Question

Check whether the following ststement are true or not :

(i) p : If x and y are odd integers, then x + y is an even integer.

(ii) q : If x, y are integers such that xy is even, then at least one of x and y is an even integer.

Answer 1

(i) Let q and r be the statements given by

q : x and y are odd integers.

r : x + y is an even integer.

The, the given statement is

if q, then r.

Direct Method : Let q be true. Then,

q is true.

$\Rightarrow$ x and y are odd integers.

$\Rightarrow$ x= 2m + 1, y = 2n + 1 for some intergers m, n

$\Rightarrow$ x + y = (2m + 1) + (2n + 1)

$\Rightarrow$ x + y = (2m+2n+2)

$\Rightarrow$ x + y = 2(m+n+1)

$\Rightarrow$ x + y is an even integer

$\Rightarrow$ r is true.

Thus, q is true $\Rightarrow$ r is true.

Hence, "if q, then r" is a true statement.

(ii) q : If x, y are integers such that xy is even, then at least one of x and y is an even integer.

Let r and s be two statements given by

r : xy an even integer.

s : At least one of x and y is an even integer.

Let S be not true. Then,

s is not true.

$\Rightarrow$ Both x and y are odd integers

Let x = 2n + 1 and y = 2m + 1 for some integers n and m. Then,

$\Rightarrow$ xy = (2n+1)+ (2m+1) for some integers n and m.

$\Rightarrow$ xy = 4nm + 2(n+m)+1 for some integers n and m.

$\Rightarrow$ xy is an odd integer

$\Rightarrow$ xy is not an even integer

$\Rightarrow$ -r is true

Thus, -s is true $\Rightarrow$ -r is true

Hence, the given statement is true.