Translating from english to a symbolic sentence?

Question

How would I translate the following from english to a symbolic sentence with quantifiers.

The universe of discussion is all real numbers.

Every integer is greater than some integer.

I did the following

$\forall x,\exists y$, (if x is an integer and y is an integers then $x>y$)

Answer 1

Since the domain of discourse is the reals, your currently suggested translation would be satisfied by taking $y$ a non-integer. And probably it is intended that you use logical symbols as much as possible.

There are many equivalent ways to do the job. Use $\text{Int}(t)$ as an abbreviation for "$t$ is an integer." Then we can use $$\forall x(\text{Int}(x)\implies \exists y(\text{Int}(y)\land x\gt y)).$$

Answer 2

You can use something like this:

$\forall x \exists y (x \in \mathbb{Z} \implies (y \in \mathbb{Z}) \land (y < x) ) $