2 3 DIFFERENT METHODS OF PROOF In this section we shall consider three different strategies for proving a statement We will also discuss a method that is used only for disproving a statement Let us start with a proof strategy based on the first rule of inference that we discussed in the previous section 2 3 1 Direct Proof This form of proof is based entirely on modus ponens Let us formally spell out the strategy Definition A direct proof of p q is a logically valid argument that begins with the assumption that p is true and in one or more applications of the law of detachment concludes that q must be true So to construct a direct proof of p q we start by assuming that P is true Then in one or more steps of the form p 9 9192 9n q we conclude that q is true Consider the following examples 1 Example 2 Give a direct proof of the statement The product of two odd integers is odd Solution Let us clearly analyse what our hypotheses are and what we have to prove We start by considering any two odd integers x and y So our hypothesis is p x and y are odd The conclusion we want to reach is q xy is odd 9 Let us first prove that P Since x is odd x 2m 1 for some integer m Similarly y 2n 1 for some integer n Then xy 2m 1 2n 1 2 2mn m n 1 Therefore xy is odd So we have shown that p q Now we can apply modus ponens to p A p q to get the required conclusion Note that the essence of this direct proof lies in showing p q b Exar 3 Give a direct proof of the theorem The square of an even integer is an even integer Solution First of all let us write the given statement symbolically as VxEZ p x q x where p x x is even and q x x is even i e q x is the same as p x The direct proof then goes as follows Let x be an even number i e we assume p x is true Then x 2n for some integer n we apply the definition of an even number