Set theory symbol

Question

I'm studying very basic set theory for a module and have come across this symbol: | quite a few times, although I have no idea what it means, can someone explain what it is and how it makes sense in the relation below?

The symbol is the: $\mid$ symbol.

$${R=\left\{(x,y)\in A\times A\mid x-y=1\right\}}\subseteq A\times A.$$

Answer 1

Essentially, it means something along the lines of "such that," "for which," "satisfying," or "with the property that" (these phrases can usually be used interchangeably). Sometimes you'll see a colon : used instead of |.

So, the relation you wrote could be read: "$R$ is the set of tuples $(x, y)$ in $A \times A$ with the property that $x - y = 1$".

Answer 2

The symbol $\mid$, when used with set-builder notation, as in the example you posted, can be "read" as "such that" [or "for which the following holds:"], so that $R$ is the set of all ordered pairs $(x, y) \in A\times A$, such that $x-y = 1$, and this set $R$ is a subset of $A\times A$.

Another symbol that is sometimes used for the same purpose is the colon :

---

In other contexts, $\mid$ means other things, for example: $a\mid b$ means "$a$ divides $b$".

Answer 3

Reading it in english it could be translated as "such that", so your set is:

All the ordered pairs $\;(x,y)\in A\times A\;$ such that the first coordinate minus the second one equals $\;1\;$ ...