A relation is a set, and a function is a specific kind of relation. Therefore a function is a set also.
However, at the levels of math I've studied (undergraduate only), I get the feeling people don't talk about functions as sets in the purest sense.
With the aim of better understanding whether there's some sort of informal line between functions and sets, here's one question I have:
Let $f(x)=\sqrt{x}$. (With domain all real numbers greater than or equal to 0)
Let $g(x)=\sqrt{|x|}$. (With domain all real numbers)
Is it correct to say $f \subseteq g$ ? How about $f(x) \subseteq g(x)$?
More broadly, am I right to notice a line between functions and sets? Can we always use the language of set theory to talk about functions?
Best Answer
When you think about a function as a set, it is a set of ordered pairs. The first element of the pair belongs to the domain, the second to the range. When you refer to the function as a whole, it should be $f$ or $g$, not $f(x)$ which should be the value of the function at $x$. Seen as a set of ordered pairs, every pair that belongs to $f$ also belongs to $g$, so you can say $f \subset g$