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This is a really simple question that I should know how to answer at this point, but I have always been sort of confused about this notation: $$f:A\mapsto B$$ Without specifying anything about $f$, what does the above statement mean? Like I'm sure that it means $$\forall x\in A, f(x)\in B$$ I guess I should give an example. If $f(x)=x^2$ for all $x\in I\subset\Bbb R$, then obviously $f(I)\subset\Bbb R$, so would it be fair to say $$f:\Bbb R\mapsto\Bbb R$$ even though $f$ is not defined for all $\Bbb R$? I think you probably see what I'm trying to ask.

clathratus
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  • What about $f:[0,\infty) \to [0,\infty), x\to \sqrt x?$ – mfl Oct 23 '18 at 16:59
  • The value of $f$ should be defined for every $a \in A$, and $f(a)$ should be an element of $B$ for every $b \in B$. So no, it would not make sense to write $f : \mathbb{R} \to \mathbb{R}$ in that case. –  Oct 23 '18 at 16:59
  • @mfl what do you mean? – clathratus Oct 23 '18 at 16:59
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    Can you say that the domain of $f(x)=\sqrt x$ is $\mathbb{R}?$ – mfl Oct 23 '18 at 17:00
  • @T.Bongers what would be that right notation in this case – clathratus Oct 23 '18 at 17:00
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    $f : I \to \mathbb{R}$ would be fine, or if you know that $J$ is a set containing all possible outputs of $f$ then $f : I \to \mathbb{J}$ would work. –  Oct 23 '18 at 17:01
  • @T.Bongers so $g:X\mapsto\Bbb R$ would be fair iff... (what?) – clathratus Oct 23 '18 at 17:03
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    If and only if $g(x) \in \mathbb{R}$ for all $x \in X$. –  Oct 23 '18 at 17:04
  • @T.Bongers okay. I think I get it. Could you type up an answer so we can both gain reputation? – clathratus Oct 23 '18 at 17:05
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    I really don't care about gaining more reputation. If you think you understand, then write your own answer expressing that and accept it. –  Oct 23 '18 at 17:07
  • @T.Bongers sure haha. :) – clathratus Oct 23 '18 at 17:10
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    If $f$ is a function sending real to reals then it is custom to use the notation $f:\mathbb R\to\mathbb R$ but not $f:\mathbb R\mapsto\mathbb R$ (I never encountered that). If more specifically $f$ is the function sending each $x\in\mathbb R$ to $x^2\in\mathbb R$ then it is custom to say that $f:\mathbb R\to\mathbb R$ is prescribed (or defined) by $x\mapsto x^2$. – drhab Oct 23 '18 at 17:16
  • @drhab so there is a difference in the meanings of $\mapsto$ and $\to$ – clathratus Oct 23 '18 at 17:17
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    Yes. Of course I am not an authority on this, but this is the way I use $\to$ and $\mapsto$ and I never engaged the notation $f:A\mapsto B$. – drhab Oct 23 '18 at 17:20
  • @drhab were do you suggest I go to learn the differences between these notations? – clathratus Oct 23 '18 at 17:24
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    Just stay here on this site :-). I think that practice and observing how others use notation is the best way to learn. It is mainly a matter of getting accustomed to it. Also you can start with developing a protocol for yourself (mine on this matter is: let $f:\mathbb Q\to\mathbb R$ be a function prescribed by $x\mapsto x^3$) and note when you see others doing it differently. Eventually you can adapt your protocol. – drhab Oct 23 '18 at 17:30

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