I don't understand what the colon $T:t\to−t$ means. I understand that it does something, is it a variable or just a symbol that has no significance?
2 Answers
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read the colon as "maps" so that $T:t \to -t$ should be read "$T$ maps $t \to -t$". So the way you parse it is the thing on the left of the colon is a function which takes the thing on the right of the colon but to the left of the arrow to the thing on the right of the arrow. So another example could be $\times : (a,b) \to ab$.
It's just another way of defining a function. It is equivalent to saying $T(t) = -t$.
This is a detail, but the way I was taught it originally, you would actually have a different arrow, so that it would be written like this $T: t \mapsto -t$. Then the regular arrow is used to specify the domain and target space, so you would write $T:\mathbb{R} \to \mathbb{R}$, $T:t \mapsto -t$. You can read more on wikipedia.
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$\begingroup$ As this question originated on Physics, I'll note that physicists do sometime make proper use of
mapsto, but usually neglect to specify the domain and range as they can be taken as understood for "the time reversal operator". $\endgroup$ May 22, 2014 at 18:10
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$T$ is a set, which contains every $(t;-t)$ ordered pair for every real number. The background logic is as it could be read as the following:
$$T \in 2^{\mathbb{R}\times\mathbb{R}}$$
(In english sentence: $T$ is a set of ordered real pairs.)
$$(x;y)\in{T}\iff{x=-y}$$
(And ordered real pair is in $T$, if and only if its second member is the negate of its first.)
