How do I read this definition of injective in English?

Question

This is a different but related question to one I asked earlier. I link to it here:

"To show that f is injective" - I don't get this statement

I am pretty new to "functions" having just went through a quick primer on "propositional logic". So the $\rightarrow$ symbol which represents a conditional statement looks very similar in the definition of injective below.

Suppose that $f: A \to B$ (Is this to be read in English as "If $A$ is true then $B$ is true"?)

To show that $f$ is injective - Show that if $f(x) = f(y)$ for arbitrary $x, y \in A$ with $x \neq y$, then $x = y$.

How do I read this in English, specifically the part where there is a comma. I am not sure if this is stating that the ordered pair $x,y$ is an element of set $A$ or just the $y$ element itself. If the author of the text (Rosen) is talking about $x, y$ as an ordered pair then it would help to use parentheses.

Answer 1

The first part should be read "Suppose that $f$ is a function from the set $A$ into the set $B.$"

The second part doesn't make sense, as written. It should say "Show that if $f(x)=f(y)$ for arbitrary $x,y\in A,$ then $x=y.$" That is, if $f(x)=f(y)$ implies that $x=y$ for any elements $x$ and $y$ of $A$ (not necessarily different elements, just with different names), then $f$ is injective. The idea, here, is that a given function output will only be the result of a single function input. Put another way, if $f$ is injective, then we will never have a situation where $f(x)=f(y)$ and $x\ne y.$

Answer 2

It reads as: "Show that if $f$ of $x$ equals $f$ of $y$ for arbitrary $x$ and $y$ in $A$ with $x$ different than $y$, then $x$ equals $y$". The comma as nothing to do with pairs, it is just a comma from common language.

Moreover $f \colon A \to B$ should be read as: $f$ is a function from the set $A$ into the set $B$.