as an arbitrary even number arid not a particular value Why don t you try an exercise ngw E6 Give a direct proof of the statement If x is a real number such that x 9 then either x 3 or x 3 Let us now consider another proof strategy 2 3 2 Indirect Proofs In this sub section we shall consider two roundabout methods for proving p q Proof by contrapositive In the first method we use the fad that the proposition p q is logically equivalent to its contrapositive q p i e p q q p For instance If Ammu does not agree with communalists then she is not orthodox is the same as If Ammu is orthodox then she agrees with communalists Because of this equivalence to prove p q wc can instead prove q p This means that we can assume that q is true and then try to prove that p is true In other words what we do to prove p q in this method is to assume that q is false and then show that p is false Let us consider an example Example 4 Prove that If x y EZ such that xy is odd then botli x and y are odd by proving its contrapositive Solution Let us name the statements involved as below P xy is odd q both x and y are odd So p xy is even and q x is even or y is even or both are even We want to prove p q by proving that q p So we start by assuming that q is true i c we suppose that x is even Then x 2ii for some n E N Therefore xy 2ny Therefore xy is even by definition That is p is true So we have shown that q p Therefore p q S f Why don t you try some related exercises now E7 Write down the contrapositive of the statement If f is a 1 1 function from a finite set X into itself then f must be surjective E8 Prove the statement If x is an integer and x is even then x is also even by proving its contrapositive And now let us consider the other way of proving a statement indirectly