"With" Meaning in Proof, Compared to "Such That"

Question

What does 'with' mean in: For all real numbers x, there is some real number y with y = $x^2$

My thought process so far is that it is similar to: $\forall$ x $\epsilon$ ${R}$, $\exists$ y $\epsilon$ ${R}$ s.t. y = $x^2$

But I thought 'with' and 'such that' would have different meanings because the other questions on my assignment uses the word 'such that' instead of 'with'.

Answer 1

In your given statement, I too would mentally replace the word with with ‘such that’.

There seems to be a tendency in middle- and high-school situations to write with or where in mathematical statements even when doing so introduces ambiguity. This recent example goes

If $p$ is prime, then $\binom p k=np$ where $n\in\mathbb Z$ and $0<k<p$

(typically, with might have been written here in lieu of where). Note that in this case, with/where is not intended to mean ‘such that’.

Apart from the ambiguity of the order of quantifiers, it is not even clear that the two qualifications after the word where are actually intended to refer alternately to existential quantification and universal quantification, respectively! The intended sentence (styled better) is

• For each prime number $p$ and each $k\in\{1,2,\ldots,p-1\},$

  there exists some $n\in\mathbb Z$ such that $\binom p k=np.$

or

• For each $p,k\in\mathbb Z,$

  $\quad$ if $p$ is prime and $0<k<p,$ then there exists some $n\in\mathbb Z$ such that $\binom p k=np.$

Related:
What does the word 'with' mean in this theorem?
Is ‘where’ interchangeable with ‘such that’?

Answer 2

There is nothing wrong with "with" when it is used in place of "such that". See for example this answer. In fact, sometimes it is convenient to use both in sentences like:

For every $k,m∈ℕ^+$ with $\gcd(k,m) = 1$, there are some $a,b∈ℕ$ such that $a·k−b·m = 1$.