Simple variable understanding question

Question

A function $f \colon X \to Y$ is a rule which assigns to each element $x \in X$ a unique element $y \in Y$.

I don't understand what it is meant by each element $x$. If it was “all $x$” or “each $x$” that would make sense to me.

Answer 1

"each element $x$" is the same thing as "each $x$". The word "element" is just to clarify that the symbol $x$ represents an element of the set $X$. In math it is common to write things like "the variable $a$", "the function $f$", "the curve $C$", and so on.

Here is the rule (definition of function) rewritten:

A function $f : X \to Y$ is a rule such that: For all $x \in X$, we can assign $x$ an element $y \in Y$. We also write $y = f(x)$.

Answer 2

What aras said:

But I'm one-upping his rewritten rule to emphasize a few things:

A function $f:X→Y$ is a rule such that: For each distinct $x∈X$, there is a specific $y\in Y$ which we consistently assign to that particular $x$.

My point being that for each and every $x_{\alpha} \in X$, then $y_{\alpha} = f(x_{\alpha})$ will always exists and will be consistent. (In other words, we can't have $f(6) = 7$ sometimes and then some other times have $f(6)=5$.)

Also it is important to not that i) although each $x$ has $y$ assigned to it, it does not have to be the case that every $y$ has an $x$ assigned to it and ii) although every $x$ has a $y$ assigned to it, the $y$s do not need to be different. It will be possible for two different $x_{\alpha}, x_{\beta}$ to be assigned to the same $y$. (In other words, it is possible to have $f(6) = 5$ and $f(7) = 5$.)