# A problem related to relation definition.

My book says:

A relation R from a non-empty set A to a non-empty set B is a subset of the cartesian product A × B. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in A × B.

The book says that relation R is a subset of A × B. But how could a relation be a set? In number 5 and 10 there is a relation that the later one is divisible by the fist one. But "division" can't be any "set" or can it be?

• Welcome to Mathematics Stack Exchange. The relation can be identified with the set of ordered pairs $(a,b)$ where $a$ divides $b$; for example, the set contains $(5,10)$ but not $(10,5)$ – J. W. Tanner Nov 1 '19 at 13:57
• It is worth pointing out that functions are also sets. In fact, depending on your construction of set theory, everything can be a set including and certainly not limited to: operations such as $+$ and $\times$, numbers such as $1$ and $\pi$, graphs such as $K_4$, and relations such as $<$ and $\mid$ as in your question, etc... – JMoravitz Nov 1 '19 at 16:17

Abstractly, a relation $$R$$ can be described as a set of ordered pairs that satisfy the relation:
$$R=\{(a,b): a\; R\; b\}\subset A\times B$$.
For your example -- the relation "divides" -- we have $$(5,10)\in R$$ but $$(10,5)\not\in R$$, for instance.
(In this case we could say $$A=B=\mathbb N$$; $$A\times B=\mathbb N^2$$.)