I am facing a problems with this question : Prove that If x and y are integers and x + y is odd , then x−y is odd. , any suggestions ?
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You can say that $x-y=x+y-2y\equiv x+y\ ({\rm mod}\ 2)\equiv 1\ ({\rm mod}\ 2)$ and thus $x-y$ is also odd. – Tuvasbien Dec 08 '20 at 07:50
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This is barely readable. – Dec 08 '20 at 08:01
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Your solution is wrong because you cannot just assume that $x$ is odd. Your proof says nothing about what happens if $x$ is even. You could separate the cases (there are four of them) and run the proof over each of those, but there is a much simpler way of proving the statement.
Hint: Use the fact that $$x-y = (x+y) - 2y.$$
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The difference between $x+y$ and $x-y$ is an even number.