I was going through some notes on Binary Relations which were handed out in class a couple of days ago and I couldn't help but notice this really interesting coincidence : (Here is the first page of the hand-out where you'll find something really interesting!)
Definition. A binary relation on a set S is a subset of the Cartesian product S × S.
This definition is so abstract that you may find it difficult to see how this is connected to the ordinary idea of things being "related". Here's the idea : A relationship between two objects is something like "x is the father of y", or "x is greater than y", or "x and y have the same color", or "x2 + y2 = 4. Look at "x is the father of y". Your experience in the real world tells you what this means — how you would verify that a given person is the father of another person.
But another way to define the "father"relationship would be to make a list of all father-child pairs.
For example, if Bonzo has a son named Wickersham and a daughter named Gordinier, then the pairs (Bonzo, Wickersham) and (Bonzo, Gordinier) would be on the "father" list. You can see that these are ordered pairs elements of the Cartesian product people × people.