I am doing IGCSE Maths, and am having a few problems with function notation. I understand the form f(x).
What does the form f: x ↦ y mean? Could you also give one or two examples?
And, if possible, state your source. Thank you.
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asked Apr 20, 2016 at 13:08
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$\begingroup$ See en.wikipedia.org/wiki/Function_(mathematics)#Notation. $\endgroup$– lhfApr 20, 2016 at 13:11
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$\begingroup$ @lhf Could you please explain this for a beginner? Thanks! $\endgroup$– Annabelle SykesApr 20, 2016 at 13:12
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1$\begingroup$ Examples: $f:x\mapsto x^2$ is the squaring function and $g:x\mapsto x+1$ is the function which adds one. $\endgroup$– arctic ternApr 20, 2016 at 13:14
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2$\begingroup$ I really prefer $f:x\in\mathbb{R}\mapsto x^2\in[0,\infty)$. It's a more complete description. Maybe for this function it's not needed, but there is LOT of cases when it's best to tell what is the domain and codomain. $\endgroup$– IntegralApr 20, 2016 at 13:26
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1$\begingroup$ Related: math.stackexchange.com/questions/1740154/… and math.stackexchange.com/questions/473247/… (The latter was posted by Deusovi in a comment under their answer.) $\endgroup$– Martin SleziakApr 20, 2016 at 13:48
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3 Answers
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It means that $f$ is a function that takes the value $x$ to the value $y$. For instance, $$f: x\mapsto x^2$$ is an alternate way of writing $f(x) = x^2$.
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$\begingroup$ Could you please state your source? Thanks :) $\endgroup$ Apr 20, 2016 at 13:21
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$\begingroup$ So, is there any scenario where you would need to use the "↦" notation instead of "=" notation? If your answer is a full explanation, it seems to me that ↦ is entirely useless. $\endgroup$– twizJul 29, 2016 at 11:15
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6$\begingroup$ @twiz: Just using = leaves open the possibility that x is a constant. Sure, we assume that it's a variable, but it's not necessarily the case. For instance, what if $f(x)$ was actually the function where $x \mapsto x³-4$? Then "$f(x) = x^2$" would be an equation to solve, not a definition of a function. That sort of thing pops up all the time - for instance, when we want to find the roots of the function, we use the equation $f(x) = 0$; that does not mean that we're redefining $f$ to be the function that always gives $0$. $\endgroup$– DeusoviJul 29, 2016 at 11:19
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$f:x \mapsto y$ means that $f$ is a function which takes in a value $x$ and gives out $y$.
But,
$f: \mathbb{N} \to \mathbb{N}$ means that $f$ is a function which takes a natural number as domain and results in a natural number as the result.
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answered Apr 20, 2016 at 13:13
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5$\begingroup$ Because you're wrong: the $\to$ and $\mapsto$ arrows mean different things. Also, $\mathbb{W}$ is not the set of positive numbers: that's $\mathbb{R}^+$. Whole numbers are not nonnegative numbers, either; they are natural numbers including 0. Oh, and $\to$ talks about the sets of the domain and range, while $\mapsto$ talks about the elements: you conflated them. $\endgroup$– DeusoviApr 20, 2016 at 13:18
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3$\begingroup$ Whole numbers are the nonnegative integers. And you conflated two different arrows: $\to$ and $\mapsto$. They have different definitions. $f(x) = x^2$ can be described as $x\mapsto x^2: \mathbb R \to \mathbb R^+$. $\endgroup$– DeusoviApr 20, 2016 at 13:27
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2$\begingroup$ @user331377: It's up to you. For reference, the commands for $\to$ and $\mapsto$ are
\toand\mapstorespectively. $\endgroup$– DeusoviApr 20, 2016 at 13:30 -
3$\begingroup$ Are you only on this site for reputation points? Expect downvotes if you aren’t going to post quality answers. $\endgroup$– Prince MMar 28, 2018 at 6:31
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As it is evident from math.stackexchange notation — the symbol $\mapsto$ reads as "maps to".
This is backed up by Wikipedia article on functions:
... the notation $\mapsto$ ("maps to", an arrow with a bar at its tail) ...
There is another arrow-symbol, which also used for mapping $\rightarrow$, which might be a bit confusing. The difference between two (as it is mentioned in the linked answer, as well as in the answer by MathEnthusiast):
- $\mapsto$ maps an element of one set to an element of another set;
- $\rightarrow$ maps a set to a set.
Example (borrowed from here):
$$f:R \rightarrow R$$ $$x \mapsto x^2$$
It means that: under $f$, any element $x \subset R$ gets mapped to the element $x∗x=x^2$ (which is also an element of $R$).