# Math Symbol for “Where”

Pretty much any math text I've read introduces notation through a similar format of equation-notation or notation-equation form e.g

ax+b+c
where a = ..., b = ..., etc.

or

Let a = ..., b = ..., etc.
ax+b+c

I tend to see the former more and was wondering if there has been an official symbol for "where" to introduce notation? If not, why not?

• Normally people just say s.t. To abbreviate "such that", i.e. $ax + b s.t. a=2, b=3$. – nosyarg May 29 '15 at 17:22
• @nosyarg I disagree; "such that" is a restriction on values, not a definition of symbols – Joffan May 29 '15 at 17:23
• Why introduce pointless symbolism? Where is short and understandable. – Brian M. Scott May 29 '15 at 17:23
• : is often used also | – Karl May 29 '15 at 17:24
• @Karl: I suspect that Joffan would also object to those as synonyms, since they, like "such that", are restrictions on values. (I tend to agree with Joffan; I think that assignment is semantically different from restriction.) – Brian Tung May 29 '15 at 17:26

In his famous paper How to write mathematics, P.R. Halmos says the following about "where"

"Where" is usually a sign of a lazy afterthought that should have been thought through before. "If $n$ is sufficiently large, then $|a_n| < \varepsilon$, where $\varepsilon$ is a preassigned positive number";

That being said it is common to write

$x = 2^n$, where $n$ is some positive integer,

although I would prefer

$x = 2^n$ for some positive integer $n$.

Now to answer your question, there is no specific symbol for "where" that I know of.

• With respect to Halmos's pronouncement, I see an interesting parallel between variable declarations and the use of "where" in mathematics. In languages such as Java, it is possible to declare them at the beginning of a method, or alternatively to declare them in a small block, for use only in that block. One might lift Halmos's restriction, then, in cases where the body of the "where" clause is used only for the nonce, and the actual denotation is unimportant. – Brian Tung May 29 '15 at 17:40
• @BrianTung: And then there is Haskell, which allows both f(n) = g(m,m) where m=n+1 and f(n) = let m=n+1 in g(m,m) – Henning Makholm May 29 '15 at 17:56

There's not really anything that makes a symbol "official" or not, and there are probably authors out there somewhere who do use a symbol of their own devising for this.

However, there is certainly no widely used and understood symbolic way to write what you want.

Mathematicians in general seem to think that using prose (such as the word "where") for this is preferable, in the sense that the extra space taken up by it is a lesser cost than it would be for everyone to have to remember a specific non-verbal symbol.

The use of natural language is often more effective when presenting an idea. In my opinion the less often a symbol is used where a few words can go, the better.

That said, there are many places where symbols are useful and simplify matters.

The word "where" can often be replaced with "such that", and corresponding to this we have a few regularly used symbols.

For instance, in set builder notation the colon (or bar) is used to indicate a condition (read "where" or "such that"):

$$\{ x \in \mathbb{R} : x > 0 \}.$$

Also $|$ and $\ni$ are used for this same purpose.

• However, the middle signs in set builder (or quantifier) notation are not really catch-all replacements for the words "such that". They are integral parts of larger notations which don't have any individual formal meaning -- the fact that expressing the entire notation in English sometimes involves saying "such that" between the things the symbol stands between in the symbolic form doesn't make those words into a definition of the symbol. – Henning Makholm May 29 '15 at 17:29
• I have seen the colon used elsewhere to mean such that. I think it is more common to see $\ni$, but I think it is even more common to see "where" or "such that". I am guilty of the replacement "st." in my lectures. @HenningMakholm – Joel May 29 '15 at 17:44
• anyhow, it is nowhere.. – Narasimham May 29 '15 at 17:45
• Or do you mean $\not\ni$? @Narasimham ? – Joel May 29 '15 at 17:46
• @Joel , like that.. – Narasimham May 29 '15 at 17:50

The general answer is "no", and the reason is more or less for human readability. Proofs are by and large written in paragraph form. Sure there will be the occasional conditional or equality chain, but such things are still often surrounded by introductory and followup text. It turns out that it's often easier to transmit a series of thoughts when they are presented more "naturally" (i.e., how humans are conditioned to receive and transmit information), and the less things that get in the way of the flow, the easier it is to understand.

That's not to say that various mechanisms haven't been introduced, but none of such things have attained enough widespread use so as to be considered standardized.