1 Intro to Induction Supplementary Notes Prepared by Raymond Wong Presented by Raymond Wong
2
e.g.1 (Page 4)
Illustration of “Proof by Contradiction”We are going to prove that a claim C is correct
Proof by Contradiction:
Suppose “NOT C”
….
Derive some results, which may contradict to 1. “NOT C”, OR
2. some facts
e.g., we derived that C is true finally
e.g., we derived that “1 = 4”
3
e.g.1
Illustration of “Proof by smallest counter example”
We are going to prove the following claim C: statement P(m) is true for each non-negative integer m, namely 0, 1, 2, …
P(0) true
P(1) true
P(2) true
P(3) true
P(4) true
If we can prove that statement P(m) is true for each non-negative integer separately, then we can prove the above claim C is correct.
Suppose that I want to prove that the above claimis correct by “Proof by Contradiction”.
… true
4
e.g.1
Illustration of “Proof by smallest counter example”
We are going to prove the following claim C: statement P(m) is true for each non-negative integer m, namely 0, 1, 2, …
P(0) true
P(1) true
P(2) true
P(3) true
P(4) true
We can assume that there exists a non-negative integer k’ such that P(k’) is false
Suppose that I want to prove that the above claimis correct by “Proof by Contradiction”.
… true
false
Suppose “NOT C”.
There may exist another non-negative integer k such that P(k) is false
false
5
e.g.1
Illustration of “Proof by smallest counter example”
We are going to prove the following claim C: statement P(m) is true for each non-negative integer m, namely 0, 1, 2, …
P(0) true
P(1) true
P(2) true
P(3) true
P(4) true
Suppose that I want to prove that the above claimis correct by “Proof by Contradiction”.
… true
false
Suppose “NOT C”.
false
We can assume that there exists a smallest non-negative integer k such that P(k) is false
Why?
This is called by “Proof by smallest counter example”.
6
e.g.2 (Page 5)Steps for “Proof by smallest counter example”
Suppose “NOT C”.
We are going to prove the following claim C: statement P(n) is true for each non-negative integer n, namely 0, 1, 2, …
P(0) true
P(1) true
P(2) true
P(3) true
P(4) true
… true
false
false
Step 1: Suppose that claim C is not true.
Step 2: there exists a smallest non-negative integer m such that P(m) is false.
Step 3: We want to show that this value m must be greater than the smallest value (i.e., 0)
Step 4: We derive that P(i) is true for 0 i < m
Step 5: We consider a special case that P(m-1) is true.
Step 6: Consider the LHS (or some components) of P(m) Prove that P(m) is true (by using P(m-1))
Step 7: We have a contradiction that P(m) is false
Step 8: Thus, by the principle of proof by contradiction,claim C is correct.
e.g., P(n) is“0+1+2+…+n = ”
n(n+1)
2
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e.g.3 (Page 14)
Illustration of “Proof by mathematical induction”
We are going to prove the following claim C: statement P(n) is true for each non-negative integer n, namely 0, 1, 2, …
P(0) true
P(1) true
P(2) true
P(3) true
P(4) true
If we can prove that statement P(n) is true for each non-negative integer separately, then we can prove the above claim C is correct.
… true
8
e.g.3
Illustration of “Proof by mathematical induction”
We are going to prove the following claim C: statement P(n) is true for each non-negative integer n, namely 0, 1, 2, …
P(0) true
P(1)
P(2)
P(3)
P(4)
…
Step 1: Prove that P(0) (i.e., the base case) is true.
Verify that P(0) is true
9
e.g.3
Illustration of “Proof by mathematical induction”
We are going to prove the following claim C: statement P(n) is true for each non-negative integer n, namely 0, 1, 2, …
P(0) true
P(1)
P(2)
P(3)
P(4)
…
Step 1: Prove that P(0) (i.e., the base case) is true.
Step 2: Prove that “P(n-1) P(n)” is true for all n > 0.
true
true
true
true
true
Step 2(a): Assume that P(n-1) is true for n > 0.
Verify that P(0) is true
Step 2(b): According to P(n-1), we deduce that P(n) is true.
Inductive Hypothesis
Inductive Step
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e.g.4 (Page 15)Prove that n 0, 2n+1 n2+2
Step 1: Prove that P(0) (i.e., the base case) is true.
Let P(n) be “2n+1 n2+2“
Consider 20+1 = 2
= 0+2
We want to show that 20+1 02+2
02+2
Thus, P(0) is true.
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e.g.4
Prove that n 0, 2n+1 n2+2
Let P(n) be “2n+1 n2+2“Step 2: Prove that “P(n-1) P(n)” is true for all n > 0.
Step 2(a): Assume that P(n-1) is true for n > 0.
Step 2(b): According to P(n-1), we deduce that P(n) is true.
That is, 2(n-1)+1 (n-1)2+2 for n > 0.
2n (n-1)2+2 for n > 0.
Consider 2n+1
We want to show that 2n+1 n2+2
= 2n . 2
[(n-1)2+2] . 2
= 2(n-1)2+2. 2
= 2(n2-2n+1)+4
= 2n2-4n+2+4
= 2n2-4n+6
= (n2+n2) – 4n + (2+4)
What should I do next?
= n2+2 + n2 – 4n + 4
= (n2+2) + (n2 – 4n + 4)
= (n2+2) + (n – 2)2
n2+2 (Since (n-2)2 0)
That is, P(n) is true.
We prove that “P(n-1) P(n)” is true for all n > 0
By Mathematical Induction, n 0, 2n+1 n2+2
Thus, 2n+1 n2+2
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e.g.5 (Page 16)Prove that n 2, 2n+1 > n2+3
Step 1: Prove that P(2) (i.e., the base case) is true.
Let P(n) be “2n+1 > n2+3“
Consider 22+1 = 23
= 8
We want to show that 22+1 > 22+3
> 7
Thus, P(2) is true.
= 22+3
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e.g.5
Prove that n 2, 2n+1 > n2+3
Let P(n) be “2n+1 > n2+3“Step 2: Prove that “P(n-1) P(n)” is true for all n > 2.
Step 2(a): Assume that P(n-1) is true for n > 2.
Step 2(b): According to P(n-1), we deduce that P(n) is true.
That is, 2(n-1)+1 > (n-1)2+3 for n > 2.
2n > (n-1)2+3 for n > 2.
Consider 2n+1
We want to show that 2n+1 > n2+3
= 2n . 2
>[(n-1)2+3] . 2
= 2(n-1)2+3. 2
= 2(n2-2n+1)+6
= 2n2-4n+2+6
= 2n2-4n+8
= (n2+n2) – 4n + (3+5)
What should I do next?
= n2+3 + n2 – 4n + 5
= (n2+3) + (n2 – 4n + 5)
= (n2+3) + (n2 – 4n + 4 + 1)
> n2+3 (Since (n-2)2 + 1 0)
Thus, P(n) is true.
We prove that “P(n-1) P(n)” is true for all n > 2
By Mathematical Induction, n 2, 2n+1 > n2+3
= (n2+3) + (n – 2)2 + 1
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e.g.6 (Page 18)Prove that k Z+, 1+3+5+…+(2k-1) = k2
Step 1: Prove that P(1) (i.e., the base case) is true.
Let P(k) be “1+3+5+…+(2k-1) = k2”
Consider 1 = 12
We want to show that 1 = 12
Thus, P(1) is true.
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e.g.6
Prove that k Z+, 1+3+5+…+(2k-1) = k2
Let P(k) be “1+3+5+…+(2k-1) = k2”Step 2: Prove that “P(n-1) P(n)” is true for all n > 1.
Step 2(a): Assume that P(n-1) is true for n > 1.
Step 2(b): According to P(n-1), we deduce that P(n) is true.
That is, 1+3+5+…+(2(n-1)-1) = (n-1)2 for n > 1.
1+3+5+…+(2n-3) = (n-1)2 for n > 1.
Consider 1+3+5+…+(2n-1)
We want to show that 1+3+5+…+(2n-1) = n2
= 1+3+5+…+(2n-3)+(2n-1)
= (n-1)2 + (2n-1)
= (n2 – 2n + 1) + (2n – 1)
= n2
Thus, P(n) is true.
We prove that “P(n-1) P(n)” is true for all n > 1 By Mathematical Induction, k Z+, 1+3+5+…+(2k-1) = k2
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e.g.7 (Page 19)For what positive integer values of n is “2n > n2”?
Step 1: Prove that P( ? ) (i.e., the base case) is true.
Let P(n) be “2n > n2”
Consider P(1) We want to see whether P(1) is true.
(i.e., whether “21 > 12” is true.)Consider 21
We don’t know the base case. Thus, we need to test the “smallest” value of n for the base case.
= 2
> 12
Thus, P(1) is true.
Consider P(2) We want to see whether P(2) is true.
(i.e., whether “22 > 22” is true.)Consider 22 = 4
Thus, P(2) is false.
> 22
P(1) is true.
P(2) is false.
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e.g.7For what positive integer values of n is “2n > n2”?
Step 1: Prove that P( ? ) (i.e., the base case) is true.
Let P(n) be “2n > n2”
Consider P(3) We want to see whether P(3) is true.
(i.e., whether “23 > 32” is true.)Consider 23
We don’t know the base case. Thus, we need to test the “smallest” value of n for the base case.
= 8
Thus, P(3) is false.
Consider P(4) We want to see whether P(4) is true.
(i.e., whether “24 > 42” is true.)Consider 24 = 16
Thus, P(4) is false.
> 42
> 32
P(1) is true.
P(2) is false.
P(3) is false.
P(4) is false.
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e.g.7For what positive integer values of n is “2n > n2”?
Step 1: Prove that P( ? ) (i.e., the base case) is true.
Let P(n) be “2n > n2”
Consider P(5) We want to see whether P(5) is true.
(i.e., whether “25 > 52” is true.)Consider 25
We don’t know the base case. Thus, we need to test the “smallest” value of n for the base case.
= 32
Thus, P(5) is true.
Consider P(6) We want to see whether P(6) is true.
(i.e., whether “26 > 62” is true.)Consider 26 = 64
Thus, P(6) is true.
> 62
> 52
Thus, we guess that P(7), P(8), …are also true.
Thus, we think that the base case is P(5).
P(5)
P(1) is true.
P(2) is false.
P(3) is false.
P(4) is false.
P(5) is true.
P(6) is true.
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e.g.7For what positive integer values of n is “2n > n2”?
Step 1: Prove that P( ? ) (i.e., the base case) is true.
Let P(n) be “2n > n2”
P(5)
Prove that n 5, 2n > n2
Thus, we think that the base case is P(5).
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e.g.7
Step 1: Prove that P( ? ) (i.e., the base case) is true.
Let P(n) be “2n > n2”
P(5)
Prove that n 5, 2n > n2
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e.g.7
Step 1: Prove that P( ? ) (i.e., the base case) is true.
Let P(n) be “2n > n2”
P(5)
Prove that n 5, 2n > n2
We want to show that “25 > 52”
Consider 25 = 32
Thus, P(5) is true.
> 52
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e.g.7
Prove that n 5, 2n > n2
Let P(n) be “2n > n2”Step 2: Prove that “P(n-1) P(n)” is true for all n > 5.
Step 2(a): Assume that P(n-1) is true for n > 5.
Step 2(b): According to P(n-1), we deduce that P(n) is true.
That is, 2n-1 > (n-1)2 for n > 5.
Consider 2n
We want to show that 2n > n2
= 2n-1 . 2
> (n-1)2 . 2
= (n2 – 2n + 1) . 2
= 2n2 – 4n + 2
Thus, P(n) is true.
We prove that “P(n-1) P(n)” is true for all n > 5 By Mathematical Induction, n 5, 2n > n2
= n2 + n2 – 4n + 2
What should I do next?
> n2 + n2 – 4n
> n2 + n2 – 5.n
> n2 + n2 – n.n (Since n > 5)= n2
Thus, 2n > n2
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e.g.7
Prove that n 5, 2n > n2
Let P(n) be “2n > n2”Step 2: Prove that “P(n-1) P(n)” is true for all n > 5.
Step 2(a): Assume that P(n-1) is true for n > 5.
Step 2(b): According to P(n-1), we deduce that P(n) is true.
That is, 2n-1 > (n-1)2 for n > 5.
Consider 2n
We want to show that 2n > n2
= 2n-1 . 2
> (n-1)2 . 2= (n2 – 2n + 1) . 2
= 2n2 – 4n + 2
Thus, P(n) is true.We prove that “P(n-1) P(n)” is true for all n > 5 By Mathematical Induction, n 5, 2n > n2
= n2 + n2 – 4n + 2
= n2 + n2 – 4n + 4 – 4 +2
= n2 + (n2 – 4n + 4) – 2
= n2 + (n – 2)2 – 2
Thus, 2n > n2
Alternative Derivation
Since (n-2)2 is increasing when n> 5,
we have (n-2)2 > (5-2)2
= 32
= 9> n2 + 9 – 2
= n2 + 7> n2
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e.g.8 (Page 21)
Illustration of “Proof by mathematical induction” (Weak Induction)
We are going to prove the following claim C: statement P(n) is true for each non-negative integer n, namely 0, 1, 2, …
P(0) true
P(1)
P(2)
P(3)
P(4)
…
Step 1: Prove that P(0) (i.e., the base case) is true.
Verify that P(0) is true
29
e.g.8
Illustration of “Proof by mathematical induction” (Weak Induction)
We are going to prove the following claim C: statement P(n) is true for each non-negative integer n, namely 0, 1, 2, …
P(0) true
P(1)
P(2)
P(3)
P(4)
…
Step 1: Prove that P(0) (i.e., the base case) is true.
Step 2: Prove that “P(n-1) P(n)” is true for all n > 0.
true
true
true
true
true
Step 2(a): Assume that P(n-1) is true for n > 0.
Verify that P(0) is true
Step 2(b): According to P(n-1), we deduce that P(n) is true.
Inductive Hypothesis
Inductive Step
30
e.g.8
Illustration of “Proof by mathematical induction” (Strong Induction)
We are going to prove the following claim C: statement P(n) is true for each non-negative integer n, namely 0, 1, 2, …P(0) true
P(1)
P(2)
P(3)
P(4)
…
Step 1: Prove that P(0) (i.e., the base case) is true.
Verify that P(0) is true
31
e.g.8
Illustration of “Proof by mathematical induction” (Strong Induction)
We are going to prove the following claim C: statement P(n) is true for each non-negative integer n, namely 0, 1, 2, …P(0) true
P(1)
P(2)
P(3)
P(4)
…
Step 1: Prove that P(0) (i.e., the base case) is true.
Step 2: Prove that “P(0)P(1)…P(n-1) P(n)” is true for all n > 0.
true
Step 2(a): Assume that P(0)P(1)…P(n-1) is true for n > 0.
Verify that P(0) is true
Step 2(b): According to P(0)P(1)…P(n-1), we deduce that P(n) is true.
Inductive Hypothesis
Inductive Step
32
e.g.8
Illustration of “Proof by mathematical induction” (Strong Induction)
We are going to prove the following claim C: statement P(n) is true for each non-negative integer n, namely 0, 1, 2, …P(0) true
P(1)
P(2)
P(3)
P(4)
…
Step 1: Prove that P(0) (i.e., the base case) is true.
Step 2: Prove that “P(0)P(1)…P(n-1) P(n)” is true for all n > 0.
true
true
Step 2(a): Assume that P(0)P(1)…P(n-1) is true for n > 0.
Verify that P(0) is true
Step 2(b): According to P(0)P(1)…P(n-1), we deduce that P(n) is true.
Inductive Hypothesis
Inductive Step
33
e.g.8
Illustration of “Proof by mathematical induction” (Strong Induction)
We are going to prove the following claim C: statement P(n) is true for each non-negative integer n, namely 0, 1, 2, …P(0) true
P(1)
P(2)
P(3)
P(4)
…
Step 1: Prove that P(0) (i.e., the base case) is true.
Step 2: Prove that “P(0)P(1)…P(n-1) P(n)” is true for all n > 0.
true
true
Step 2(a): Assume that P(0)P(1)…P(n-1) is true for n > 0.
Verify that P(0) is true
Step 2(b): According to P(0)P(1)…P(n-1), we deduce that P(n) is true.
Inductive Hypothesis
Inductive Step
true
34
e.g.8
Illustration of “Proof by mathematical induction” (Strong Induction)
We are going to prove the following claim C: statement P(n) is true for each non-negative integer n, namely 0, 1, 2, …P(0) true
P(1)
P(2)
P(3)
P(4)
…
Step 1: Prove that P(0) (i.e., the base case) is true.
Step 2: Prove that “P(0)P(1)…P(n-1) P(n)” is true for all n > 0.
true
true
Step 2(a): Assume that P(0)P(1)…P(n-1) is true for n > 0.
Verify that P(0) is true
Step 2(b): According to P(0)P(1)…P(n-1), we deduce that P(n) is true.
Inductive Hypothesis
Inductive Step
true
true
35
e.g.9 (Page 26)Prove that every positive integer n is a power of a prime number OR a product of powers of prime numbers
Step 1: Prove that P(1) (i.e., the base case) is true.
Consider 1 = 20
which is a power of a prime number (i.e., 2).
Thus, P(1) is true.
Let P(n) be “n is a power of a prime number OR a product of powers of prime numbers.”We want to show that 1 is a power of a prime number OR a product of powers of prime numbers
e.g., 4 = 22
12 = 22 . 3
36
e.g.9Prove that every positive integer n is a power of a prime number OR a product of powers of prime numbers
Let P(n) be “n is a power of a prime number OR a product of powers of prime numbers.”
37
e.g.9
Prove that every positive integer n is a power of a prime number OR a product of powers of prime numbers
Let P(n) be “n is a power of a prime number OR a product of powers of prime numbers.”Step 2: Prove that “P(1)P(2)…P(n-1) P(n)” is true for all n > 1.
Step 2(a): Assume that P(1)P(2)…P(n-1) is true for n > 1.
Step 2(b): According to P(0)P(1)…P(n-1), we deduce that P(n) is true.
That is, 1 is a power of a prime number OR a product of powers of prime numbers. 2 is a power of a prime number OR a product of powers of prime numbers. … n-1 is a power of a prime number OR a product of powers of prime numbers.
We want to show that n is a power of a prime number OR a product of powers of prime numbers.Consider two cases.
Case 1: n is a prime number.
Case 2: n is not a prime number.
n is a power of a prime numbern is a product of two smaller numbers, namely a and b (where a < n and b < n)Thus, n = a . bNote that a is a power of a prime number or a product of powers of prime
numbers.Note that b is a power of a prime number or a product of powers of prime numbersThus, n is a power of a prime number or a product of powers of prime numbers
38
e.g.9
Prove that every positive integer n is a power of a prime number OR a product of powers of prime numbers
Let P(n) be “n is a power of a prime number OR a product of powers of prime numbers.”Step 2: Prove that “P(1)P(2)…P(n-1) P(n)” is true for all n > 1.
Step 2(a): Assume that P(1)P(2)…P(n-1) is true for n > 1.
Step 2(b): According to P(0)P(1)…P(n-1), we deduce that P(n) is true.
That is, 1 is a power of a prime number OR a product of powers of prime numbers. 2 is a power of a prime number OR a product of powers of prime numbers. … n-1 is a power of a prime number OR a product of powers of prime numbers.
Thus, P(n) is true.
We prove that “P(1)P(2)…P(n-1) P(n)” is true for all n > 1
By Strong Mathematical Induction, every positive integer n is a power of a prime number OR a product of powers of prime numbers