Question

Check whether the following statement 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) p: If x and y are odd integers, then x + y is an even integer.
Let q and r be two statements.
Here,
q: x and y are odd integers.
r: x + y is an even integer.
Let q be true.
Then, q is true.
Now,
x and y are odd integers.
x = 2m +1 and y = 2n + 1 for some integers m and n.
$\Rightarrow$x + y = (2m + 1) + (2n + 1)
$\Rightarrow$x + y = 2m + 2n + 2 = 2(m + n + 1)
So, x + y is an even integer.
Hence, the statement is true.

ii) p: If x and y are integers such that xy is even, then at least one of x and y is an even integer.
Let q and r be two statements.
Here,
q: xy is an even integer.
r: At least one of x and y is an even integer.
Let r be not true.
Then, r is not true.
It is false that at least one of x and y is an even integer.
Now,
x and y are odd integers.
x = 2m +1 and y = 2n + 1 for some integers m and n.
$\Rightarrow$xy =(2m + 1)(2n + 1)
$\Rightarrow$xy = 2(2mn + m + n) + 1
So, xy is not an even integer. Thus, xy is not true.
Hence, the statement is true.