈ N be so that k ∼ m and m∼ n. Thenevery power of 2 that divides k divides m and every power of 2that divides m divides n. Hence every power of 2 that divides kdivides n. The “vice versa” part is proved similarly, so k∼ n.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 32: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/32.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Example. The relation ∼⊆ N×N defined by m∼ n iff everypower of 2 that divides m also divides n and vice versa is anequivalence relation.
Proof. Reflexivity. Let n ∈ N. Every power of 2 that divides n,divides n. So n∼ n.Symmetry. Let m,n ∈ N be so that m∼ n. Then every power of2 that divides m divides n and vice versa.
Hence every power of2 that divides n divides m and vice versa. Therefore n∼ m.Transitivity. Let k,m,n ∈ N be so that k ∼ m and m∼ n. Thenevery power of 2 that divides k divides m and every power of 2that divides m divides n. Hence every power of 2 that divides kdivides n. The “vice versa” part is proved similarly, so k∼ n.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 33: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/33.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Example. The relation ∼⊆ N×N defined by m∼ n iff everypower of 2 that divides m also divides n and vice versa is anequivalence relation.
Proof. Reflexivity. Let n ∈ N. Every power of 2 that divides n,divides n. So n∼ n.Symmetry. Let m,n ∈ N be so that m∼ n. Then every power of2 that divides m divides n and vice versa. Hence every power of2 that divides n divides m and vice versa.
Therefore n∼ m.Transitivity. Let k,m,n ∈ N be so that k ∼ m and m∼ n. Thenevery power of 2 that divides k divides m and every power of 2that divides m divides n. Hence every power of 2 that divides kdivides n. The “vice versa” part is proved similarly, so k∼ n.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 34: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/34.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Example. The relation ∼⊆ N×N defined by m∼ n iff everypower of 2 that divides m also divides n and vice versa is anequivalence relation.
Proof. Reflexivity. Let n ∈ N. Every power of 2 that divides n,divides n. So n∼ n.Symmetry. Let m,n ∈ N be so that m∼ n. Then every power of2 that divides m divides n and vice versa. Hence every power of2 that divides n divides m and vice versa. Therefore n∼ m.
Transitivity. Let k,m,n ∈ N be so that k ∼ m and m∼ n. Thenevery power of 2 that divides k divides m and every power of 2that divides m divides n. Hence every power of 2 that divides kdivides n. The “vice versa” part is proved similarly, so k∼ n.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 35: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/35.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Example. The relation ∼⊆ N×N defined by m∼ n iff everypower of 2 that divides m also divides n and vice versa is anequivalence relation.
Proof. Reflexivity. Let n ∈ N. Every power of 2 that divides n,divides n. So n∼ n.Symmetry. Let m,n ∈ N be so that m∼ n. Then every power of2 that divides m divides n and vice versa. Hence every power of2 that divides n divides m and vice versa. Therefore n∼ m.Transitivity.
Let k,m,n ∈ N be so that k ∼ m and m∼ n. Thenevery power of 2 that divides k divides m and every power of 2that divides m divides n. Hence every power of 2 that divides kdivides n. The “vice versa” part is proved similarly, so k∼ n.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 36: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/36.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Example. The relation ∼⊆ N×N defined by m∼ n iff everypower of 2 that divides m also divides n and vice versa is anequivalence relation.
Proof. Reflexivity. Let n ∈ N. Every power of 2 that divides n,divides n. So n∼ n.Symmetry. Let m,n ∈ N be so that m∼ n. Then every power of2 that divides m divides n and vice versa. Hence every power of2 that divides n divides m and vice versa. Therefore n∼ m.Transitivity. Let k,m,n ∈ N be so that k ∼ m and m∼ n.
Thenevery power of 2 that divides k divides m and every power of 2that divides m divides n. Hence every power of 2 that divides kdivides n. The “vice versa” part is proved similarly, so k∼ n.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 37: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/37.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Example. The relation ∼⊆ N×N defined by m∼ n iff everypower of 2 that divides m also divides n and vice versa is anequivalence relation.
Proof. Reflexivity. Let n ∈ N. Every power of 2 that divides n,divides n. So n∼ n.Symmetry. Let m,n ∈ N be so that m∼ n. Then every power of2 that divides m divides n and vice versa. Hence every power of2 that divides n divides m and vice versa. Therefore n∼ m.Transitivity. Let k,m,n ∈ N be so that k ∼ m and m∼ n. Thenevery power of 2 that divides k divides m
and every power of 2that divides m divides n. Hence every power of 2 that divides kdivides n. The “vice versa” part is proved similarly, so k∼ n.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 38: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/38.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Example. The relation ∼⊆ N×N defined by m∼ n iff everypower of 2 that divides m also divides n and vice versa is anequivalence relation.
Proof. Reflexivity. Let n ∈ N. Every power of 2 that divides n,divides n. So n∼ n.Symmetry. Let m,n ∈ N be so that m∼ n. Then every power of2 that divides m divides n and vice versa. Hence every power of2 that divides n divides m and vice versa. Therefore n∼ m.Transitivity. Let k,m,n ∈ N be so that k ∼ m and m∼ n. Thenevery power of 2 that divides k divides m and every power of 2that divides m divides n.
Hence every power of 2 that divides kdivides n. The “vice versa” part is proved similarly, so k∼ n.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 39: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/39.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Example. The relation ∼⊆ N×N defined by m∼ n iff everypower of 2 that divides m also divides n and vice versa is anequivalence relation.
Proof. Reflexivity. Let n ∈ N. Every power of 2 that divides n,divides n. So n∼ n.Symmetry. Let m,n ∈ N be so that m∼ n. Then every power of2 that divides m divides n and vice versa. Hence every power of2 that divides n divides m and vice versa. Therefore n∼ m.Transitivity. Let k,m,n ∈ N be so that k ∼ m and m∼ n. Thenevery power of 2 that divides k divides m and every power of 2that divides m divides n. Hence every power of 2 that divides kdivides n.
The “vice versa” part is proved similarly, so k∼ n.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 40: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/40.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Example. The relation ∼⊆ N×N defined by m∼ n iff everypower of 2 that divides m also divides n and vice versa is anequivalence relation.
Proof. Reflexivity. Let n ∈ N. Every power of 2 that divides n,divides n. So n∼ n.Symmetry. Let m,n ∈ N be so that m∼ n. Then every power of2 that divides m divides n and vice versa. Hence every power of2 that divides n divides m and vice versa. Therefore n∼ m.Transitivity. Let k,m,n ∈ N be so that k ∼ m and m∼ n. Thenevery power of 2 that divides k divides m and every power of 2that divides m divides n. Hence every power of 2 that divides kdivides n. The “vice versa” part is proved similarly
, so k∼ n.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 41: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/41.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Example. The relation ∼⊆ N×N defined by m∼ n iff everypower of 2 that divides m also divides n and vice versa is anequivalence relation.
Proof. Reflexivity. Let n ∈ N. Every power of 2 that divides n,divides n. So n∼ n.Symmetry. Let m,n ∈ N be so that m∼ n. Then every power of2 that divides m divides n and vice versa. Hence every power of2 that divides n divides m and vice versa. Therefore n∼ m.Transitivity. Let k,m,n ∈ N be so that k ∼ m and m∼ n. Thenevery power of 2 that divides k divides m and every power of 2that divides m divides n. Hence every power of 2 that divides kdivides n. The “vice versa” part is proved similarly, so k∼ n.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 42: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/42.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Example. The relation ∼⊆ N×N defined by m∼ n iff everypower of 2 that divides m also divides n and vice versa is anequivalence relation.
Proof. Reflexivity. Let n ∈ N. Every power of 2 that divides n,divides n. So n∼ n.Symmetry. Let m,n ∈ N be so that m∼ n. Then every power of2 that divides m divides n and vice versa. Hence every power of2 that divides n divides m and vice versa. Therefore n∼ m.Transitivity. Let k,m,n ∈ N be so that k ∼ m and m∼ n. Thenevery power of 2 that divides k divides m and every power of 2that divides m divides n. Hence every power of 2 that divides kdivides n. The “vice versa” part is proved similarly, so k∼ n.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 43: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/43.jpg)
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Introduction Equivalence Relations Equivalence Classes Partitions
Definition.
Let ∼⊆ X×X be an equivalence relation on the setX. For each x ∈ X, the set [x] := {y ∈ X : y∼ x} is called theequivalence class of x.
Example. The equivalence classes of the relation ∼⊆ N×Ndefined by m∼ n iff every power of 2 that divides m also dividesn and vice versa are of the form
{2jk : k ∈ N,2 - k
}, where
j ∈ N0.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 44: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/44.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Definition. Let ∼⊆ X×X be an equivalence relation on the setX.
For each x ∈ X, the set [x] := {y ∈ X : y∼ x} is called theequivalence class of x.
Example. The equivalence classes of the relation ∼⊆ N×Ndefined by m∼ n iff every power of 2 that divides m also dividesn and vice versa are of the form
{2jk : k ∈ N,2 - k
}, where
j ∈ N0.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 45: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/45.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Definition. Let ∼⊆ X×X be an equivalence relation on the setX. For each x ∈ X, the set [x] := {y ∈ X : y∼ x} is called theequivalence class of x.
Example. The equivalence classes of the relation ∼⊆ N×Ndefined by m∼ n iff every power of 2 that divides m also dividesn and vice versa are of the form
{2jk : k ∈ N,2 - k
}, where
j ∈ N0.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 46: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/46.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Definition. Let ∼⊆ X×X be an equivalence relation on the setX. For each x ∈ X, the set [x] := {y ∈ X : y∼ x} is called theequivalence class of x.
Example.
The equivalence classes of the relation ∼⊆ N×Ndefined by m∼ n iff every power of 2 that divides m also dividesn and vice versa are of the form
{2jk : k ∈ N,2 - k
}, where
j ∈ N0.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 47: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/47.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Definition. Let ∼⊆ X×X be an equivalence relation on the setX. For each x ∈ X, the set [x] := {y ∈ X : y∼ x} is called theequivalence class of x.
Example. The equivalence classes of the relation ∼⊆ N×Ndefined by m∼ n iff every power of 2 that divides m also dividesn and vice versa
are of the form{
2jk : k ∈ N,2 - k}
, wherej ∈ N0.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 48: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/48.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Definition. Let ∼⊆ X×X be an equivalence relation on the setX. For each x ∈ X, the set [x] := {y ∈ X : y∼ x} is called theequivalence class of x.
Example. The equivalence classes of the relation ∼⊆ N×Ndefined by m∼ n iff every power of 2 that divides m also dividesn and vice versa are of the form
{2jk : k ∈ N,2 - k
}, where
j ∈ N0.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 49: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/49.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Definition.
Let X be a set. Then a family {Xi}i∈I of subsets of Xis called a partition of X iff
1. All Xi are nonempty.2. The Xi are pairwise disjoint, that is, for i 6= j we have that
Xi∩Xj = /0.3.
⋃i∈I
Xi = X.
Proposition. Let X be a set. If ∼ is an equivalence relation onX, then the equivalence classes of ∼ form a partition of X.Conversely, if X is a set and {Xi}i∈I is a partition of X, thena∼ b iff there is an i ∈ I so that a,b ∈ Xi defines an equivalencerelation on X.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 50: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/50.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Definition. Let X be a set.
Then a family {Xi}i∈I of subsets of Xis called a partition of X iff
1. All Xi are nonempty.2. The Xi are pairwise disjoint, that is, for i 6= j we have that
Xi∩Xj = /0.3.
⋃i∈I
Xi = X.
Proposition. Let X be a set. If ∼ is an equivalence relation onX, then the equivalence classes of ∼ form a partition of X.Conversely, if X is a set and {Xi}i∈I is a partition of X, thena∼ b iff there is an i ∈ I so that a,b ∈ Xi defines an equivalencerelation on X.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 51: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/51.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Definition. Let X be a set. Then a family {Xi}i∈I of subsets of Xis called a partition of X iff
1. All Xi are nonempty.2. The Xi are pairwise disjoint, that is, for i 6= j we have that
Xi∩Xj = /0.3.
⋃i∈I
Xi = X.
Proposition. Let X be a set. If ∼ is an equivalence relation onX, then the equivalence classes of ∼ form a partition of X.Conversely, if X is a set and {Xi}i∈I is a partition of X, thena∼ b iff there is an i ∈ I so that a,b ∈ Xi defines an equivalencerelation on X.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 52: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/52.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Definition. Let X be a set. Then a family {Xi}i∈I of subsets of Xis called a partition of X iff
1. All Xi are nonempty.
2. The Xi are pairwise disjoint, that is, for i 6= j we have thatXi∩Xj = /0.
3.⋃i∈I
Xi = X.
Proposition. Let X be a set. If ∼ is an equivalence relation onX, then the equivalence classes of ∼ form a partition of X.Conversely, if X is a set and {Xi}i∈I is a partition of X, thena∼ b iff there is an i ∈ I so that a,b ∈ Xi defines an equivalencerelation on X.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 53: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/53.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Definition. Let X be a set. Then a family {Xi}i∈I of subsets of Xis called a partition of X iff
1. All Xi are nonempty.2. The Xi are pairwise disjoint, that is, for i 6= j we have that
Xi∩Xj = /0.
3.⋃i∈I
Xi = X.
Proposition. Let X be a set. If ∼ is an equivalence relation onX, then the equivalence classes of ∼ form a partition of X.Conversely, if X is a set and {Xi}i∈I is a partition of X, thena∼ b iff there is an i ∈ I so that a,b ∈ Xi defines an equivalencerelation on X.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 54: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/54.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Definition. Let X be a set. Then a family {Xi}i∈I of subsets of Xis called a partition of X iff
1. All Xi are nonempty.2. The Xi are pairwise disjoint, that is, for i 6= j we have that
Xi∩Xj = /0.3.
⋃i∈I
Xi = X.
Proposition. Let X be a set. If ∼ is an equivalence relation onX, then the equivalence classes of ∼ form a partition of X.Conversely, if X is a set and {Xi}i∈I is a partition of X, thena∼ b iff there is an i ∈ I so that a,b ∈ Xi defines an equivalencerelation on X.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 55: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/55.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Definition. Let X be a set. Then a family {Xi}i∈I of subsets of Xis called a partition of X iff
1. All Xi are nonempty.2. The Xi are pairwise disjoint, that is, for i 6= j we have that
Xi∩Xj = /0.3.
⋃i∈I
Xi = X.
Proposition.
Let X be a set. If ∼ is an equivalence relation onX, then the equivalence classes of ∼ form a partition of X.Conversely, if X is a set and {Xi}i∈I is a partition of X, thena∼ b iff there is an i ∈ I so that a,b ∈ Xi defines an equivalencerelation on X.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 56: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/56.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Definition. Let X be a set. Then a family {Xi}i∈I of subsets of Xis called a partition of X iff
1. All Xi are nonempty.2. The Xi are pairwise disjoint, that is, for i 6= j we have that
Xi∩Xj = /0.3.
⋃i∈I
Xi = X.
Proposition. Let X be a set.
If ∼ is an equivalence relation onX, then the equivalence classes of ∼ form a partition of X.Conversely, if X is a set and {Xi}i∈I is a partition of X, thena∼ b iff there is an i ∈ I so that a,b ∈ Xi defines an equivalencerelation on X.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 57: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/57.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Definition. Let X be a set. Then a family {Xi}i∈I of subsets of Xis called a partition of X iff
1. All Xi are nonempty.2. The Xi are pairwise disjoint, that is, for i 6= j we have that
Xi∩Xj = /0.3.
⋃i∈I
Xi = X.
Proposition. Let X be a set. If ∼ is an equivalence relation onX, then the equivalence classes of ∼ form a partition of X.
Conversely, if X is a set and {Xi}i∈I is a partition of X, thena∼ b iff there is an i ∈ I so that a,b ∈ Xi defines an equivalencerelation on X.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 58: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/58.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Definition. Let X be a set. Then a family {Xi}i∈I of subsets of Xis called a partition of X iff
1. All Xi are nonempty.2. The Xi are pairwise disjoint, that is, for i 6= j we have that
Xi∩Xj = /0.3.
⋃i∈I
Xi = X.
Proposition. Let X be a set. If ∼ is an equivalence relation onX, then the equivalence classes of ∼ form a partition of X.Conversely, if X is a set and {Xi}i∈I is a partition of X, thena∼ b iff there is an i ∈ I so that a,b ∈ Xi defines an equivalencerelation on X.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 59: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/59.jpg)
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Introduction Equivalence Relations Equivalence Classes Partitions
Proof (equivalence relations induce partitions).
Let ∼ be anequivalence relation on X and let
{[x] : x ∈ X
}be the family of
equivalence classes of ∼. Then [x] 6= /0 for all x, because x ∈ [x].For pairwise disjointness, we prove that [x]∩ [y] 6= /0 implies[x] = [y]. Let [x], [y] be so that there is a z ∈ [x]∩ [y]. Then x∼ zand z∼ y, which implies x∼ y. Now let u ∈ [x]. Then u∼ x andx∼ y, so u∼ y and u ∈ [y]. Hence [x]⊆ [y], and we prove[y]⊆ [x] similarly. Thus the equivalence classes of ∼ arepairwise disjoint. Finally, for
⋃{[x] : x ∈ X
}= X first note that⋃{
[x] : x ∈ X}⊆ X is clear. For the reverse containment, let
x ∈ X. Then x ∈ [x] and [x]⊆⋃{
[x] : x ∈ X}, so
X ⊆⋃{
[x] : x ∈ X}.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 60: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/60.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Proof (equivalence relations induce partitions). Let ∼ be anequivalence relation on X and let
{[x] : x ∈ X
}be the family of
equivalence classes of ∼.
Then [x] 6= /0 for all x, because x ∈ [x].For pairwise disjointness, we prove that [x]∩ [y] 6= /0 implies[x] = [y]. Let [x], [y] be so that there is a z ∈ [x]∩ [y]. Then x∼ zand z∼ y, which implies x∼ y. Now let u ∈ [x]. Then u∼ x andx∼ y, so u∼ y and u ∈ [y]. Hence [x]⊆ [y], and we prove[y]⊆ [x] similarly. Thus the equivalence classes of ∼ arepairwise disjoint. Finally, for
⋃{[x] : x ∈ X
}= X first note that⋃{
[x] : x ∈ X}⊆ X is clear. For the reverse containment, let
x ∈ X. Then x ∈ [x] and [x]⊆⋃{
[x] : x ∈ X}, so
X ⊆⋃{
[x] : x ∈ X}.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 61: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/61.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Proof (equivalence relations induce partitions). Let ∼ be anequivalence relation on X and let
{[x] : x ∈ X
}be the family of
equivalence classes of ∼. Then [x] 6= /0 for all x, because x ∈ [x].
For pairwise disjointness, we prove that [x]∩ [y] 6= /0 implies[x] = [y]. Let [x], [y] be so that there is a z ∈ [x]∩ [y]. Then x∼ zand z∼ y, which implies x∼ y. Now let u ∈ [x]. Then u∼ x andx∼ y, so u∼ y and u ∈ [y]. Hence [x]⊆ [y], and we prove[y]⊆ [x] similarly. Thus the equivalence classes of ∼ arepairwise disjoint. Finally, for
⋃{[x] : x ∈ X
}= X first note that⋃{
[x] : x ∈ X}⊆ X is clear. For the reverse containment, let
x ∈ X. Then x ∈ [x] and [x]⊆⋃{
[x] : x ∈ X}, so
X ⊆⋃{
[x] : x ∈ X}.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 62: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/62.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Proof (equivalence relations induce partitions). Let ∼ be anequivalence relation on X and let
{[x] : x ∈ X
}be the family of
equivalence classes of ∼. Then [x] 6= /0 for all x, because x ∈ [x].For pairwise disjointness, we prove that [x]∩ [y] 6= /0 implies[x] = [y].
Let [x], [y] be so that there is a z ∈ [x]∩ [y]. Then x∼ zand z∼ y, which implies x∼ y. Now let u ∈ [x]. Then u∼ x andx∼ y, so u∼ y and u ∈ [y]. Hence [x]⊆ [y], and we prove[y]⊆ [x] similarly. Thus the equivalence classes of ∼ arepairwise disjoint. Finally, for
⋃{[x] : x ∈ X
}= X first note that⋃{
[x] : x ∈ X}⊆ X is clear. For the reverse containment, let
x ∈ X. Then x ∈ [x] and [x]⊆⋃{
[x] : x ∈ X}, so
X ⊆⋃{
[x] : x ∈ X}.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 63: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/63.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Proof (equivalence relations induce partitions). Let ∼ be anequivalence relation on X and let
{[x] : x ∈ X
}be the family of
equivalence classes of ∼. Then [x] 6= /0 for all x, because x ∈ [x].For pairwise disjointness, we prove that [x]∩ [y] 6= /0 implies[x] = [y]. Let [x], [y] be so that there is a z ∈ [x]∩ [y].
Then x∼ zand z∼ y, which implies x∼ y. Now let u ∈ [x]. Then u∼ x andx∼ y, so u∼ y and u ∈ [y]. Hence [x]⊆ [y], and we prove[y]⊆ [x] similarly. Thus the equivalence classes of ∼ arepairwise disjoint. Finally, for
⋃{[x] : x ∈ X
}= X first note that⋃{
[x] : x ∈ X}⊆ X is clear. For the reverse containment, let
x ∈ X. Then x ∈ [x] and [x]⊆⋃{
[x] : x ∈ X}, so
X ⊆⋃{
[x] : x ∈ X}.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 64: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/64.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Proof (equivalence relations induce partitions). Let ∼ be anequivalence relation on X and let
{[x] : x ∈ X
}be the family of
equivalence classes of ∼. Then [x] 6= /0 for all x, because x ∈ [x].For pairwise disjointness, we prove that [x]∩ [y] 6= /0 implies[x] = [y]. Let [x], [y] be so that there is a z ∈ [x]∩ [y]. Then x∼ zand z∼ y, which implies x∼ y.
Now let u ∈ [x]. Then u∼ x andx∼ y, so u∼ y and u ∈ [y]. Hence [x]⊆ [y], and we prove[y]⊆ [x] similarly. Thus the equivalence classes of ∼ arepairwise disjoint. Finally, for
⋃{[x] : x ∈ X
}= X first note that⋃{
[x] : x ∈ X}⊆ X is clear. For the reverse containment, let
x ∈ X. Then x ∈ [x] and [x]⊆⋃{
[x] : x ∈ X}, so
X ⊆⋃{
[x] : x ∈ X}.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 65: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/65.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Proof (equivalence relations induce partitions). Let ∼ be anequivalence relation on X and let
{[x] : x ∈ X
}be the family of
equivalence classes of ∼. Then [x] 6= /0 for all x, because x ∈ [x].For pairwise disjointness, we prove that [x]∩ [y] 6= /0 implies[x] = [y]. Let [x], [y] be so that there is a z ∈ [x]∩ [y]. Then x∼ zand z∼ y, which implies x∼ y. Now let u ∈ [x].
Then u∼ x andx∼ y, so u∼ y and u ∈ [y]. Hence [x]⊆ [y], and we prove[y]⊆ [x] similarly. Thus the equivalence classes of ∼ arepairwise disjoint. Finally, for
⋃{[x] : x ∈ X
}= X first note that⋃{
[x] : x ∈ X}⊆ X is clear. For the reverse containment, let
x ∈ X. Then x ∈ [x] and [x]⊆⋃{
[x] : x ∈ X}, so
X ⊆⋃{
[x] : x ∈ X}.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 66: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/66.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Proof (equivalence relations induce partitions). Let ∼ be anequivalence relation on X and let
{[x] : x ∈ X
}be the family of
equivalence classes of ∼. Then [x] 6= /0 for all x, because x ∈ [x].For pairwise disjointness, we prove that [x]∩ [y] 6= /0 implies[x] = [y]. Let [x], [y] be so that there is a z ∈ [x]∩ [y]. Then x∼ zand z∼ y, which implies x∼ y. Now let u ∈ [x]. Then u∼ x andx∼ y, so u∼ y and u ∈ [y].
Hence [x]⊆ [y], and we prove[y]⊆ [x] similarly. Thus the equivalence classes of ∼ arepairwise disjoint. Finally, for
⋃{[x] : x ∈ X
}= X first note that⋃{
[x] : x ∈ X}⊆ X is clear. For the reverse containment, let
x ∈ X. Then x ∈ [x] and [x]⊆⋃{
[x] : x ∈ X}, so
X ⊆⋃{
[x] : x ∈ X}.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 67: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/67.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Proof (equivalence relations induce partitions). Let ∼ be anequivalence relation on X and let
{[x] : x ∈ X
}be the family of
equivalence classes of ∼. Then [x] 6= /0 for all x, because x ∈ [x].For pairwise disjointness, we prove that [x]∩ [y] 6= /0 implies[x] = [y]. Let [x], [y] be so that there is a z ∈ [x]∩ [y]. Then x∼ zand z∼ y, which implies x∼ y. Now let u ∈ [x]. Then u∼ x andx∼ y, so u∼ y and u ∈ [y]. Hence [x]⊆ [y]
, and we prove[y]⊆ [x] similarly. Thus the equivalence classes of ∼ arepairwise disjoint. Finally, for
⋃{[x] : x ∈ X
}= X first note that⋃{
[x] : x ∈ X}⊆ X is clear. For the reverse containment, let
x ∈ X. Then x ∈ [x] and [x]⊆⋃{
[x] : x ∈ X}, so
X ⊆⋃{
[x] : x ∈ X}.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 68: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/68.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Proof (equivalence relations induce partitions). Let ∼ be anequivalence relation on X and let
{[x] : x ∈ X
}be the family of
equivalence classes of ∼. Then [x] 6= /0 for all x, because x ∈ [x].For pairwise disjointness, we prove that [x]∩ [y] 6= /0 implies[x] = [y]. Let [x], [y] be so that there is a z ∈ [x]∩ [y]. Then x∼ zand z∼ y, which implies x∼ y. Now let u ∈ [x]. Then u∼ x andx∼ y, so u∼ y and u ∈ [y]. Hence [x]⊆ [y], and we prove[y]⊆ [x] similarly.
Thus the equivalence classes of ∼ arepairwise disjoint. Finally, for
⋃{[x] : x ∈ X
}= X first note that⋃{
[x] : x ∈ X}⊆ X is clear. For the reverse containment, let
x ∈ X. Then x ∈ [x] and [x]⊆⋃{
[x] : x ∈ X}, so
X ⊆⋃{
[x] : x ∈ X}.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 69: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/69.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Proof (equivalence relations induce partitions). Let ∼ be anequivalence relation on X and let
{[x] : x ∈ X
}be the family of
equivalence classes of ∼. Then [x] 6= /0 for all x, because x ∈ [x].For pairwise disjointness, we prove that [x]∩ [y] 6= /0 implies[x] = [y]. Let [x], [y] be so that there is a z ∈ [x]∩ [y]. Then x∼ zand z∼ y, which implies x∼ y. Now let u ∈ [x]. Then u∼ x andx∼ y, so u∼ y and u ∈ [y]. Hence [x]⊆ [y], and we prove[y]⊆ [x] similarly. Thus the equivalence classes of ∼ arepairwise disjoint.
Finally, for⋃{
[x] : x ∈ X}
= X first note that⋃{[x] : x ∈ X
}⊆ X is clear. For the reverse containment, let
x ∈ X. Then x ∈ [x] and [x]⊆⋃{
[x] : x ∈ X}, so
X ⊆⋃{
[x] : x ∈ X}.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 70: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/70.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Proof (equivalence relations induce partitions). Let ∼ be anequivalence relation on X and let
{[x] : x ∈ X
}be the family of
equivalence classes of ∼. Then [x] 6= /0 for all x, because x ∈ [x].For pairwise disjointness, we prove that [x]∩ [y] 6= /0 implies[x] = [y]. Let [x], [y] be so that there is a z ∈ [x]∩ [y]. Then x∼ zand z∼ y, which implies x∼ y. Now let u ∈ [x]. Then u∼ x andx∼ y, so u∼ y and u ∈ [y]. Hence [x]⊆ [y], and we prove[y]⊆ [x] similarly. Thus the equivalence classes of ∼ arepairwise disjoint. Finally, for
⋃{[x] : x ∈ X
}= X first note that⋃{
[x] : x ∈ X}⊆ X is clear.
For the reverse containment, let
x ∈ X. Then x ∈ [x] and [x]⊆⋃{
[x] : x ∈ X}, so
X ⊆⋃{
[x] : x ∈ X}.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 71: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/71.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Proof (equivalence relations induce partitions). Let ∼ be anequivalence relation on X and let
{[x] : x ∈ X
}be the family of
equivalence classes of ∼. Then [x] 6= /0 for all x, because x ∈ [x].For pairwise disjointness, we prove that [x]∩ [y] 6= /0 implies[x] = [y]. Let [x], [y] be so that there is a z ∈ [x]∩ [y]. Then x∼ zand z∼ y, which implies x∼ y. Now let u ∈ [x]. Then u∼ x andx∼ y, so u∼ y and u ∈ [y]. Hence [x]⊆ [y], and we prove[y]⊆ [x] similarly. Thus the equivalence classes of ∼ arepairwise disjoint. Finally, for
⋃{[x] : x ∈ X
}= X first note that⋃{
[x] : x ∈ X}⊆ X is clear. For the reverse containment, let
x ∈ X.
Then x ∈ [x] and [x]⊆⋃{
[x] : x ∈ X}, so
X ⊆⋃{
[x] : x ∈ X}.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 72: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/72.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Proof (equivalence relations induce partitions). Let ∼ be anequivalence relation on X and let
{[x] : x ∈ X
}be the family of
equivalence classes of ∼. Then [x] 6= /0 for all x, because x ∈ [x].For pairwise disjointness, we prove that [x]∩ [y] 6= /0 implies[x] = [y]. Let [x], [y] be so that there is a z ∈ [x]∩ [y]. Then x∼ zand z∼ y, which implies x∼ y. Now let u ∈ [x]. Then u∼ x andx∼ y, so u∼ y and u ∈ [y]. Hence [x]⊆ [y], and we prove[y]⊆ [x] similarly. Thus the equivalence classes of ∼ arepairwise disjoint. Finally, for
⋃{[x] : x ∈ X
}= X first note that⋃{
[x] : x ∈ X}⊆ X is clear. For the reverse containment, let
x ∈ X. Then x ∈ [x]
and [x]⊆⋃{
[x] : x ∈ X}, so
X ⊆⋃{
[x] : x ∈ X}.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 73: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/73.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Proof (equivalence relations induce partitions). Let ∼ be anequivalence relation on X and let
{[x] : x ∈ X
}be the family of
equivalence classes of ∼. Then [x] 6= /0 for all x, because x ∈ [x].For pairwise disjointness, we prove that [x]∩ [y] 6= /0 implies[x] = [y]. Let [x], [y] be so that there is a z ∈ [x]∩ [y]. Then x∼ zand z∼ y, which implies x∼ y. Now let u ∈ [x]. Then u∼ x andx∼ y, so u∼ y and u ∈ [y]. Hence [x]⊆ [y], and we prove[y]⊆ [x] similarly. Thus the equivalence classes of ∼ arepairwise disjoint. Finally, for
⋃{[x] : x ∈ X
}= X first note that⋃{
[x] : x ∈ X}⊆ X is clear. For the reverse containment, let
x ∈ X. Then x ∈ [x] and [x]⊆⋃{
[x] : x ∈ X},
so
X ⊆⋃{
[x] : x ∈ X}.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 74: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/74.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Proof (equivalence relations induce partitions). Let ∼ be anequivalence relation on X and let
{[x] : x ∈ X
}be the family of
equivalence classes of ∼. Then [x] 6= /0 for all x, because x ∈ [x].For pairwise disjointness, we prove that [x]∩ [y] 6= /0 implies[x] = [y]. Let [x], [y] be so that there is a z ∈ [x]∩ [y]. Then x∼ zand z∼ y, which implies x∼ y. Now let u ∈ [x]. Then u∼ x andx∼ y, so u∼ y and u ∈ [y]. Hence [x]⊆ [y], and we prove[y]⊆ [x] similarly. Thus the equivalence classes of ∼ arepairwise disjoint. Finally, for
⋃{[x] : x ∈ X
}= X first note that⋃{
[x] : x ∈ X}⊆ X is clear. For the reverse containment, let
x ∈ X. Then x ∈ [x] and [x]⊆⋃{
[x] : x ∈ X}, so
X ⊆⋃{
[x] : x ∈ X}.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 75: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/75.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Proof (partitions induce equivalence relations).
Let {Xi}i∈Ibe a partition of X, and define a∼ b iff there is an i ∈ I so thata,b ∈ Xi. For every x ∈ X there is an Xi with x ∈ Xi and hencex∼ x. If x∼ y, then there is an Xi with x,y ∈ Xi and hencey∼ x. If x∼ y and y∼ z, then there are Xi and Xj so thatx,y ∈ Xi and y,z ∈ Xj. Therefore Xi∩Xj 3 y, which impliesXi = Xj. But then x,z ∈ Xi = Xj and hence x∼ z.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 76: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/76.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Proof (partitions induce equivalence relations). Let {Xi}i∈Ibe a partition of X, and define a∼ b iff there is an i ∈ I so thata,b ∈ Xi.
For every x ∈ X there is an Xi with x ∈ Xi and hencex∼ x. If x∼ y, then there is an Xi with x,y ∈ Xi and hencey∼ x. If x∼ y and y∼ z, then there are Xi and Xj so thatx,y ∈ Xi and y,z ∈ Xj. Therefore Xi∩Xj 3 y, which impliesXi = Xj. But then x,z ∈ Xi = Xj and hence x∼ z.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 77: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/77.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Proof (partitions induce equivalence relations). Let {Xi}i∈Ibe a partition of X, and define a∼ b iff there is an i ∈ I so thata,b ∈ Xi. For every x ∈ X there is an Xi with x ∈ Xi and hencex∼ x.
If x∼ y, then there is an Xi with x,y ∈ Xi and hencey∼ x. If x∼ y and y∼ z, then there are Xi and Xj so thatx,y ∈ Xi and y,z ∈ Xj. Therefore Xi∩Xj 3 y, which impliesXi = Xj. But then x,z ∈ Xi = Xj and hence x∼ z.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 78: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/78.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Proof (partitions induce equivalence relations). Let {Xi}i∈Ibe a partition of X, and define a∼ b iff there is an i ∈ I so thata,b ∈ Xi. For every x ∈ X there is an Xi with x ∈ Xi and hencex∼ x. If x∼ y, then there is an Xi with x,y ∈ Xi and hencey∼ x.
If x∼ y and y∼ z, then there are Xi and Xj so thatx,y ∈ Xi and y,z ∈ Xj. Therefore Xi∩Xj 3 y, which impliesXi = Xj. But then x,z ∈ Xi = Xj and hence x∼ z.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 79: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/79.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Proof (partitions induce equivalence relations). Let {Xi}i∈Ibe a partition of X, and define a∼ b iff there is an i ∈ I so thata,b ∈ Xi. For every x ∈ X there is an Xi with x ∈ Xi and hencex∼ x. If x∼ y, then there is an Xi with x,y ∈ Xi and hencey∼ x. If x∼ y and y∼ z, then there are Xi and Xj so thatx,y ∈ Xi and y,z ∈ Xj.
Therefore Xi∩Xj 3 y, which impliesXi = Xj. But then x,z ∈ Xi = Xj and hence x∼ z.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 80: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/80.jpg)
logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Proof (partitions induce equivalence relations). Let {Xi}i∈Ibe a partition of X, and define a∼ b iff there is an i ∈ I so thata,b ∈ Xi. For every x ∈ X there is an Xi with x ∈ Xi and hencex∼ x. If x∼ y, then there is an Xi with x,y ∈ Xi and hencey∼ x. If x∼ y and y∼ z, then there are Xi and Xj so thatx,y ∈ Xi and y,z ∈ Xj. Therefore Xi∩Xj 3 y, which impliesXi = Xj.
But then x,z ∈ Xi = Xj and hence x∼ z.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
![Page 81: Equivalence Relations · Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Equivalence Relations. ... Classification helps organize different entities](https://reader030.vdocument.in/reader030/viewer/2022041003/5ea632ec3d6169654b265a68/html5/thumbnails/81.jpg)
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Introduction Equivalence Relations Equivalence Classes Partitions
Proof (partitions induce equivalence relations). Let {Xi}i∈Ibe a partition of X, and define a∼ b iff there is an i ∈ I so thata,b ∈ Xi. For every x ∈ X there is an Xi with x ∈ Xi and hencex∼ x. If x∼ y, then there is an Xi with x,y ∈ Xi and hencey∼ x. If x∼ y and y∼ z, then there are Xi and Xj so thatx,y ∈ Xi and y,z ∈ Xj. Therefore Xi∩Xj 3 y, which impliesXi = Xj. But then x,z ∈ Xi = Xj and hence x∼ z.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations
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logo1
Introduction Equivalence Relations Equivalence Classes Partitions
Proof (partitions induce equivalence relations). Let {Xi}i∈Ibe a partition of X, and define a∼ b iff there is an i ∈ I so thata,b ∈ Xi. For every x ∈ X there is an Xi with x ∈ Xi and hencex∼ x. If x∼ y, then there is an Xi with x,y ∈ Xi and hencey∼ x. If x∼ y and y∼ z, then there are Xi and Xj so thatx,y ∈ Xi and y,z ∈ Xj. Therefore Xi∩Xj 3 y, which impliesXi = Xj. But then x,z ∈ Xi = Xj and hence x∼ z.
Bernd Schroder Louisiana Tech University, College of Engineering and Science
Equivalence Relations