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The colon mark sometimes stands for "such that".

When writing functions, it is customary to write, for example:

$f:[0,1]\rightarrow[0,1]$.

Can the colon in this context be interpreted as "such that"?

If not, is it just a historical quirk that ":" is used in this context, but sometimes also means "such that"?

Asaf Karagila
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    I don't think "such that" makes much sense in this context. I would read it as "$f$ is a mapping from $[0,1]$ into $[0,1]$". –  Mar 16 '16 at 06:28
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    There are only so many symbols. It's only natural many of them get reused. Just like how $\pi$ is both a constant and a very common symbol for a group homomorphism, or a topological projection, or just a function. – Asaf Karagila Mar 16 '16 at 06:37
  • No. ... $f:A\to B$ is "$f$ is a function from (the domain) $A$ into (the range) $B$." Or we say "to $B$" instead of "into $B$", with no difference in the meaning – DanielWainfleet Mar 16 '16 at 13:11

2 Answers2

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In type theory the same notation is used, where f:x->y means that f is of the function type x->y, which in set-theoretical terms means that f is an element of the function set x^y. Type theorists would for instance also write z:x×y to say that z of of the product type, i.e. that z is an element of the set x×y in set theory.

Though I'm not sure if the type theorists copied the notation from the mathematicians or the other way around.

Dan Saattrup Nielsen
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It's just a separator between the name of the function and its domain/range. In other contexts, like $\exists x : P(x)$ it can be read exactly that way (there exists an $x$ such that $P(x)$). A function definition like $f:\mathbb{N} \rightarrow \mathbb{N}$ can be read aloud as $f$ is a function mapping $\mathbb{N}$ to itself. There is a contrived way: $f$ is a function such that its domain is $\mathbb{N}$ and its range is too. But that takes more words.

Dan Brumleve
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