Is it right to say that: if $2a+1=2b$ we have a contradiction?

Question

I am trying to prove by contradiction and I have reached the conclusion that $2a+1=2b$. Now I am tempted to say it's a contradiction and call it a night. Is it a contradiction? because one is even and the other is odd?

Thanks!

Answer 1

Yes, it's correct if $a$ and $b$ are integers: You can rearrange to write

$$1 = 2(b - a)$$

Hence either $1 = 0$ or $2 | 1$, either of which is bad.

Answer 2

Yes. You're right. It's a contradiction exactly because of what you said: "one is even and one is odd". This gives rise to a contradiction because the sets of odd numbers and even numbers are disjoint. In other words, a number can't be both even and odd at the same time.

Answer 3

It's not a contradiction if it's not an integral domain...