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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