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.