I am having trouble understanding what a relation actually is.
The way I understand it is, if $A$ and $B$ are sets, then a relation from $A$ to $B$, $R$, is simply any subset of $A \times B$.
This totally makes sense to me, all we are doing is mapping elements from $A$ to elements of $B$. So the relation '=' on $\mathbb{R}$ is actually just the subset of $\mathbb{R} \times \mathbb{R}$.
What confuses me here:
Does the '=' have anything to do with the relation, or are relations really just sets of tuples? Do we call it the 'rule' of the relation or something along those lines?
How do I interpret the question: "How many relations are there on a set of size n?". Is it simply the cardinality of the power set of size $n^2$?
What would I call a relation which is constructed from randomly choosing 2 numbers from $A\times B$ and relating them?
