Lecture 11

1.5, 3.1 Methods of Proof

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Last time in 1.5

To prove theorems we use rules of inference such as:

p, pq, therefore, q

NOT q, pq, therefore NOT p.

p AND q, therefore p

FORALL x P(x), therefore for arbitrary c, P(c)

EXISTS x P(x), therefore for some c, P(c)

It is easy to make mistakes, make sure that:

1) All premises pi are true when you prove (p1 AND p2 AND...pn) q

2) Every rule of inference you use is correct.

Some proof strategies:

To proof pq

1) direct proof: assume p is true, use rules to prove that q is true.

2) indirect proof, assume q is NOT true, use rules to prove p is NOT true.

To prove p is true:

3) By contradiction: assume p is NOT true, use rules to show that NOT pF

i.e. it leads to a contradiction.

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Vacuous –Trivial Proofs

Lets say we want to prove pq but the premise p can be shown to be false!

Then pq is always true because (FT) = T and (FF) = T.

This is a vacuous prove.

Old example: prove that for any set S:

Proof: The following must be shown to be true:

However: the empty set does not contain any elements

and the premise is always false. Therefore the implication

must always be true!

S∅⊆

()xx x S∀ ∈∅→ ∈

Trivial Proof: We want to prove pq, and we can show that q is true.

Then, because (TT) = T and (FT) = T we have proven the implication.

Example: P(n): a>=b a^n >= b^n for postive integers.

Is P(0) true?

P(0): a^0 >= b^0 is equivalent to 1>=1. Therefore, q is true and thus pq is true.

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Example Indirect Proof

Prove that: if n is an integer and n^2 is odd, then n is odd.

Direct prove is hard in this case.

Indirect proof: Assume NOT q : n is even.

n = 2k

n^2 = 4k^2 = 2(2k^2) is even, is not odd.

Thus NOT q NOT p, pq

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