induction - complete inductioncommunity.wvu.edu/~krsubramani/courses/sp13/dm2/lecnotes/...complete...
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![Page 1: Induction - Complete Inductioncommunity.wvu.edu/~krsubramani/courses/sp13/dm2/lecnotes/...Complete Induction Example Show that every number greater than or equal to 8 can be expressed](https://reader035.vdocument.in/reader035/viewer/2022062416/61125c6954d0df5e5a4e689f/html5/thumbnails/1.jpg)
Outline
Induction - Complete Induction
K. Subramani1
1Lane Department of Computer Science and Electrical EngineeringWest Virginia University
April 1 2013
Subramani Mathematical Induction
![Page 2: Induction - Complete Inductioncommunity.wvu.edu/~krsubramani/courses/sp13/dm2/lecnotes/...Complete Induction Example Show that every number greater than or equal to 8 can be expressed](https://reader035.vdocument.in/reader035/viewer/2022062416/61125c6954d0df5e5a4e689f/html5/thumbnails/2.jpg)
Outline
Outline
1 Complete Induction
Subramani Mathematical Induction
![Page 3: Induction - Complete Inductioncommunity.wvu.edu/~krsubramani/courses/sp13/dm2/lecnotes/...Complete Induction Example Show that every number greater than or equal to 8 can be expressed](https://reader035.vdocument.in/reader035/viewer/2022062416/61125c6954d0df5e5a4e689f/html5/thumbnails/3.jpg)
Complete Induction
Complete Induction or Strong Induction
Axiom Schema
[(∀n) (∀n′) ((n′ < n)→ P(n′))→ P(n)]→ (∀x) P(x).
Note
Do we need a base case? Has it been addressed?
Subramani Mathematical Induction
![Page 4: Induction - Complete Inductioncommunity.wvu.edu/~krsubramani/courses/sp13/dm2/lecnotes/...Complete Induction Example Show that every number greater than or equal to 8 can be expressed](https://reader035.vdocument.in/reader035/viewer/2022062416/61125c6954d0df5e5a4e689f/html5/thumbnails/4.jpg)
Complete Induction
Complete Induction or Strong Induction
Axiom Schema
[(∀n) (∀n′) ((n′ < n)→ P(n′))→ P(n)]→ (∀x) P(x).
Note
Do we need a base case? Has it been addressed?
Subramani Mathematical Induction
![Page 5: Induction - Complete Inductioncommunity.wvu.edu/~krsubramani/courses/sp13/dm2/lecnotes/...Complete Induction Example Show that every number greater than or equal to 8 can be expressed](https://reader035.vdocument.in/reader035/viewer/2022062416/61125c6954d0df5e5a4e689f/html5/thumbnails/5.jpg)
Complete Induction
Complete Induction or Strong Induction
Axiom Schema
[(∀n) (∀n′) ((n′ < n)→ P(n′))→ P(n)]→ (∀x) P(x).
Note
Do we need a base case? Has it been addressed?
Subramani Mathematical Induction
![Page 6: Induction - Complete Inductioncommunity.wvu.edu/~krsubramani/courses/sp13/dm2/lecnotes/...Complete Induction Example Show that every number greater than or equal to 8 can be expressed](https://reader035.vdocument.in/reader035/viewer/2022062416/61125c6954d0df5e5a4e689f/html5/thumbnails/6.jpg)
Complete Induction
Complete Induction or Strong Induction
Axiom Schema
[(∀n) (∀n′) ((n′ < n)→ P(n′))→ P(n)]→ (∀x) P(x).
Note
Do we need a base case?
Has it been addressed?
Subramani Mathematical Induction
![Page 7: Induction - Complete Inductioncommunity.wvu.edu/~krsubramani/courses/sp13/dm2/lecnotes/...Complete Induction Example Show that every number greater than or equal to 8 can be expressed](https://reader035.vdocument.in/reader035/viewer/2022062416/61125c6954d0df5e5a4e689f/html5/thumbnails/7.jpg)
Complete Induction
Complete Induction or Strong Induction
Axiom Schema
[(∀n) (∀n′) ((n′ < n)→ P(n′))→ P(n)]→ (∀x) P(x).
Note
Do we need a base case? Has it been addressed?
Subramani Mathematical Induction
![Page 8: Induction - Complete Inductioncommunity.wvu.edu/~krsubramani/courses/sp13/dm2/lecnotes/...Complete Induction Example Show that every number greater than or equal to 8 can be expressed](https://reader035.vdocument.in/reader035/viewer/2022062416/61125c6954d0df5e5a4e689f/html5/thumbnails/8.jpg)
Complete Induction
Example
Show that every number greater than or equal to 8 can be expressed in the form5 · a + 3 · b, for suitably chosen a and b.
Proof.
The conjecture is clearly true for 8, 9 and 10. Assume that the conjecture holds for allr , 8 ≤ r ≤ k . Consider the integer k + 1. Without loss of generality, we assume that(k + 1) ≥ 11. Observe that (k + 1)− 3 = k − 2 is at least 8 and less than k . As perthe inductive hypothesis, k − 2 can be expressed in the form 3a + 5b, for suitablychosen a and b. It follows that (k + 1) = 3 · (a + 1) + 5 · b, can also be so expressed.Applying the second principle of mathematical induction, we conclude that theconjecture is true.
Subramani Mathematical Induction
![Page 9: Induction - Complete Inductioncommunity.wvu.edu/~krsubramani/courses/sp13/dm2/lecnotes/...Complete Induction Example Show that every number greater than or equal to 8 can be expressed](https://reader035.vdocument.in/reader035/viewer/2022062416/61125c6954d0df5e5a4e689f/html5/thumbnails/9.jpg)
Complete Induction
Example
Show that every number greater than or equal to 8 can be expressed in the form5 · a + 3 · b, for suitably chosen a and b.
Proof.
The conjecture is clearly true for 8, 9 and 10.
Assume that the conjecture holds for allr , 8 ≤ r ≤ k . Consider the integer k + 1. Without loss of generality, we assume that(k + 1) ≥ 11. Observe that (k + 1)− 3 = k − 2 is at least 8 and less than k . As perthe inductive hypothesis, k − 2 can be expressed in the form 3a + 5b, for suitablychosen a and b. It follows that (k + 1) = 3 · (a + 1) + 5 · b, can also be so expressed.Applying the second principle of mathematical induction, we conclude that theconjecture is true.
Subramani Mathematical Induction
![Page 10: Induction - Complete Inductioncommunity.wvu.edu/~krsubramani/courses/sp13/dm2/lecnotes/...Complete Induction Example Show that every number greater than or equal to 8 can be expressed](https://reader035.vdocument.in/reader035/viewer/2022062416/61125c6954d0df5e5a4e689f/html5/thumbnails/10.jpg)
Complete Induction
Example
Show that every number greater than or equal to 8 can be expressed in the form5 · a + 3 · b, for suitably chosen a and b.
Proof.
The conjecture is clearly true for 8, 9 and 10. Assume that the conjecture holds for allr , 8 ≤ r ≤ k .
Consider the integer k + 1. Without loss of generality, we assume that(k + 1) ≥ 11. Observe that (k + 1)− 3 = k − 2 is at least 8 and less than k . As perthe inductive hypothesis, k − 2 can be expressed in the form 3a + 5b, for suitablychosen a and b. It follows that (k + 1) = 3 · (a + 1) + 5 · b, can also be so expressed.Applying the second principle of mathematical induction, we conclude that theconjecture is true.
Subramani Mathematical Induction
![Page 11: Induction - Complete Inductioncommunity.wvu.edu/~krsubramani/courses/sp13/dm2/lecnotes/...Complete Induction Example Show that every number greater than or equal to 8 can be expressed](https://reader035.vdocument.in/reader035/viewer/2022062416/61125c6954d0df5e5a4e689f/html5/thumbnails/11.jpg)
Complete Induction
Example
Show that every number greater than or equal to 8 can be expressed in the form5 · a + 3 · b, for suitably chosen a and b.
Proof.
The conjecture is clearly true for 8, 9 and 10. Assume that the conjecture holds for allr , 8 ≤ r ≤ k . Consider the integer k + 1.
Without loss of generality, we assume that(k + 1) ≥ 11. Observe that (k + 1)− 3 = k − 2 is at least 8 and less than k . As perthe inductive hypothesis, k − 2 can be expressed in the form 3a + 5b, for suitablychosen a and b. It follows that (k + 1) = 3 · (a + 1) + 5 · b, can also be so expressed.Applying the second principle of mathematical induction, we conclude that theconjecture is true.
Subramani Mathematical Induction
![Page 12: Induction - Complete Inductioncommunity.wvu.edu/~krsubramani/courses/sp13/dm2/lecnotes/...Complete Induction Example Show that every number greater than or equal to 8 can be expressed](https://reader035.vdocument.in/reader035/viewer/2022062416/61125c6954d0df5e5a4e689f/html5/thumbnails/12.jpg)
Complete Induction
Example
Show that every number greater than or equal to 8 can be expressed in the form5 · a + 3 · b, for suitably chosen a and b.
Proof.
The conjecture is clearly true for 8, 9 and 10. Assume that the conjecture holds for allr , 8 ≤ r ≤ k . Consider the integer k + 1. Without loss of generality, we assume that(k + 1) ≥ 11.
Observe that (k + 1)− 3 = k − 2 is at least 8 and less than k . As perthe inductive hypothesis, k − 2 can be expressed in the form 3a + 5b, for suitablychosen a and b. It follows that (k + 1) = 3 · (a + 1) + 5 · b, can also be so expressed.Applying the second principle of mathematical induction, we conclude that theconjecture is true.
Subramani Mathematical Induction
![Page 13: Induction - Complete Inductioncommunity.wvu.edu/~krsubramani/courses/sp13/dm2/lecnotes/...Complete Induction Example Show that every number greater than or equal to 8 can be expressed](https://reader035.vdocument.in/reader035/viewer/2022062416/61125c6954d0df5e5a4e689f/html5/thumbnails/13.jpg)
Complete Induction
Example
Show that every number greater than or equal to 8 can be expressed in the form5 · a + 3 · b, for suitably chosen a and b.
Proof.
The conjecture is clearly true for 8, 9 and 10. Assume that the conjecture holds for allr , 8 ≤ r ≤ k . Consider the integer k + 1. Without loss of generality, we assume that(k + 1) ≥ 11. Observe that (k + 1)− 3 = k − 2 is at least 8 and less than k .
As perthe inductive hypothesis, k − 2 can be expressed in the form 3a + 5b, for suitablychosen a and b. It follows that (k + 1) = 3 · (a + 1) + 5 · b, can also be so expressed.Applying the second principle of mathematical induction, we conclude that theconjecture is true.
Subramani Mathematical Induction
![Page 14: Induction - Complete Inductioncommunity.wvu.edu/~krsubramani/courses/sp13/dm2/lecnotes/...Complete Induction Example Show that every number greater than or equal to 8 can be expressed](https://reader035.vdocument.in/reader035/viewer/2022062416/61125c6954d0df5e5a4e689f/html5/thumbnails/14.jpg)
Complete Induction
Example
Show that every number greater than or equal to 8 can be expressed in the form5 · a + 3 · b, for suitably chosen a and b.
Proof.
The conjecture is clearly true for 8, 9 and 10. Assume that the conjecture holds for allr , 8 ≤ r ≤ k . Consider the integer k + 1. Without loss of generality, we assume that(k + 1) ≥ 11. Observe that (k + 1)− 3 = k − 2 is at least 8 and less than k . As perthe inductive hypothesis, k − 2 can be expressed in the form 3a + 5b, for suitablychosen a and b.
It follows that (k + 1) = 3 · (a + 1) + 5 · b, can also be so expressed.Applying the second principle of mathematical induction, we conclude that theconjecture is true.
Subramani Mathematical Induction
![Page 15: Induction - Complete Inductioncommunity.wvu.edu/~krsubramani/courses/sp13/dm2/lecnotes/...Complete Induction Example Show that every number greater than or equal to 8 can be expressed](https://reader035.vdocument.in/reader035/viewer/2022062416/61125c6954d0df5e5a4e689f/html5/thumbnails/15.jpg)
Complete Induction
Example
Show that every number greater than or equal to 8 can be expressed in the form5 · a + 3 · b, for suitably chosen a and b.
Proof.
The conjecture is clearly true for 8, 9 and 10. Assume that the conjecture holds for allr , 8 ≤ r ≤ k . Consider the integer k + 1. Without loss of generality, we assume that(k + 1) ≥ 11. Observe that (k + 1)− 3 = k − 2 is at least 8 and less than k . As perthe inductive hypothesis, k − 2 can be expressed in the form 3a + 5b, for suitablychosen a and b. It follows that (k + 1) = 3 · (a + 1) + 5 · b, can also be so expressed.
Applying the second principle of mathematical induction, we conclude that theconjecture is true.
Subramani Mathematical Induction
![Page 16: Induction - Complete Inductioncommunity.wvu.edu/~krsubramani/courses/sp13/dm2/lecnotes/...Complete Induction Example Show that every number greater than or equal to 8 can be expressed](https://reader035.vdocument.in/reader035/viewer/2022062416/61125c6954d0df5e5a4e689f/html5/thumbnails/16.jpg)
Complete Induction
Example
Show that every number greater than or equal to 8 can be expressed in the form5 · a + 3 · b, for suitably chosen a and b.
Proof.
The conjecture is clearly true for 8, 9 and 10. Assume that the conjecture holds for allr , 8 ≤ r ≤ k . Consider the integer k + 1. Without loss of generality, we assume that(k + 1) ≥ 11. Observe that (k + 1)− 3 = k − 2 is at least 8 and less than k . As perthe inductive hypothesis, k − 2 can be expressed in the form 3a + 5b, for suitablychosen a and b. It follows that (k + 1) = 3 · (a + 1) + 5 · b, can also be so expressed.Applying the second principle of mathematical induction, we conclude that theconjecture is true.
Subramani Mathematical Induction
![Page 17: Induction - Complete Inductioncommunity.wvu.edu/~krsubramani/courses/sp13/dm2/lecnotes/...Complete Induction Example Show that every number greater than or equal to 8 can be expressed](https://reader035.vdocument.in/reader035/viewer/2022062416/61125c6954d0df5e5a4e689f/html5/thumbnails/17.jpg)
Complete Induction
The theory T ∗PA
Peano Arithmetic with division
Consider the theory of T∗PA, which is the theory of Peano arithmetic TPA, augmented bythe following axioms:
A1. (∀x)(∀y) (x < y)→ [quot(x , y) = 0].
A2. (∀x)(∀y) (y > 0)→ [quot(x + y , y) = quot(x , y) + 1].
A3. (∀x)(∀y) (x < y)→ [rem(x , y) = x ].
A4. (∀x)(∀y) (y > 0)→ [rem(x + y , y) = rem(x , y)].
Example
Argue that(∀x)(∀y) (y > 0)→ [rem(x , y) < y ].
Subramani Mathematical Induction
![Page 18: Induction - Complete Inductioncommunity.wvu.edu/~krsubramani/courses/sp13/dm2/lecnotes/...Complete Induction Example Show that every number greater than or equal to 8 can be expressed](https://reader035.vdocument.in/reader035/viewer/2022062416/61125c6954d0df5e5a4e689f/html5/thumbnails/18.jpg)
Complete Induction
The theory T ∗PA
Peano Arithmetic with division
Consider the theory of T∗PA, which is the theory of Peano arithmetic TPA, augmented bythe following axioms:
A1. (∀x)(∀y) (x < y)→ [quot(x , y) = 0].
A2. (∀x)(∀y) (y > 0)→ [quot(x + y , y) = quot(x , y) + 1].
A3. (∀x)(∀y) (x < y)→ [rem(x , y) = x ].
A4. (∀x)(∀y) (y > 0)→ [rem(x + y , y) = rem(x , y)].
Example
Argue that(∀x)(∀y) (y > 0)→ [rem(x , y) < y ].
Subramani Mathematical Induction
![Page 19: Induction - Complete Inductioncommunity.wvu.edu/~krsubramani/courses/sp13/dm2/lecnotes/...Complete Induction Example Show that every number greater than or equal to 8 can be expressed](https://reader035.vdocument.in/reader035/viewer/2022062416/61125c6954d0df5e5a4e689f/html5/thumbnails/19.jpg)
Complete Induction
The theory T ∗PA
Peano Arithmetic with division
Consider the theory of T∗PA, which is the theory of Peano arithmetic TPA, augmented bythe following axioms:
A1. (∀x)(∀y) (x < y)→ [quot(x , y) = 0].
A2. (∀x)(∀y) (y > 0)→ [quot(x + y , y) = quot(x , y) + 1].
A3. (∀x)(∀y) (x < y)→ [rem(x , y) = x ].
A4. (∀x)(∀y) (y > 0)→ [rem(x + y , y) = rem(x , y)].
Example
Argue that(∀x)(∀y) (y > 0)→ [rem(x , y) < y ].
Subramani Mathematical Induction
![Page 20: Induction - Complete Inductioncommunity.wvu.edu/~krsubramani/courses/sp13/dm2/lecnotes/...Complete Induction Example Show that every number greater than or equal to 8 can be expressed](https://reader035.vdocument.in/reader035/viewer/2022062416/61125c6954d0df5e5a4e689f/html5/thumbnails/20.jpg)
Complete Induction
The theory T ∗PA
Peano Arithmetic with division
Consider the theory of T∗PA, which is the theory of Peano arithmetic TPA, augmented bythe following axioms:
A1. (∀x)(∀y) (x < y)→ [quot(x , y) = 0].
A2. (∀x)(∀y) (y > 0)→ [quot(x + y , y) = quot(x , y) + 1].
A3. (∀x)(∀y) (x < y)→ [rem(x , y) = x ].
A4. (∀x)(∀y) (y > 0)→ [rem(x + y , y) = rem(x , y)].
Example
Argue that(∀x)(∀y) (y > 0)→ [rem(x , y) < y ].
Subramani Mathematical Induction
![Page 21: Induction - Complete Inductioncommunity.wvu.edu/~krsubramani/courses/sp13/dm2/lecnotes/...Complete Induction Example Show that every number greater than or equal to 8 can be expressed](https://reader035.vdocument.in/reader035/viewer/2022062416/61125c6954d0df5e5a4e689f/html5/thumbnails/21.jpg)
Complete Induction
The theory T ∗PA
Peano Arithmetic with division
Consider the theory of T∗PA, which is the theory of Peano arithmetic TPA, augmented bythe following axioms:
A1. (∀x)(∀y) (x < y)→ [quot(x , y) =
0].
A2. (∀x)(∀y) (y > 0)→ [quot(x + y , y) = quot(x , y) + 1].
A3. (∀x)(∀y) (x < y)→ [rem(x , y) = x ].
A4. (∀x)(∀y) (y > 0)→ [rem(x + y , y) = rem(x , y)].
Example
Argue that(∀x)(∀y) (y > 0)→ [rem(x , y) < y ].
Subramani Mathematical Induction
![Page 22: Induction - Complete Inductioncommunity.wvu.edu/~krsubramani/courses/sp13/dm2/lecnotes/...Complete Induction Example Show that every number greater than or equal to 8 can be expressed](https://reader035.vdocument.in/reader035/viewer/2022062416/61125c6954d0df5e5a4e689f/html5/thumbnails/22.jpg)
Complete Induction
The theory T ∗PA
Peano Arithmetic with division
Consider the theory of T∗PA, which is the theory of Peano arithmetic TPA, augmented bythe following axioms:
A1. (∀x)(∀y) (x < y)→ [quot(x , y) = 0].
A2. (∀x)(∀y) (y > 0)→ [quot(x + y , y) = quot(x , y) + 1].
A3. (∀x)(∀y) (x < y)→ [rem(x , y) = x ].
A4. (∀x)(∀y) (y > 0)→ [rem(x + y , y) = rem(x , y)].
Example
Argue that(∀x)(∀y) (y > 0)→ [rem(x , y) < y ].
Subramani Mathematical Induction
![Page 23: Induction - Complete Inductioncommunity.wvu.edu/~krsubramani/courses/sp13/dm2/lecnotes/...Complete Induction Example Show that every number greater than or equal to 8 can be expressed](https://reader035.vdocument.in/reader035/viewer/2022062416/61125c6954d0df5e5a4e689f/html5/thumbnails/23.jpg)
Complete Induction
The theory T ∗PA
Peano Arithmetic with division
Consider the theory of T∗PA, which is the theory of Peano arithmetic TPA, augmented bythe following axioms:
A1. (∀x)(∀y) (x < y)→ [quot(x , y) = 0].
A2. (∀x)(∀y) (y > 0)→ [quot(x + y , y) =
quot(x , y) + 1].
A3. (∀x)(∀y) (x < y)→ [rem(x , y) = x ].
A4. (∀x)(∀y) (y > 0)→ [rem(x + y , y) = rem(x , y)].
Example
Argue that(∀x)(∀y) (y > 0)→ [rem(x , y) < y ].
Subramani Mathematical Induction
![Page 24: Induction - Complete Inductioncommunity.wvu.edu/~krsubramani/courses/sp13/dm2/lecnotes/...Complete Induction Example Show that every number greater than or equal to 8 can be expressed](https://reader035.vdocument.in/reader035/viewer/2022062416/61125c6954d0df5e5a4e689f/html5/thumbnails/24.jpg)
Complete Induction
The theory T ∗PA
Peano Arithmetic with division
Consider the theory of T∗PA, which is the theory of Peano arithmetic TPA, augmented bythe following axioms:
A1. (∀x)(∀y) (x < y)→ [quot(x , y) = 0].
A2. (∀x)(∀y) (y > 0)→ [quot(x + y , y) = quot(x , y) + 1].
A3. (∀x)(∀y) (x < y)→ [rem(x , y) = x ].
A4. (∀x)(∀y) (y > 0)→ [rem(x + y , y) = rem(x , y)].
Example
Argue that(∀x)(∀y) (y > 0)→ [rem(x , y) < y ].
Subramani Mathematical Induction
![Page 25: Induction - Complete Inductioncommunity.wvu.edu/~krsubramani/courses/sp13/dm2/lecnotes/...Complete Induction Example Show that every number greater than or equal to 8 can be expressed](https://reader035.vdocument.in/reader035/viewer/2022062416/61125c6954d0df5e5a4e689f/html5/thumbnails/25.jpg)
Complete Induction
The theory T ∗PA
Peano Arithmetic with division
Consider the theory of T∗PA, which is the theory of Peano arithmetic TPA, augmented bythe following axioms:
A1. (∀x)(∀y) (x < y)→ [quot(x , y) = 0].
A2. (∀x)(∀y) (y > 0)→ [quot(x + y , y) = quot(x , y) + 1].
A3. (∀x)(∀y) (x < y)→ [rem(x , y) =
x ].
A4. (∀x)(∀y) (y > 0)→ [rem(x + y , y) = rem(x , y)].
Example
Argue that(∀x)(∀y) (y > 0)→ [rem(x , y) < y ].
Subramani Mathematical Induction
![Page 26: Induction - Complete Inductioncommunity.wvu.edu/~krsubramani/courses/sp13/dm2/lecnotes/...Complete Induction Example Show that every number greater than or equal to 8 can be expressed](https://reader035.vdocument.in/reader035/viewer/2022062416/61125c6954d0df5e5a4e689f/html5/thumbnails/26.jpg)
Complete Induction
The theory T ∗PA
Peano Arithmetic with division
Consider the theory of T∗PA, which is the theory of Peano arithmetic TPA, augmented bythe following axioms:
A1. (∀x)(∀y) (x < y)→ [quot(x , y) = 0].
A2. (∀x)(∀y) (y > 0)→ [quot(x + y , y) = quot(x , y) + 1].
A3. (∀x)(∀y) (x < y)→ [rem(x , y) = x ].
A4. (∀x)(∀y) (y > 0)→ [rem(x + y , y) = rem(x , y)].
Example
Argue that(∀x)(∀y) (y > 0)→ [rem(x , y) < y ].
Subramani Mathematical Induction
![Page 27: Induction - Complete Inductioncommunity.wvu.edu/~krsubramani/courses/sp13/dm2/lecnotes/...Complete Induction Example Show that every number greater than or equal to 8 can be expressed](https://reader035.vdocument.in/reader035/viewer/2022062416/61125c6954d0df5e5a4e689f/html5/thumbnails/27.jpg)
Complete Induction
The theory T ∗PA
Peano Arithmetic with division
Consider the theory of T∗PA, which is the theory of Peano arithmetic TPA, augmented bythe following axioms:
A1. (∀x)(∀y) (x < y)→ [quot(x , y) = 0].
A2. (∀x)(∀y) (y > 0)→ [quot(x + y , y) = quot(x , y) + 1].
A3. (∀x)(∀y) (x < y)→ [rem(x , y) = x ].
A4. (∀x)(∀y) (y > 0)→ [rem(x + y , y) =
rem(x , y)].
Example
Argue that(∀x)(∀y) (y > 0)→ [rem(x , y) < y ].
Subramani Mathematical Induction
![Page 28: Induction - Complete Inductioncommunity.wvu.edu/~krsubramani/courses/sp13/dm2/lecnotes/...Complete Induction Example Show that every number greater than or equal to 8 can be expressed](https://reader035.vdocument.in/reader035/viewer/2022062416/61125c6954d0df5e5a4e689f/html5/thumbnails/28.jpg)
Complete Induction
The theory T ∗PA
Peano Arithmetic with division
Consider the theory of T∗PA, which is the theory of Peano arithmetic TPA, augmented bythe following axioms:
A1. (∀x)(∀y) (x < y)→ [quot(x , y) = 0].
A2. (∀x)(∀y) (y > 0)→ [quot(x + y , y) = quot(x , y) + 1].
A3. (∀x)(∀y) (x < y)→ [rem(x , y) = x ].
A4. (∀x)(∀y) (y > 0)→ [rem(x + y , y) = rem(x , y)].
Example
Argue that(∀x)(∀y) (y > 0)→ [rem(x , y) < y ].
Subramani Mathematical Induction
![Page 29: Induction - Complete Inductioncommunity.wvu.edu/~krsubramani/courses/sp13/dm2/lecnotes/...Complete Induction Example Show that every number greater than or equal to 8 can be expressed](https://reader035.vdocument.in/reader035/viewer/2022062416/61125c6954d0df5e5a4e689f/html5/thumbnails/29.jpg)
Complete Induction
The theory T ∗PA
Peano Arithmetic with division
Consider the theory of T∗PA, which is the theory of Peano arithmetic TPA, augmented bythe following axioms:
A1. (∀x)(∀y) (x < y)→ [quot(x , y) = 0].
A2. (∀x)(∀y) (y > 0)→ [quot(x + y , y) = quot(x , y) + 1].
A3. (∀x)(∀y) (x < y)→ [rem(x , y) = x ].
A4. (∀x)(∀y) (y > 0)→ [rem(x + y , y) = rem(x , y)].
Example
Argue that(∀x)(∀y) (y > 0)→ [rem(x , y) < y ].
Subramani Mathematical Induction