Mathematical Induction

Assume that we are given an infinite supply of stamps of two different denominations, 3 cents and and 5 cents. Prove using mathematical induction that it is possible to make up stamps of any value 8 cents.

Basis: We can make an 8 cents stamp by using one 3-cents and one 5-cents stamp. Inductive hypothesis: Assume that we can make stamps of values k = 8, 9, 10, … n. Inductive step: Show that we can compose a stamp of value n + 1.

Composing Stamps: Example 3

Illustration

8 = 3 + 5

9 = 3 + 3 + 3 = 3 * 3

10 = 5 + 5 = 2 * 5

11 = 1 * 5 + 2 * 3

Outline of Proof

Composing Stamps – Actual Proof

Inductive step: Show that we can compose a stamp of value n + 1.In composing n, several (or none) 3-cents and several (or none) 5-cents stamps have been used.To go from n to n + 1, we consider two cases.Case 1 : If there is at least one 5-cents stamp in the collection, replace it by two 3-cents stamps. This gives us stamps of value n + 1 cents.

Case 2 : Suppose that the current collection uses only 3-cents stamps. Since n 8, there must three 3-cents stamps in the collection. Replace these three stamps by two 5-cents stamps. This gives us a stamp of value n + 1 cents.The proof is complete. Another Proof

Another Proof

Another Proof

n + 1 = 3 * (p + 2 ) + 5 * (q – 1) if q > 1

n + 1 = 3 * ( p – 3 ) + 5 * (q + 2 ) if p > 3

For all n > 8, n can be expressed as a linear combination of 3 and 5, that is n = 3 * p + 5 * q for n >= 8

Review Propositional Logic – Page 1

Implication In propositional logic "if p then q" is written as p q and read as "p implies q". p q p q same as p q

F F T F T T T F F T T T

How to remember this definition?Implication is false only when the premise is true and the consequence is false.

Propositional Logic – Page 2

Bi-conditional p q means p iff q

p q p q same as

F F T

F T F

T F F

T T T

How to remember this definition?Bi-conditional is a like a magnitude comparator, it is true when both inputs are identical.

More Terminology

ContradictionA logical expression that is always false, regardless of what truth values are assigned to its statement variables, is called a contradiction. The statement p p is a contradiction.TheoremIf A and B are logical statements and if the statements A and A B are true, then the statement B is true.

A logical expression that is always true, regardless of what truth values are assigned to its statement variables, is called a tautology. The statement p p is a tautology.

Tautology

Simple Logic Proofs

A B A B A (A B) A (A B) B F F T F T F T T F T T F F F T T T T T T

Prove that A (A B) B is a contradiction. A (A B) B = A (A B) B= (A A) (A B) B using distributive law.= F (A B) B = (A B) B = A (B B) using associative law = A F = F Contradiction. Hence B must be true.

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An Important Theorem

Among the four statementsp q statementq p conversep q inverseq p contra-positive

1. The statement and its contra-positive are equivalent.

2. The converse and inverse are equivalent.3. No other pairs in the statements given

above are equivalent.Proof: Make a truth table to see that 1 and 2 are tautologies.

Proof By Contradiction

Prove that is not a rational number. (Example 1.7, page 13)2

Proof: Assume is a rational number. Let m and n be two integers, with no common factor, such that

2

m

n2

Or 222 nm

2n

22 42 km 22 2km

This shows that is even.2m

Since is a multiple of 2, it is an even number. Therefore, n is even must be of the form 2k for some integer k.

1

1

Proof Continued …

We have concluded that both and are even, therefore, n and m are also even They must have a common factor. This contradicts our assumption. Hence is not a rational number.

2n 2m

2

Proof of Proposition 1

p q p q ~p

~q

~q ~p (p q)(~q ~p)

F F T T T T T

F T T T F T T

T F F F T F T

T T T T T T T