# Let $x \in \mathbb{Z}$. Prove that $5x-11$ is even IFF $x$ is odd.

Let $$x \in \mathbb{Z}$$. Prove that $$5x-11$$ is even IFF $$x$$ is odd.

I know that for an IFF proof we prove it directly, and then prove it again by the contrapositive. I don't know how to prove this directly, so in a normal direct proof I would use the contrapositive since that is logically equivalent to the direct proof.

The contrapositive would be:

Assume $$x$$ is even. By definition $$x=2k$$ for $$k \in\mathbb{Z}$$. Therefore we can see $$5x-11=5(2k)-11$$. This becomes $$5x-11= 2(5k-6)+1$$. Since $$5k-6 \in\mathbb{Z}$$, then $$5x-11$$ is odd. Which is true for the contrapositive.

I am confused on how to prove this as an IFF proof.

• to prove $5x-1$ is even $\iff x$ is odd, prove (A) $5x-1$ is even $\implies x$ is odd, and (B) $x$ is odd $\implies 5x-1$ is even Oct 6, 2020 at 2:13
• How do I prove part A. If $5x-1$ is even then $5x-1=2k$ for $k\in\mathbb{Z}$ and then how do you prove $x$ is odd from that? Oct 6, 2020 at 2:21
• for that part, you could prove the contrapositive: if $x$ is not odd, then $x$ is even, so $5x$ is even, so $5x-1$ is odd, so $5x-1$ is not even Oct 6, 2020 at 2:26
• Okay, that makes sense. Thank you! Oct 6, 2020 at 2:27

So here by proving the contrapositive, you proved the if direction. i.e. $$\forall x \in \mathbb{Z}~5x - 11\text{ even} \implies x \text{ odd}.$$ You need to prove the other direction $$\forall x \in \mathbb{Z}~5x - 11 \text{ even}\impliedby x \text{ odd}.$$
And the other direction is easy. $$x$$ odd implies that $$x=2k+1$$ for some $$k\in \mathbb{Z}$$. So $$5x - 11 = 10k - 6 = 2(5k -3)$$, which is even.
• you typed $x$ instead of $k$ in a couple of places in the last line Oct 6, 2020 at 2:15
You could prove $$A\iff B$$ by proving $$A\implies B$$ and the inverse $$\lnot A \implies \lnot B$$. (Since the inverse is the contrapositive of the converse, the inverse $$\lnot A \implies \lnot B$$ and converse $$B \implies A$$ are logically equivalent to each other.)
So, to prove what you want, it suffices to show that, if $$x$$ is odd, then $$5x-11$$ is even, and, if $$x$$ is not odd (i.e., $$x$$ is even), then $$5x-11$$ is not even (i.e., $$5x-1$$ is odd). Those proofs should be fairly straightforward.