How do I symbolically express a statement of the form "Something, given that something is true, will be something"?

Question

I am focusing especially on the "something, given that something is true" part.

An example would be "An equation in the form $ax^2$ given that $a \ne 0$ will have a derivative greater than $0$."

Instead of saying "$ax^2$ given that $a \ne 0$", can I instead say one of the below?

1. "$ax^2$ | $a \ne 0$"
2. "$ax^2$ : $a \ne 0$"
3. "$ax^2 \ni a \ne 0$"

If not, what would be the right way of saying it? If more than one are right, then what makes them different? Or, instead of using symbols, should I just stick with natural language instead?

Thank you for the answer.

Answer 1

It's best to use natural language instead of symbols here. The suggestions 1 through 3 above are not standard notation at all. In fact, "given" sounds quite awkward in this context and I would replace it with "if", for example, "If $a \ne 0$, an equation in the form $ax^2$..."

Answer 2

Instead of saying "$ax^2$ given that $a \ne 0$", can I instead say

1. $ax^2$ | $a \ne 0$ $\quad ax^2$ will have a derivative greater than $0.$
2. $ax^2$ : $a \ne 0$ $\quad ax^2$ will have a derivative greater than $0.$
3. $ax^2 \ni a \ne 0$ $\quad ax^2$ will have a derivative greater than $0.$

If more than one are right, then what makes them different?

Your three symbols all literally read as such that (observe that its logical meaning varies with context: “for each $x$ such that $P(x)$ is true, $Q(x)$ is also true” is a conditional (‘if’ statement), whereas “$P(x)$ is true such that $Q(x)$ is true” is a conjunction (‘and’ statement)).

Among the three symbols, only | and : are common (by the way, ∋ also means “contains the element”). Generally, because peppering mathematical prose with symbols decreases readability, the phrase such that should be spelt out, or abbreviated as “s.t.”, except in set notation in which case either of the first two symbols is fine. In fact, writing : in formal sentences can lead to mistranslation including when it's not clear whether the symbol is functioning as a delimiter.

Given that, on the other hand, usually means ‘if’, like in “Given that $a\ne0,\;ax^2$ has a derivative greater than $0.$”