What is this notation for a function? I've never seen it written like this before.

Question

What does this mean? $$ f=\{ (x,y): y= x+2 \}$$

I don't understand what "$(x,y):$" means in regard to the problem. Also why is the $y$ inside of the $f(x)$ function. Isn't it supposed to be outside?

Answer 1

If you would have the function (in your usual sense) $f(x) = x+2$, the set $f$ you described above would be precisely the graph of the function. In other words, instead of giving you the value of $f(x)$ for each $x$ (which is indicated by the equality $f(x) = x+2$), you are being given the set of points in the plane which are on its graph, that is, the couples $(x,y)$ such that $y = f(x)$ (i.e. $y=x+2$).

Hope that helps,

Answer 2

Everything is a set. Particularly, a function itself is a set. When we write $$f: A \to B$$ we have that $f \subseteq A \times B$, where $\times$ denotes the cartesian product of $A$ and $B$. But, to say that $f$ is a function, it must satisfy the following condition: $$\forall \ x \in A ~((x,y),(x,z)\in f \implies y = z)$$ This is just saying that the function is well-defined. Since the element in the second entry of the ordered pair is uniquely defined for $x \in A$, we call that element $f(x)$. In details, that set $f$ is the graph of the function itself.