Cartesian Product and Relations

Question

If a relation is defined between the first coordinate and the second coordinate of an ordered pair(aRb), then how come R = {(1,2),(5,6)} is a relation ? How can there be more than 1 ordered pair in a relation ?

Answer 1

A relation on $A$ is defined as any subset of $A\times A$. As such, $\{(1,2), (5,6)\}$ is a relation on $\mathbb R$ (and on $\mathbb Q$, and $\mathbb N$, for that matter). In other words, a relation is a set of pairs, not just a single pair.

In your particular relation, you have $1R2$ and you have $5R6$, because writing $aRb$ is just shorthand for saying $(a,b)\in R$.