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Question
Check whether the statement are true or not:
q : If x, y are integers such that xy is even, then at least one of x and y is an even integer.
Solution
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.
⇒ xy =(2m + 1)(2n + 1)
⇒ xy = 2(2mn + m + n) + 1
So, xy is not an even integer. Thus, xy is not true.
Hence, the statement is true.
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