# Definition: Function

Definition: function from a set A to a set B is a mapping that assigns each element of A to exactly one element of B.

Notation:  To remind ourselves that functions have a sort of “one-directionality” to them, we use the following notation for a function f from the set A to the set B: $f:A\rightarrow B$.  This notation reminds us where the function goes “from” and “to” thanks to the directionality of the arrow.  We should also read this just like we would English.  Thus, if I write “let $g:C \rightarrow D$”, then we should read this as “let g be a function from the set C to the set D”. We also use the notation $f(a)=b$ to denote the action of f on individual elements, where in this case we’ve expressed that “a is sent to b under f”.

Note that if a function f is from A to B, then every element of A is sent to something in B, but it is not the case that everything in B has a number sent to it.  Take the set of numbers $A=\{1,2,3,...10\}$ and the set of numbers $B=\{1,2,3,4,...,20\}$.  Now let f be the function that sends each element of A to the corresponding element in B (1 goes to 1, 2 goes to 2, etc.).   As you can check, this satisfies everything in the definition of function, yet $\{11, 12,..., 20\}$ in B do not have anything “going to them”.

This might all seems like a bunch of abstract rubbish at this point, and if you find no beauty in this definition don’t fret.  It is a relatively dry definition, but it is extremely important and hopefully after a little time going through the lessons on this site you’ll get a sense of this importance.  Indeed, for more on functions go check out lesson 6.

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