Can someone please define the meaning of the two following arrows and tell me what the difference is between them? When would you use one or the other?
$\mapsto$ and $\to$
Thank you so much!
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1$f: \mathbb{R} \to \mathbb{R}$ via $f: x \mapsto x^2$. – Randall Jun 25 '19 at 13:05
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It's just convenience – Cloud JR K Jun 25 '19 at 13:09
3 Answers
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It's a notational convention to use $\to$ to denote a mapping of objects and $\mapsto$ a mapping of elements.
e.g. if $f$ is the map on the real line that sends a real number $x$ to $x + 1$ then we may denote this as $$ f : \Bbb R \to \Bbb R,\; x \mapsto x + 1. $$
Ruben
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The answer was more or less already given by @Randall in the comment section. The first arrow is called mapsto and as its name suggests it means that the object to its left is mapped to the object on its right side.
Thus $x \mapsto x^2$ means that any inserted value for $x$ will get mapped to its squared value.
The second arrow typically is used to describe the relationship between the spaces a function operates on, so $f \colon \mathcal{X} \to \mathcal{Y} $ means that any value inserted into $f$ stemming from the space $\mathcal{X}$ will be mapped by some mechanism (cf. $\mapsto$) into the space $\mathcal{Y}$.
Thomas Lang
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This is very helpful! I just have one followup question: How would I read each of these symbols outloud. – Sparkles the unicorn Jun 25 '19 at 13:26
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I suppose just like their TeX symbols: mapso for $\mapsto$ and to for $\to$. So for $f\colon \mathbb{R}\to \mathbb{R}, \quad x \mapsto x^2$ you could say "$f$ is a function that goes from $\mathbb{R}$ into $\mathbb{R}$ where each element is mapped to its squared value." Note that you can also say that $f$ maps from $\mathbb{R}$ to $\mathbb{R}$, so be careful. – Thomas Lang Jun 25 '19 at 13:33
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There is a clear distinction when it comes to functions. For instance, $$\exp: {\Bbb R} \rightarrow {\Bbb R}_{>0}: x\mapsto e^x.$$.
Wuestenfux
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