Binary relations and notation

Question

Is it mathematically possible to "confound" a binary relation with its notation? Or it became ambiguous? For example, let $A=\{1,\,2, \,3\}$ and $R⊂A^{2}=\{(x, \,y):x $ is less than $y$$\}$. Thus, $R=\{(1, 2), (1, 3), (2, 3)\}$ and $1R2, 1R3 $ and $2R3$. If we define the symbol $<$ to denote the relation "less than", it is right to claim that $<$ $=$ $\{(x, \,y):x $ is less than $y$$\}$, where $<$ $=$ $R$? Another example, defining $R_{1}⊂B^{2}$ $:=$ $\{(m,\,n)$: $m$ divides $n$$\}$ and $B=\{2, 4, 6\}$, we have $R_1=\{(2,4), (2, 6)\}$ and, defining $m$ divides $n$ by $m\,|\,n$, then $m\,|\,n=$ $\{(m,\,n)$: $m$ divides $n$$\}$. Can I do this? Thanks in advance!

Answer 1

It's a bit cumbersome using a symbol instead of a letter to name a set, but of course there is nothing wrong in it. Imagine what would happen using "$=$" as the symbol for the relation: some ambiguities arise.

Consider the following if you need it, anyway. You can use the fact that restriction of a relation to a subset of its domain can be represented as $R_{|S}$ meaning: $$R_{|S}=\{(x, y): x \text{ is less than } y \text{ and } x\in S\}$$ where $S\subset \operatorname{Dom}(R)=A$. So being $R=R_{|A}$ and then you can have no ambiguity writing: $$<_{|A}=\{(x, y): x \text{ is less than } y\}$$

Alternatively if you want to distinghuish between the relation and its graph you can use the notation $(A, B, R)$, that is, the relation is represented as the ordered triple of the set of departure, the codomain, and the graph.

In this case you would not have cluttered notation for your definition: $$(A, A, <)=(A, A, \{(x, y): x \text{ is less than } y\})$$