math 110 sec 2-1 , 2-2 lecture on sets and comparing sets
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MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets A set is a collection of objects.
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets A set is a collection of objects.
For example: the set of seasons S = {Spring, Summer, Fall, Winter}
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets A set is a collection of objects.
For example: the set of seasons S = {Spring, Summer, Fall, Winter}
element
Each object is called an element of the set
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets A set is a collection of objects.
element
Each object is called an element of the set
For example: the set of seasons S = {Spring, Summer, Fall, Winter}
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets A set is a collection of objects.
element
Each object is called an element of the set
For example: the set of seasons S = {Spring, Summer, Fall, Winter}
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets A set is a collection of objects.
element
Each object is called an element of the set
For example: the set of seasons S = {Spring, Summer, Fall, Winter}
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets A set is a collection of objects.
For example: the set of seasons S = {Spring, Summer, Fall, Winter}
There is a standard notation for indicating the number of elements in a set.
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets A set is a collection of objects.
For example: the set of seasons S = {Spring, Summer, Fall, Winter}
There is a standard notation for indicating the number of elements in a set.
1 2 3 4
The set S above has 4 elements
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets A set is a collection of objects.
For example: the set of seasons S = {Spring, Summer, Fall, Winter}
There is a standard notation for indicating the number of elements in a set.
1 2 3 4
The set S above has 4 elementsso we write
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets A set is a collection of objects.
For example: the set of seasons S = {Spring, Summer, Fall, Winter}
There is a standard notation for indicating the number of elements in a set.
1 2 3 4
The set S above has 4 elements
n(S) = 4so we write
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets A set is a collection of objects.
More examples of sets:
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets A set is a collection of objects.
More examples of sets:T = {1, 2, 3, 4, 5}
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets A set is a collection of objects.
More examples of sets:T = {1, 2, 3, 4, 5}U = {1, 2, 3, … , 1000}
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets A set is a collection of objects.
More examples of sets:T = {1, 2, 3, 4, 5}U = {1, 2, 3, … , 1000}V = {1, 2, 3, 4, …}
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets A set is a collection of objects.
More examples of sets:T = {1, 2, 3, 4, 5}U = {1, 2, 3, … , 1000}V = {1, 2, 3, 4, …}W = {x : x is a 2 legged animal}
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets A set is a collection of objects.
More examples of sets:T = {1, 2, 3, 4, 5}U = {1, 2, 3, … , 1000}V = {1, 2, 3, 4, …}W = {x : x is a 2 legged animal}
W is written in what we call ‘set builder’ notation Read as: “The set of all x, such that x is a 2 legged animal.”
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets A set is a collection of objects.
More examples of sets:T = {1, 2, 3, 4, 5}U = {1, 2, 3, … , 1000}V = {1, 2, 3, 4, …}W = {x : x is a 2 legged animal}
W is written in what we call ‘set builder’ notation Read as: “The set of all x, such that x is a 2 legged animal.”
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets A set is a collection of objects.
More examples of sets:T = {1, 2, 3, 4, 5}U = {1, 2, 3, … , 1000}V = {1, 2, 3, 4, …}W = {x : x is a 2 legged animal}
W is written in what we call ‘set builder’ notation Read as: “The set of all x, such that x is a 2 legged animal.”
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets A set is a collection of objects.
More examples of sets:T = {1, 2, 3, 4, 5}U = {1, 2, 3, … , 1000}V = {1, 2, 3, 4, …}W = {x : x is a 2 legged animal}
W is written in what we call ‘set builder’ notation Read as: “The set of all x, such that x is a 2 legged animal.”
W is written in what we call ‘set builder’ notation Read as: “The set of all x, such that x is a 2 legged animal.”
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets A set is a collection of objects.
More examples of sets:T = {1, 2, 3, 4, 5}U = {1, 2, 3, … , 1000}V = {1, 2, 3, 4, …}W = {x : x is a 2 legged animal}
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets A set is a collection of objects.
More examples of sets:T = {1, 2, 3, 4, 5}U = {1, 2, 3, … , 1000}V = {1, 2, 3, 4, …}W = {x : x is a 2 legged animal}
W is written in what we call ‘set builder’ notation Read as: “The set of all x, such that x is a 2 legged animal.”
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets A set is well defined if it is possible to
definitively determine whether or not any particular object is a member of the set.
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets A set is well defined if it is possible to
definitively determine whether or not any particular object is a member of the set.
A = {1, 2, 3, 4, 5} WELL DEFINEDB= {x : x is tall} NOT WELL DEFINED
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets A set is well defined if it is possible to
definitively determine whether or not any particular object is a member of the set.
A = {1, 2, 3, 4, 5} WELL DEFINEDB= {x : x is tall} NOT WELL DEFINED
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets A set is well defined if it is possible to
definitively determine whether or not any particular object is a member of the set.
A = {1, 2, 3, 4, 5} WELL DEFINEDB= {x : x is tall} NOT WELL DEFINED
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets A set is well defined if it is possible to
definitively determine whether or not any particular object is a member of the set.
A = {1, 2, 3, 4, 5} WELL DEFINEDB= {x : x is tall} NOT WELL DEFINED
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets SYMBOLS
Є is the symbol for “is an element of”
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets SYMBOLS
Є is the symbol for “is an element of”
If A = {1, 2, 3, 4, 5}, then 2 Є A.
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets SYMBOLS
Є is the symbol for “is an element of”
If A = {1, 2, 3, 4, 5}, then 2 Є A.In general the symbol for “not” something is the symbol for that thing with a diagonal line through it.
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets SYMBOLS
Є is the symbol for “is an element of”
If A = {1, 2, 3, 4, 5}, then 2 Є A.In general the symbol for “not” something is the symbol for that thing with a diagonal line through it.
For example, 7 A.
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets SYMBOLS
The set containing no elements is calledthe empty set (or sometimes, the null set).
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets SYMBOLS
The set containing no elements is calledthe empty set (or sometimes, the null set).
Let M = {x: x is a female U.S. President before 2010}
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets SYMBOLS
The set containing no elements is calledthe empty set (or sometimes, the null set).
Let M = {x: x is a female U.S. President before 2010}
Because this set is EMPTY, we can write
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets SYMBOLS
The set containing no elements is calledthe empty set (or sometimes, the null set).
Let M = {x: x is a female U.S. President before 2010}
Because this set is EMPTY, we can write
Ø or { }
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets SYMBOLS
The set of all elements under consideration for a particular problem is called the universal set (U).
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets SYMBOLS
The set of all elements under consideration for a particular problem is called the universal set (U).
If you are choosing a 3-person committee from a 50 member club, the Universal set consists of the names of all 50 members.
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets SYMBOLS
The set of all elements under consideration for a particular problem is called the universal set (U).
If you are choosing a 3-person committee from a 50 member club, the Universal set consists of the names of all 50 members.If you are looking at course grades in a class where the only
grades possible are A, B, C, D, F, W, then U = { A, B, C, D, F, W}.
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets SYMBOLS
The set of all elements under consideration for a particular problem is called the universal set (U).
If you are choosing a 3-person committee from a 50 member club, the Universal set consists of the names of all 50 members.If you are looking at course grades in a class where the only
grades possible are A, B, C, D, F, W, then U = { A, B, C, D, F, W}.If you roll a die twice & count how many fives you get
U = {0, 1, 2}.
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets SYMBOLS
The set of all elements under consideration for a particular problem is called the universal set (U).
If you are choosing a 3-person committee from a 50 member club, the Universal set consists of the names of all 50 members.If you are looking at course grades in a class where the only
grades possible are A, B, C, D, F, W, then U = { A, B, C, D, F, W}.If you roll a die twice & count how many fives you get
U = {0, 1, 2}.
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets SYMBOLS
The set of all elements under consideration for a particular problem is called the universal set (U).
THE UNIVERSAL SET IS CONTEXTUAL…IT DEPENDS COMPLETELY ON THE CONTEXT OF THE PROBLEM.
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets SYMBOLS
The set of all elements under consideration for a particular problem is called the universal set (U).
THE UNIVERSAL SET IS CONTEXTUAL…IT DEPENDS COMPLETELY ON THE CONTEXT OF THE PROBLEM.
For example, if we are showing the results of a coin flip, U = { HEAD , TAIL }
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets SYMBOLS
The set of all elements under consideration for a particular problem is called the universal set (U).
THE UNIVERSAL SET IS CONTEXTUAL…IT DEPENDS COMPLETELY ON THE CONTEXT OF THE PROBLEM.
For example, if we are showing the results of a coin flip, U = { HEAD , TAIL }
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets SYMBOLS
The set of all elements under consideration for a particular problem is called the universal set (U).
THE UNIVERSAL SET IS CONTEXTUAL…IT DEPENDS COMPLETELY ON THE CONTEXT OF THE PROBLEM.
For example, if we are showing the results of a coin flip, U = { HEAD , TAIL }
If we roll a single ordinary die, then U =
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets SYMBOLS
The set of all elements under consideration for a particular problem is called the universal set (U).
THE UNIVERSAL SET IS CONTEXTUAL…IT DEPENDS COMPLETELY ON THE CONTEXT OF THE PROBLEM.
For example, if we are showing the results of a coin flip, U = { HEAD , TAIL }
If we roll a single ordinary die, then U = { 1 , 2 , 3 , 4 , 5 , 6 }
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets SYMBOLS
The number of elements in set A is called the cardinal number of the set.
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets SYMBOLS
The number of elements in set A is called the cardinal number of the set.
n(A) is read
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets SYMBOLS
The number of elements in set A is called the cardinal number of the set.
n(A) is read ‘the cardinal number of A’ or more informally, ‘the number of elements of A’.
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets SYMBOLS
The number of elements in set A is called the cardinal number of the set.
n(A) is read ‘the cardinal number of A’ or more informally, ‘the number of elements of A’.
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets SYMBOLS
The number of elements in set A is called the cardinal number of the set.
n(A) is read ‘the cardinal number of A’ or more informally, ‘the number of elements of A’.
If A = { 1 , 2 , 4 , 6 , 8 , 10 }, then n(A) = 6.
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets SYMBOLS
The number of elements in set A is called the cardinal number of the set.
n(A) is read ‘the cardinal number of A’ or more informally, ‘the number of elements of A’.
A set is finite if its cardinal number is a whole number and infinite if its cardinal number is not a whole number.
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets SYMBOLS
The number of elements in set A is called the cardinal number of the set.
n(A) is read ‘the cardinal number of A’ or more informally, ‘the number of elements of A’.
A set is finite if its cardinal number is a whole number and infinite if its cardinal number is not a whole number.
A = { 1 , 2 , 4 }
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets SYMBOLS
The number of elements in set A is called the cardinal number of the set.
n(A) is read ‘the cardinal number of A’ or more informally, ‘the number of elements of A’.
A set is finite if its cardinal number is a whole number and infinite if its cardinal number is not a whole number.
FINITEA = { 1 , 2 , 4 }
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets SYMBOLS
The number of elements in set A is called the cardinal number of the set.
n(A) is read ‘the cardinal number of A’ or more informally, ‘the number of elements of A’.
A set is finite if its cardinal number is a whole number and infinite if its cardinal number is not a whole number.
FINITE A = { 2 , 4 , 6 , 8 , …}A = { 1 , 2 , 4 }
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets SYMBOLS
The number of elements in set A is called the cardinal number of the set.
n(A) is read ‘the cardinal number of A’ or more informally, ‘the number of elements of A’.
A set is finite if its cardinal number is a whole number and infinite if its cardinal number is not a whole number.
FINITEINFINITEA = { 2 , 4 , 6 , 8 , …}A = { 1 , 2 , 4 }
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets COMPARING SETS
DEFINITION: EQUALITY OF 2 SETSTwo sets are equal (A = B) if they have exactly the
same elements. Otherwise, we write A ≠ B.
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets COMPARING SETS
DEFINITION: EQUALITY OF 2 SETSTwo sets are equal (A = B) if they have exactly the
same elements. Otherwise, we write A ≠ B.
{1, 2, 3, 7} = {1, 7, 3, 2} (Order Is not important.)
Example: {a, b, c} ≠ {a, c, e}
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets COMPARING SETS
DEFINITION: EQUALITY OF 2 SETSTwo sets are equal (A = B) if they have exactly the
same elements. Otherwise, we write A ≠ B.
{1, 2, 3, 7} = {1, 7, 3, 2} (Order Is not important.)
Example: {a, b, c} ≠ {a, c, e}
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets COMPARING SETS
{1, 2, 3, 7} = {1, 7, 3, 2} (Order Is not important.)
Example: {a, b, c} ≠ {a, c, e}
DEFINITION: EQUALITY OF 2 SETSTwo sets are equal (A = B) if they have exactly the
same elements. Otherwise, we write A ≠ B.
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets COMPARING SETS
DEFINITION: SUBSETSet A is a subset of set B (written A B)
if every element of A is also an element of B.
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets COMPARING SETS
DEFINITION: SUBSETSet A is a subset of set B (written A B)
if every element of A is also an element of B.
{a, b, e} {a, b, c, e, g} {a, b, c, e, g} {a, b, c, e, g} {dog, cat} is NOT a subset of {dog, pig, goat}
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets COMPARING SETS
DEFINITION: SUBSETSet A is a subset of set B (written A B)
if every element of A is also an element of B.
{a, b, e} {a, b, c, e, g} {a, b, c, e, g} {a, b, c, e, g} {dog, cat} is NOT a subset of {dog, pig, goat}
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets COMPARING SETS
DEFINITION: SUBSETSet A is a subset of set B (written A B)
if every element of A is also an element of B.
{a, b, e} {a, b, c, e, g} {a, b, c, e, g} {a, b, c, e, g} {dog, cat} is NOT a subset of {dog, pig, goat}
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets COMPARING SETS
DEFINITION: SUBSETSet A is a subset of set B (written A B)
if every element of A is also an element of B.
{a, b, e} {a, b, c, e, g} {a, b, c, e, g} {a, b, c, e, g} {dog, cat} is NOT a subset of {dog, pig, goat}
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets COMPARING SETS
DEFINITION: SUBSETSet A is a subset of set B (written A B)
if every element of A is also an element of B.
{a, b, e} {a, b, c, e, g} {a, b, c, e, g} {a, b, c, e, g} {dog, cat} {dog, pig, goat}
{a, b, e} {a, b, c, e, g} {a, b, c, e, g} {a, b, c, e, g} {dog, cat} {dog, pig, goat}
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets COMPARING SETS
DEFINITION: SUBSETSet A is a subset of set B (written A B)
if every element of A is also an element of B.
/⊆No ‘cat’
{a, b, e} {a, b, c, e, g} {a, b, c, e, g} {a, b, c, e, g} {dog, cat} {dog, pig, goat}
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets COMPARING SETS
DEFINITION: SUBSETSet A is a subset of set B (written A B)
if every element of A is also an element of B.
/⊆
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets COMPARING SETS
DEFINITION: PROPER SUBSETSet A is a proper subset of set B (written A B)
if every element of A is also an element of B but A ≠ B.
{a, b, e} {a, b, c, e, g} {a, b, c, e, g} {a, b, c, e, g} {dog, cat} {dog, pig, goat}
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets COMPARING SETS
/
DEFINITION: PROPER SUBSETSet A is a proper subset of set B (written A B)
if every element of A is also an element of B but A ≠ B.
⊆/
{a, b, e} {a, b, c, e, g} {a, b, c, e, g} {a, b, c, e, g} {dog, cat} {dog, pig, goat}
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets COMPARING SETS
/
DEFINITION: PROPER SUBSETSet A is a proper subset of set B (written A B)
if every element of A is also an element of B but A ≠ B.
⊆/
{a, b, e} {a, b, c, e, g} {a, b, c, e, g} {a, b, c, e, g} {dog, cat} {dog, pig, goat}
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets COMPARING SETS
/
DEFINITION: PROPER SUBSETSet A is a proper subset of set B (written A B)
if every element of A is also an element of B but A ≠ B.
⊆/
{a, b, e} {a, b, c, e, g} {a, b, c, e, g} {a, b, c, e, g} {dog, cat} {dog, pig, goat}
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets COMPARING SETS
/
DEFINITION: PROPER SUBSETSet A is a proper subset of set B (written A B)
if every element of A is also an element of B but A ≠ B.
⊆/ Here A = B
{a, b, e} {a, b, c, e, g} {a, b, c, e, g} {a, b, c, e, g} {dog, cat} {dog, pig, goat}
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets COMPARING SETS
/
DEFINITION: PROPER SUBSETSet A is a proper subset of set B (written A B)
if every element of A is also an element of B but A ≠ B.
/⊂ Here A = B
{a, b, e} {a, b, c, e, g} {a, b, c, e, g} {a, b, c, e, g} {dog, cat} {dog, pig, goat}
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets COMPARING SETS
/
DEFINITION: PROPER SUBSETSet A is a proper subset of set B (written A B)
if every element of A is also an element of B but A ≠ B.
/⊂
{a, b, e} {a, b, c, e, g} {a, b, c, e, g} {a, b, c, e, g} {dog, cat} {dog, pig, goat}
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets COMPARING SETS
⊆/
DEFINITION: PROPER SUBSETSet A is a proper subset of set B (written A B)
if every element of A is also an element of B but A ≠ B.
/⊂
{a, b, e} {a, b, c, e, g} {a, b, c, e, g} {a, b, c, e, g} {dog, cat} {dog, pig, goat}
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets COMPARING SETS
/⊂
DEFINITION: PROPER SUBSETSet A is a proper subset of set B (written A B)
if every element of A is also an element of B but A ≠ B.
/⊂No ‘cat’
{a, b, e} {a, b, c, e, g} {a, b, c, e, g} {a, b, c, e, g} {dog, cat} {dog, pig, goat}
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets COMPARING SETS
/⊂
DEFINITION: PROPER SUBSETSet A is a proper subset of set B (written A B)
if every element of A is also an element of B but A ≠ B.
/⊂
{a, b, e} {a, b, c, e, g} {a, b, c, e, g} {a, b, c, e, g} {dog, cat} {dog, pig, goat}
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets COMPARING SETS
/⊂
DEFINITION: PROPER SUBSETSet A is a proper subset of set B (written A B)
if every element of A is also an element of B but A ≠ B.
/⊂IMPORTANT
Pay close attentionto the difference between subset ()
and proper subset().
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets COMPARING SETS
DEFINITION: PROPER SUBSETSet A is a proper subset of set B (written A B)
if every element of A is also an element of B but A ≠ B.
An analogy to something with which you are more familiar:Difference between less than (‘<‘) & less than or equal to (‘<‘)
Just as ‘<‘ allows the possibility of equality: (5 < 5 is true) “” also allows for the possibility of equality:
({1, 4, 7} {1, 4, 7} is true)A proper subset doesn’t allow the sets to be equal just as less than doesn’t allow for equality.
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets COMPARING SETS
DEFINITION: PROPER SUBSETSet A is a proper subset of set B (written A B)
if every element of A is also an element of B but A ≠ B.
An analogy to something with which you are more familiar:Difference between less than (‘<‘) & less than or equal to (‘<‘)
Just as ‘<‘ allows the possibility of equality: (5 < 5 is true) “” also allows for the possibility of equality:
({1, 4, 7} {1, 4, 7} is true)A proper subset doesn’t allow the sets to be equal just as less than doesn’t allow for equality.
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets COMPARING SETS
DEFINITION: PROPER SUBSETSet A is a proper subset of set B (written A B)
if every element of A is also an element of B but A ≠ B.
An analogy to something with which you are more familiar:Difference between less than (‘<‘) & less than or equal to (‘<‘)
Just as ‘<‘ allows the possibility of equality: (5 < 5 is true) “” also allows for the possibility of equality:
({1, 4, 7} {1, 4, 7} is true)A proper subset doesn’t allow the sets to be equal just as less than doesn’t
allow for equality.
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets COMPARING SETS
DEFINITION: PROPER SUBSETSet A is a proper subset of set B (written A B)
if every element of A is also an element of B but A ≠ B.
An analogy to something with which you are more familiar:Difference between less than (‘<‘) & less than or equal to (‘<‘)
Just as ‘<‘ allows the possibility of equality: (5 < 5 is true) “” also allows for the possibility of equality:
({1, 4, 7} {1, 4, 7} is true)A proper subset doesn’t allow the sets to be equal just as less than doesn’t
allow for equality.
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets COMPARING SETS
DEFINITION: PROPER SUBSETSet A is a proper subset of set B (written A B)
if every element of A is also an element of B but A ≠ B.
An analogy to something with which you are more familiar:Difference between less than (‘<‘) & less than or equal to (‘<‘)
Just as ‘<‘ allows the possibility of equality: (5 < 5 is true) “” also allows for the possibility of equality:
({1, 4, 7} {1, 4, 7} is true)But a proper subset doesn’t allow the sets to be equal just as less than doesn’t allow for equality.
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets COMPARING SETS
DEFINITION: PROPER SUBSETSet A is a proper subset of set B (written A B)
if every element of A is also an element of B but A ≠ B.
An analogy to something with which you are more familiar:Difference between less than (‘<‘) & less than or equal to (‘<‘)
Just as ‘<‘ allows the possibility of equality: (5 < 5 is true) “” also allows for the possibility of equality:
({1, 4, 7} {1, 4, 7} is true)But a proper subset doesn’t allow the sets to be equal just as less than doesn’t allow for equality.
So, both 5 < 5 and {1, 4, 7} {1, 4, 7} are FALSE!
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets COMPARING SETS
Number of subsets of a set:A set with k elements has subsets.
For the set {a, c, f}, there are = 8 elements.
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets COMPARING SETS
Number of subsets of a set:A set with k elements has subsets.
For the set {a, c, f}, there are = 8 elements.
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets COMPARING SETS
Number of subsets of a set:A set with k elements has subsets.
For the set {a, c, f}, there are = 8 elements.Let’s list them:
Subsets with:0 elements 1 element 2 elements 3 elements
Ø
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets COMPARING SETS
Number of subsets of a set:A set with k elements has subsets.
For the set {a, c, f}, there are = 8 elements.Let’s list them:
Subsets with:0 elements 1 element 2 elements 3 elements
Ø
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets COMPARING SETS
Number of subsets of a set:A set with k elements has subsets.
For the set {a, c, f}, there are = 8 elements.Let’s list them:
Subsets with:0 elements 1 element 2 elements 3 elements
Ø {a}, {c}, {f}
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets COMPARING SETS
Number of subsets of a set:A set with k elements has subsets.
For the set {a, c, f}, there are = 8 elements.Let’s list them:
Subsets with:0 elements 1 element 2 elements 3 elements
Ø {a}, {c}, {f}
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets COMPARING SETS
Number of subsets of a set:A set with k elements has subsets.
For the set {a, c, f}, there are = 8 elements.Let’s list them:
Subsets with:0 elements 1 element 2 elements 3 elements
Ø {a}, {c}, {f}
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets COMPARING SETS
Number of subsets of a set:A set with k elements has subsets.
For the set {a, c, f}, there are = 8 elements.Let’s list them:
Subsets with:0 elements 1 element 2 elements 3 elements
Ø {a}, {c}, {f} {a,c},{a,f},{c,f}
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets COMPARING SETS
Number of subsets of a set:A set with k elements has subsets.
For the set {a, c, f}, there are = 8 elements.Let’s list them:
Subsets with:0 elements 1 element 2 elements 3 elements
Ø {a}, {c}, {f} {a,c},{a,f},{c,f}
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets COMPARING SETS
Number of subsets of a set:A set with k elements has subsets.
For the set {a, c, f}, there are = 8 elements.Let’s list them:
Subsets with:0 elements 1 element 2 elements 3 elements
Ø {a}, {c}, {f} {a,c},{a,f},{c,f}
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets COMPARING SETS
Number of subsets of a set:A set with k elements has subsets.
For the set {a, c, f}, there are = 8 elements.Let’s list them:
Subsets with:0 elements 1 element 2 elements 3 elements
Ø {a}, {c}, {f} {a,c},{a,f},{c,f} {a, c, f}
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets COMPARING SETS
EQUIVALENCE of two sets:Two sets A and B are equivalent if the two sets have the same
number of elements…that is, if n(A) = n(B).
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets COMPARING SETS
EQUIVALENCE of two sets:Two sets A and B are equivalent if the two sets have the same
number of elements…that is, if n(A) = n(B).
Perhaps the best way to think about the equivalence of two sets is to think of the elements as simply names or labels.Some analog clock use ordinary numbers while some use Roman numerals:A={1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} B={I, II, III, IV, V, VI, VII, VIII, IX, X, XI, XII}
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets COMPARING SETS
EQUIVALENCE of two sets:Two sets A and B are equivalent if the two sets have the same
number of elements…that is, if n(A) = n(B).
Perhaps the best way to think about the equivalence of two sets is to think of the elements as simply names or labels.Some analog clock use ordinary numbers while some use Roman numerals:A={1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} B={I, II, III, IV, V, VI, VII, VIII, IX, X, XI, XII}
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets COMPARING SETS
EQUIVALENCE of two sets:Two sets A and B are equivalent if the two sets have the same
number of elements…that is, if n(A) = n(B).
Perhaps the best way to think about the equivalence of two sets is to think of the elements as simply names or labels.Some analog clock use ordinary numbers while some use Roman numerals:A={1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} B={I, II, III, IV, V, VI, VII, VIII, IX, X, XI, XII}
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets COMPARING SETS
EQUIVALENCE of two sets:Two sets A and B are equivalent if the two sets have the same
number of elements…that is, if n(A) = n(B).
Perhaps the best way to think about the equivalence of two sets is to think of the elements as simply names or labels.Some analog clock use ordinary numbers while some use Roman numerals:A={1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} B={I, II, III, IV, V, VI, VII, VIII, IX, X, XI, XII}It is in this sense that we say that set A is equivalent to set B.
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets COMPARING SETS
EQUIVALENCE of two sets:Two sets A and B are equivalent if the two sets have the same
number of elements…that is, if n(A) = n(B).
Perhaps the best way to think about the equivalence of two sets is to think of the elements as simply names or labels.Some analog clock use ordinary numbers while some use Roman numerals:A={1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} B={I, II, III, IV, V, VI, VII, VIII, IX, X, XI, XII}It is in this sense that we say that set A is equivalent to set B.
{a, 2, 5, f} is equivalent to {dog, cat, bird, pig}
MATH 110 Sec 2-1 , 2-2 Lecture on Sets and Comparing Sets COMPARING SETS
EQUIVALENCE of two sets:Two sets A and B are equivalent if the two sets have the same
number of elements…that is, if n(A) = n(B).
Perhaps the best way to think about the equivalence of two sets is to think of the elements as simply names or labels.Some analog clock use ordinary numbers while some use Roman numerals:A={1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} B={I, II, III, IV, V, VI, VII, VIII, IX, X, XI, XII}It is in this sense that we say that set A is equivalent to set B.
{a, 2, 5, f} is equivalent to {dog, cat, bird, pig}
{p, q, x} is NOT equivalent to {a, 2, 5, f}
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