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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.

Peashooter8890
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    Your example is problematic, because $ax^2$ is not an equation. Equations have an equals sign – Thomas Andrews Feb 05 '23 at 05:12
  • Anyway, I agree with Ted's answer, don't use symbolic language unless you are specifically trying to learn the precision of the formal language, or you are working on a blackboard in a lecture. The only place the symbol "such that" should be used is on a blackboard - it is not a formal symbol. – Thomas Andrews Feb 05 '23 at 05:19
  • In any event, none of these are valid use of standard formal languages. – Thomas Andrews Feb 05 '23 at 05:22
  • Alright, I'll denote it as an expression from now – Peashooter8890 Feb 05 '23 at 06:26

2 Answers2

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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 $ax^2$..."

Ted
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  • Thanks a lot, I will do that from now on. On a related note, if there are multiple conditions, then should I use "and" or "$\land$"? For example "$b > 0$ $\land$ $b \ne \infty$" or "$b > 0$ and $b \ne \infty$"? – Peashooter8890 Feb 05 '23 at 05:12
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    You should write "and", not $\land$. In general, do not use symbols for logical connectives like "if", "and", etc. Use words. – Ted Feb 05 '23 at 05:14
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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 $0.$”

ryang
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