I'm currently working through an real analysis text and came across a definition that seemed a little strange to me. When defining a function the text states:
For a function, $$f:\,S\rightarrow T$$ $S$ is called the Domain of $f$, and $T$ is called the Range of $f$.
Shouldn't $T$ be called the Codomain of $f$? Isn't $T$ the range of $f$ only if $f$ is onto?
I'd appreciate any clarification. Thanks.
Sometimes the words "codomain" and "range" get used interchangeably. In this book it would seem that this is the case.
However, sometimes the term codomain refers to the set in which all the outputs of a function fall and range ends up as a subset of the codomain into which the images that have been evaluated in a particular case fall. Check out this link: http://mathworld.wolfram.com/Codomain.html
This image was taken from the link: http://www.mathsisfun.com/sets/domain-range-codomain.html.
Hope this helps!