Logic Translating into Symbols

Question

Can someone help translate “$(x^2)−1$ is even for every odd integer $x$” into symbols.

I'm a bit lost, my train of though led me to:

$$∀x ∈ Z, x = ? ⇒ ?$$

Answer 1

There are multiple ways to do that depending on which symbols you prefer and if you need a low order logic or a more high level description:

• $\forall x\in\mathbb Z,\ x\equiv 1\pmod 2\implies x^2-1\equiv 0\pmod 2$
• $x\in2\mathbb Z+1\implies x^2-1\in2\mathbb Z$
• $\forall x\bigg( \exists k\in\mathbb Z\ :\ x=2k+1\implies \exists m\in\mathbb Z\ :\ x^2-1=2m\bigg)$
• $\forall x\in\mathbb Z,\ (-1)^x<0\implies(-1)^{x^2-1}>0$
• let $f:x\mapsto x^2-1$ then $f(2\mathbb Z+1)\subset2\mathbb Z$

And you can probably invent some more...

Answer 2

$$ \forall x:\bigg(\underbrace{(\exists m \in \mathbb Z:x = 2m+1)}_{x\text{ is odd integer}} \to \underbrace{(\exists n\in \mathbb Z:x^2-1=2n)}_{x^2-1\text{ is even integer}}\bigg) $$