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Proof Techniques - Stanford University Computer Science

Proof TechniquesJessica SuNovember 12, 20161 Proof techniquesHere we will learn to prove universal mathematical statements, like the square of any oddnumber is odd . It s easy enough to show that this is true in specific cases for example,32= 9, which is an odd number, and 52= 25, which is another odd number. However, toprove the statement, we must show that it works forallodd numbers, which is hard becauseyou can t try every single one of that if we want todisprovea universal statement, we only need to find one counterex-ample. For instance, if we want to disprove the statement the square of any odd number iseven , it suffices to provide a specific example of an odd number whose square is not even.(For instance, 32= 9, which is not an even number.)Rule of thumb: Toprovea universal statement, you must show it works in all cases. Todisprovea universal statement, it suffices to find one counterexample.

In proof by contradiction, you assume your statement is not true, and then derive a con-tradiction. This is really a special case of proof by contrapositive (where your \if" is all of mathematics, and your \then" is the statement you are trying to prove). 2. 1.2 Proof by induction 1 PROOF TECHNIQUES

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