Binary Relations - Definition

Question

I am familiar with the definition of a binary relation from set $A$ to set $B$ as a subset of their Cartesian product $A × B$. I do not understand, however, how one can view certain mathematical symbols, viz. $=$, $<$, $>$, as being binary relations. In what way are they sets (and particularly, subsets of $A × B$)?

The only reasoning I have produced is that such symbols impose conditions whose fulfillment produces a $n-$tuple. (A bit wordy...) For illustration, suppose we have $x^2 + y^2 = 1$. If $x$, and $y$ fulfill the condition imposed by $=$, then that $(x,y) \in R_=$. I find this a bit bothersome, since it implies that $=$, $<$, etc. are means by which to produce subsets of $A × B$ , and not subsets of $A × B$ themselves. Any suggestions?

Answer 1

You would have to specify a set (or two sets) on which the relation is applied. For instance $<$ is a relation on $\mathbb{R}$ via $\{(x,y)\in \mathbb{R}\times\mathbb{R} : x<y\}$ being the set of ordered pairs making up the relation.

Answer 2

You can think of a binary relation a tuple, if you like, each element of the tuple may not come from the same set. So, for example, $+:(a,b)\to a+b$ can be represented by a tuple $(a,b,c)$ where $c=a+b$ and all three sets are integers.

But, you might also define $<:(a,b)\to\begin{cases} \text{true} & \text{ if } a<b\\\text{false}&\text { if } a\ge b\end{cases}$ and the third element of your tuples would be drawn from the set $\{\text{true},\text{false}\}$.

The point is, it's a binary relation because it is a relation between the first two elements and does not care about what the kind of "thing" elements of the third set are.