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Jimmy King
Registered User regular

This is for homework so please, no direct answers. I just can't figure out how to start this in the first place, so clearly I'm not even really trying to prove the right thing. The problem I have is this.

Prove that if x + y is odd then x -y is also odd. The teacher gives a hint of use two cases, one where y is odd and one where y is even. I also have to assume this means integers based on what we are doing in class.

I have no idea what to do after that, though. I make my first case...

Let y be an odd integer.

Then I have tried these 2 things...

x + y = 2k + 1 -> then what? I can prove that x = 2k + 1 - y, but that doesn't get me anywhere

x + (2k +1) = 2m + 1 -> ok, now what? I can solve for x, which proves nothing and just proves that x is even (x=2(m-k)).

Do I take this further and take that even value I got for x = 2(m-k) and plug that in for 2(m-k) - (2k + 1) and show that it creates another odd integer? This is sounding correct now that I type it out.

Then I go back and do the same when y is even, of course.

Prove that if x + y is odd then x -y is also odd. The teacher gives a hint of use two cases, one where y is odd and one where y is even. I also have to assume this means integers based on what we are doing in class.

I have no idea what to do after that, though. I make my first case...

Let y be an odd integer.

Then I have tried these 2 things...

x + y = 2k + 1 -> then what? I can prove that x = 2k + 1 - y, but that doesn't get me anywhere

x + (2k +1) = 2m + 1 -> ok, now what? I can solve for x, which proves nothing and just proves that x is even (x=2(m-k)).

Do I take this further and take that even value I got for x = 2(m-k) and plug that in for 2(m-k) - (2k + 1) and show that it creates another odd integer? This is sounding correct now that I type it out.

Then I go back and do the same when y is even, of course.

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## Posts

You've got it. There's no way for 2m - 4k - 1 to be even if m and k are integers. If your teacher is a real stickler, formulate it as 2(m - 2k - 1) + 1 to fit the textbook definition of an odd integer.

Then do the even case, which is probably going to go in a similar direction.

enlightenedbumon