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I have come across this symbol many times, but I am unsure as to how to correctly use it.

So I can read up on it, what is the name of this mapping function?

When would it be correct to use and when wouldn't you use it?

I think it may be used when you haven't specified a function for the mapping, but just a guess.

Example: Any wellordered set $\langle X,\prec\rangle$ is order isomorphic to the set of its segments ordered by $\subset$

Proof: Let $Y=\{X_a\vert a\in X\}$. Then $a\rightarrowtail X_a$ is a (1-1) mapping onto $Y$, and since $a\prec b\Leftrightarrow X_a\subset X_b$ the mapping is order preserving.

Is it necessary to use $\rightarrowtail$ here, could you just use $\mapsto$ or $\rightarrow$, in this example why did we use this symbol?

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    $\begingroup$ Actually, you are supposed to use $\mapsto$ here. $\endgroup$ – Zhen Lin Jan 1 '15 at 10:58
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The symbol $\rightarrowtail$, meaning a one-to-one map, or injection, must be placed between the domain of the map and its codomain; the symbol $\mapsto$ is used between the corresponding elements. Thus, for example, $$f:\Bbb R_{\geqslant 0}\rightarrowtail\Bbb R:x\mapsto x^2.$$

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It is usually used to denote an injection.

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    $\begingroup$ Thanks for your answer, so it is never necessary and we could just use $\rightarrow$? $\endgroup$ – Sam Houston Jan 1 '15 at 10:54
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    $\begingroup$ Knowing a map is an injection is often better than only knowing it is a map. See: en.wikipedia.org/wiki/Injective_function $\endgroup$ – Huy Jan 1 '15 at 10:55

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