Question -
Answer -
(i) p: If x and y are odd integers, then x + y is an even integer.
Let us assume that ‘p’ and ‘q’ be the statements given by
p: x and y are odd integers.
q: x + y is an even integer
the given statement can be written as :
if p, then q.
Let p be true. Then, 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.
So, p is true and q is true.
Hence, “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.
Let us assume that p and q be the statements given by
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.
So,
xy = 2(n + 1)
Now,
Let x = 2(k + 1)
Since, x is an even integer, xy = 2(k + 1). y is also an even integer.
Now take x = 2(k + 1) and y = 2(m + 1)
xy = 2(k + 1).2(m + 1) = 2.2(k + 1)(m + 1)
So, it is also true.
Hence, the statement is true.