Mathematical Induction

1
Mathematical Induction

• Introductory Discrete Mathematics (CS/MAT165)

2
Sequences

• P. 213, Nos. 2, 6, 11
• Work by yourself
• 10 minutes

3
Principle of Induction

• Let P(n) be a property that is defined for
  integers n, and let a be a fixed integer
• Suppose the following is true
• P(a) is true
• ?(k?a), P(k) ? P(k1)
• Then
• ?(n?a), P(n)

What is P(n)? Is it a Predicate or a Statement?
4
Proof by Induction

• Basis stepShow that the property is true for n
  a
• Induction stepShow that for all integers k ? a,
  if the property is true for n k, then it is
  true for n k 1

5
Proof by InductionExample

• Prove
• For all integers n ? 13 (22n - 1)
• Basis
• n 1
• 22(1) - 1 3, and 33
• Induction
• 22(k1) - 1 22k?22 - 1 22k?4 - 1
• 22k?4 - 1 - (22k - 1) 22k?3

Divisible by 3
6
Try this

• Prove by induction
• For all integers n ? 32n 1 lt 2n
• Work in pairs
• You have 10 minutes!

7
And this

• Prove by induction
• For all n ? 04 (5n - 1)
• For all n ? 0 4n ? 1 3n
• For all n ? 1(x - y) (xn - yn), where x ? y

8
And this

• P. 226, Nos. 7, 9, 12

9
Summing Series

• Q What is the sum of the numbers 1 through 10?
• Sum 1 2 3 4 5 6 7 8 9 10
  55
• Or (110) (29) (38) (47) (56)
  11 11 11 11 11 5(11) 55

10
Summing Series

• Sum 1 2 3 4 5 6 7 8 9 10
  5(11) (10/2)(11) (n/2)(n1), where n
  10 (n(n1))/2
• Does this formula work for any n?
• Can you prove this using induction?
• Try it!

Work in pairs 10 minutes!
11
Series Product

• Q What is the product of the numbers 1 through
  3?
• Product 1 x 2 x 3 6
• How about 1 through 4?
• Product 6 x 4 24
• 1 through 5?
• Any pattern emerging yet?
• Tuck this away for a future date!

12
Strong Mathematical Induction

• Basis
• Show P(a), P(a1), , P(b)
• Inductive
• First For any integer k gt b, if P(i) is true
  for all integers with k gt i ? a, then P(k)
• Second For all integers n ? a, P(n)

13
Strong Mathematical InductionExample

• Prove that for all integers n ? 2, pn, where p
  is a prime number.
• Basis
• n22 is a prime number, therefore its divisible
  by itself

14
Strong Mathematical InductionExample

• Inductive
• Assume P(i) for 2 ? i lt k
• Prove P(k)
• If k is prime, then its divisible by itself
• If k is not prime then kabBut 2 ? a lt k, which
  means pa and thus pk

15
Try this

• Given any integer n and positive integer d, there
  exists integers q and r such that
• n dq rand0 ? r lt d

16
And this

• P. 242, Nos. 1, 3 5