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I'm reading a book, I quote from it:
"" A function assigns an element of one set, called the domain, to elements of another set, called the codomain. The notation $f: A \to B$ indicates that $f$ is a function with domain, $A$, and codomain, $B$. The familiar notation $f(a)=b$ indicates that f assigns the element $b\in B$ to $a$. Here $b$ would be called the value of $f$ at argument $a$. ""

I always thought $f(a)=b$ assings the element $a\in A$ to $b$, since $a$ is the input and $b$ is the output.

Am I wrong? If yes explain please.

kptlronyttcn
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  • The function assigns the output b to the input a. – user84413 May 16 '14 at 18:03
  • One source of confusion with the "assigning" language is that in the everyday use of the word "assign", a thing is assigned to only one thing. (For example, if a job is assigned to a particular person, or to a particular team, the assignment is exclusive — that work is their responsibility now, not anybody else's.) But in this mathematical usage, one $b$ can be "assigned" by $f$ to many different values in $A$. This might be part of why you thought "$f$ assigns $a$ to $b$" made more sense. –  May 16 '14 at 18:40

2 Answers2

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$f : A \to B$ means that $f$ maps elements in $A$ to elements in $B$. So if $f(a) = b$, $a$ is mapped to $b$ by way of $f$, sometimes also written as $a \stackrel{f}{\to} b$ or $a \mapsto f(a)$.

The function doesn't "assign" $a$ to $b$, but rather the image of $a$, under the mapping $f$, is equal to $b$. Note that $A$ and $B$ might be different sets, so "assigning" a value from one set to another doesn't always make much sense; $a$ need not be related to $b$ in any way except for the existence of the function $f$.

Emily
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Standard terminology is that a function assigns an output value to each input value. What you quoted from your book uses the opposite convention in the first sentence and the standard convention in the third sentence. The author is being sloppy.

Andreas Blass
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