User:IssaRice/Stringing together relations and binary operations

If $R$ is a relation on a set $X$, and $x,y,z$ are elements of $X$, we sometimes write $xRyRz$ as an abbreviation of "$xRy$ and $yRz$. This makes sense especially when $R$ is a transitive relation, because in that case we also have $xRz$, which is suggested by the notation "$xRyRz$".

For instance, if we have three real numbers $x,y,z$ and the relation $\le$, then $x\le y\le z$ means that $x\le y$ and $y\le z$. Since the relation is transitive, we also have $x\le z$.

Another example is given sets $A,B,C$ we can write $A\subseteq B\subseteq C$ or $A\supseteq B\supseteq C$.

In fact, the relation that is used does not have to be the same in both places. We might write $p\in B\subseteq U$ to mean "$p\in B$ and $B\subseteq U$".