How many different relations can exist between two sets?

"Now if I define a relation between X and Y, I’m simply specifying that certain of these ordered pairs are in the relation, and certain ones are not. For example, I could define a relation R that contains only {(Harry, Mt. Dew), (Ron, Mt. Dew)}. I could define another relation S that contains {(Hermione, Mt. Dew), (Hermione, Dr. Pepper), (Harry, Dr. Pepper)}. I could define another relation T that has none of the ordered pairs; in other words, T = ∅. A question that should occur to you is: how many different relations are there between two sets X and Y? Every one of the ordered pairs in X × Y either is, or is not, in a particular relation between X and Y. Since there are a total of |X| · |Y| ordered pairs, and each one of them can be either present or absent from each relation, there must be a total of 2^(|X|·|Y|) different relations between them. Put another way, the set of all relations between X and Y is the power set of X × Y. In the example above, then, there are a whopping 2^6, or 64, different relations between those two little sets. One of those relations is the empty set. Another one has all six ordered pairs in it. The rest fall somewhere in the middle. Food for thought: how many of these relations have exactly one ordered pair? How many have exactly five?"