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 $x\mathrel {R} y\mathrel {R} z$ as an abbreviation of "$x\mathrel {R} y$ and $y\mathrel {R} z$. This makes sense especially when $R$ is a transitive relation, because in that case we also have $x\mathrel {R} z$, which is suggested by the notation "$x\mathrel {R} y\mathrel {R} z$".

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

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 $B\subseteq U$".

On the other hand, if $*$ is some binary operation on a set $S$, and $a,b,c\in S$, then $a*b*c$ means $(a*b)*c$ if $*$ associates to the left and means $a*(b*c)$ if $*$ associates to the right. If $*$ is associative, then these two are the same, so $a*b*c$ means either/both.

A relation has type $S\times S\to \{T,F\}$, and a binary operation has type $S\times S\to S$. Things can get confusing when we take $S=\{T,F\}$, because now there are two possible interpretations (depending on whether we take the relation one or the binary operation one).

One interesting exception to this is when we use $\implies$ between propositions. Let $p,q,r$ be three propositions. What does $p\implies q\implies r$ mean? Some possibilities are:

• $p\implies q$ and $q\implies r$
• $p\implies (q\implies r)$
• $(p\implies q)\implies r$

other examples to look at:

$x\in A\times B$ means $x\in (A\times B)$, not $\{T\}\times B$ or $\{F\}\times B$. The interpretation "$x\in A$ and $A\times B$" does not even parse since $A\times B$ is not a statement.

$n\cdot m\mid k$