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Show that if x and y are two integers whose product is even, then at least one of the two must be even (use the contrapositive argument)

now the thing is I understand how to prove this with a contrapositive argument and without. what I don't get is why proving it with a contrapositive argument works. The prove would be that assume x an y are both odd. you do some calculations and get that xy is odd. I don't understand how this proves the other 2 possibilities. Whats to say that xy isn't ALWAYS odd and we just walked into this nicely. I can only see one way to properly prove this and this is to go through every single (3) case and work it out. I get that contrapositive arguments are meant to take the idea that x-> q so not x -> not q but i don't see why cant BOTH results be "wrong"

RandomGuy57
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1 Answers1

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$P\implies Q$ has a contraposition $\neg Q\implies\neg P$.

We can use a simpler proposition to make understanding why easier.

We know that all squares are rectangles. This can be rewritten as "if this shape is a square, then this shape is a rectangle". The contraposition is "if this shape is not a rectangle, then there is no way that this shape is a square".

Now we draw a shape. Using geometry, we identify that this shape is not a rectangle. Therefore, there is no way that this shape is a square.

Now we draw a second shape. Using geometry, we identify that this shape is a rectangle. Therefore, there is a possibility that this shape is a square (but we do not care, nor do we know).

Going back to your statement, if $xy$ is always odd, since your statement does not have anything to do with $xy$ being odd, you have no conclusion. You simply do not care. You only care that if you have $xy$ is even.

Andrew Chin
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