1.57
Definition (Function.)
Let
![$A,B$](img208.gif)
be sets, and let
![$f$](img38.gif)
be a rule that assigns to each element
![$a$](img24.gif)
in
![$A$](img50.gif)
a
unique element (denoted by
![$f(a)$](img209.gif)
) in
![$B$](img68.gif)
. The ordered triple
![$(A,B,f)$](img210.gif)
is called a
function with domain
and codomain ![$B$](img68.gif)
. We write
to indicate that
![$(A,B,f)$](img210.gif)
is a function. It follows from the definition that two
functions are equal if and only if they have the same domain, the same codomain, and
the same rule: If
![$f\colon A\to B$](img212.gif)
and
![$g\colon A\to B$](img213.gif)
, I say that the rule
![$f$](img38.gif)
and
the rule
![$g$](img214.gif)
are the same if and only if
![$f(a)=g(a)$](img215.gif)
for all
![$a\in A$](img216.gif)
. We usually
say `` the function
![$f$](img38.gif)
" when we mean `` the function
![$(A,B,f)$](img210.gif)
," i.e., we
name a function by giving just the name for its rule.