Answers:

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

Conisder,

p: x and y are odd integers.

q: x + y is an even integer

If p, then q.

Let p be true. [x and y are odd integers]

x = 2m+1, y = 2n+1 for some integers m, n

x + y = (2m+1) + (2n+1)

x + y = (2m+2n+2)

x + y = 2(m+n+1)

x + y is an integer

q is true.

p is true and q is true.

“if p, then q “is a true statement.”

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

Consider,

p: x and y are integers and xy is an even integer.

q: At least one of x and y is even.

Let p be true, and then xy is an even integer.

xy = 2(n + 1)

Let,

x = 2(k + 1)

x is an even integer, xy = 2(k + 1). y is also an even integer.

Let us take,

x = 2(k + 1) and y = 2(m + 1)

xy = 2(k + 1).2(m + 1) = 2.2(k + 1)(m + 1)

The statement is true.