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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.

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    $\begingroup$ The term range is sometimes used to refer to the codomain, i.e. $Y$ in $f : X \to Y$, and also sometimes used to refer to the image, i.e. $f(X)$. $\endgroup$ Commented Dec 30, 2015 at 0:12
  • $\begingroup$ So we have to use context to determine what is implied? Would it not be more general to use the term Codomain rather than range in definitions such as this? $\endgroup$ Commented Dec 30, 2015 at 0:44
  • $\begingroup$ What text are you reading? Is this definition clarified in the beginning of the book? $\endgroup$ Commented Dec 30, 2015 at 1:05
  • $\begingroup$ Analysis texts are often quite sloppy on this sort of thing: another similar example is that $f(x)$ is often called a function. It isn't, of course: it's the value of the function $f$ at the point $x$. I would also conjecture that until recently, most (non-Bourbaki...) analysis texts had not fallen under the influence of Hard Set Theory: the tradition of the Cour d'Analyse is rather older than both set theory and mathematical logic, so there will still be plenty of older texts that do it the old-fashioned way, so to speak. $\endgroup$ Commented Dec 30, 2015 at 1:12
  • $\begingroup$ I've never heard of "range" as being anything other than $f(S).$ My teachers of yesteryear would sometimes call $T$ the "target space", which I rather liked. "Codomain" is a relatively newfangled term (to me anyway) that is intended to address the issue I guess. $\endgroup$ Commented Dec 30, 2015 at 4:55

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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

enter image description here

This image was taken from the link: http://www.mathsisfun.com/sets/domain-range-codomain.html.

Hope this helps!

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