we consider some examples. Example Check whether the following statement is true or not. If x , y ∈ Z are such that x and y are odd, then xy is odd. Solution Let p : x , y ∈ Z such that x and y are odd q : xy is odd To check the validity of the given statement, we apply Case of Rule .
That is assume that if p is true, then q is true. p is true means that x and y are odd integers. Then x = m + , for some integer m. y = n + , for some integer n .
Thus xy = ( m + ) ( n + ) = ( mn + m + n ) + This shows that xy is odd. Therefore, the given statement is true. Suppose we want to check this by using Case of Rule , then we will proceed as follows. We assume that q is not true.
This implies that we need to consider the negation of the statement q . This gives the statement ∼ q : Product xy is even. This is possible only if either x or y is even. This shows that p is not true.
Thus we have shown that ∼ q ⇒ ∼ p A Note The above example illustrates that to prove p ⇒ q , it is enough to show ∼ q ⇒ ∼ p which is the contrapositive of the statement p ⇒ q. Example Check whether the following statement is true or false by proving its contrapositive. If x , y ∈ ΖΖΖΖΖ such that xy is odd, then both x and y are odd. Solution Let us name the statements as below MATHEMATICAL REASONING p : xy is odd.
q : both x and y are odd. We have to check whether the statement p ⇒ q is true or not, that is, by checking its contrapositive statement i.e., ∼ q ⇒ ∼ p Now ∼ q : It is false