Mathematical Induction - Discrete Math - Lecture Slides, Slides of Discrete Mathematics

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Discrete Maths

• Objective
  • to introduce mathematical induction through examples

1. Mathematical

Induction

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Overview

1. Motivation
2. Induction Defined
3. Maths Notation Reminder
4. Four Examples
5. More General Induction Proofs
6. A Fun Tiling Problem
7. Further Information

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• Induction is often used to derive mathematical estimates of program running time - timings based on the size of input data - e.g. time increases linearly with the number of data items processed - timings based on the number of times a loop executes

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1. Induction Defined

• Induction is used to solve problems such as:
  • is S(n) correct/true for all n values?
    • usually for all n >= 0 or all n >=
• Example:
  • let S(n) be "n^2 + 1 > 0"
  • is S(n) true for all n >= 1?

continued

S(n) can be much more complicated, such as a program that reads in a n value.

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Induction to the Rescue

• Induction is a technique for quickly proving S(n) true or false for all n - we only have to do two things
• First show that S(1) is true
  • do that by calculation as before

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• Second, assume that S(n) is true, and use it to show that S(n+1) is true - mathematically, we show that S(n) --> S(n+1)
• Now we know S(n) is true for all n>=1.
• Why?

continued

"--> stands for "implies"

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Let’s do it

• Prove S(n): "n^2 + 1 > 0" for all n >= 1.
• First task: show S(1) is true
  • S(1) == 1^2 + 1 == 2, which is > 0
  • so S(1) is true
• Second task: show S(n+1) is true by assuming S(n) is true

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• Assume S(n) is true, so n^2 +1 > 0
• Prove S(n+1)
  • S(n+1) == (n+1)^2 + 1 == n^2 + 2n + 1 + == (n^2 + 1) + 2n + 1
  • since n^2 + 1 > 0, then (n^2 + 1) + 2n + 1 > 0
  • so S(n+1) is true, by assuming S(n) is true
  • so S(n) --> S(n+1)

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Induction More Formally

• Three pieces:
  • 1. A statement S(n) to be proved
    • the statement must be about an integer n
  • 1. A basis for the proof. This is the statement S(b) for some integer. Often b = 0 or b = 1.

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• 1. An inductive step for the proof. We prove the statement “S(n) --> S(n+1)” for all n.
• The statement S(n), used in this proof, is called the inductive hypothesis
• We conclude that S(n) is true for all n >= b
• S(n) might not be true for some n < b

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1. Example 1

• Prove the statement S(n):

for all n >= 1

• e.g. 1+2+3+4 = (4*5)/2 = 10
• Basis. S(1), n = 1

so 1 = (1*2)/

2

( 1 ) 1

∑ =

n n i

n

i

∑

=

1 1

1 i

i

continued

(1)

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• Induction. Assume S(n) to be true.

Prove S(n+1), which is:

Notice that:

2

( 1 )( 2 ) 2

(^11) ( 1 1 )

1

=

∑ =

=

n n n n i

n

i

(2)

∑ ∑

= =

= + +

1

1 1

( 1 )

n

i

n

i

i i n (3)

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Example 2

• Prove the statement S(n):

for all n >= 0

• e.g. 1+2+4+8 = 16-
• Basis. S(0), n = 0

so 2^0 = 2^1 -

2 2 1 1 0

= + −

∑

n n

i

i

∑

=

0 0

2 20 i

i

continued

(1)

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• Induction. Assume S(n) to be true.

Prove S(n+1), which is:

Notice that:

2 2 2 1

1

0

= + −

=

∑

n n

i

i (^) (2)

∑ ∑

= =

(^1) = + +

0 0

n 2 2 2 1

i

n

i

i i n (3)

continued Docsity.com