Are "where" and "such that" interchangeable? I am learning some Discrete Math and was wondering whether, in a compound statement, I can use "where" and "such that" interchangeably without any problem?
For example, I am quite sure that

*

*$ax^2+b=7$ such that $a,b\in\mathbb C$
and

*

*$ax^2+b=7$ where $a,b\in\mathbb C$
mean the same, but is interchanging "where" and "such that" in a compound sentence always valid?
Also, is there is a symbol for "where" or "such that"? I just use ":" for such that.
 A: *

*The word ‘where’ sometimes means if (i.e., wherever/whenever/when):
$$Q(x)\text{ is true where }P(x)\text{ is true}\tag1$$ might implicitly mean $$\forall x\:\Big(\textbf{if }P(x)\text{ is true, then }Q(x)\text{ is true}\Big),$$ that is, “for each $x$ such that $P(x)$ is true, $Q(x)$ is also true”.


*The word ‘where’ sometimes means and:
$$P(x)\text{ is true, where }Q(x)\text{ is true}\tag2$$ might mean $$P(x)\textbf{ and }Q(x)\text{  are both true},$$ which can be phrased as “$P(x)$ is true such that $Q(x)$ is true”.
The tiny difference between between sentences $(1)$ and $(2)$ (the comma) is so technical that in practice, context is the only way to disambiguate their meanings.


*The word ‘where’ sometimes literally means ‘in which’, connoting both and and iff.
For example: $\text“S$ is the set of reals where each, for some natural $k,$ equals $2k\text”,$ which can be phrased as $\text“S$ is the set of reals such that each, for some natural $k,$ equals $2k\text”.$
Since the word ‘where’, when used to introduce a clause in mathematical/technical writing, is potentially ambiguous, I'd use it sparingly and carefully; Paul Halmos agrees.
In this example, the author has, in one instance of the word ‘where’, unintentionally and confusingly invoked both the first two meanings!

is interchanging "where" and "such that" in a compound sentence always valid?

‘Where’ cannot be replaced with ‘such that’ in sentence $(1).$
