"Just as with sets, we can define a relation extensionally or intensionally. To do it extensionally, it’s just like the examples above — we simply list the ordered pairs: { (Hermione, Mt. Dew), (Hermione, Dr. Pepper), (Harry, Dr. Pepper) }. Most of the time, however, we want a relation to mean something. In other words, it’s not just some arbitrary selection of the possible ordered pairs, but rather reflects some larger notion of how the elements of the two sets are related. For example, suppose I wanted to define a relation called “hasTasted” between the sets X and Y, above. This relation might have the five of the possible six ordered pairs in it: (Harry, Dr. Pepper) (Ron, Dr. Pepper) (Ron, Mt. Dew) (Hermione, Dr. Pepper) (Hermione, Mt. Dew) Another way of expressing the same information would be to write: Harry hasTasted Dr. Pepper Harry hasTasted Mt. Dew Ron hasTasted Dr. Pepper Ron hasTasted Mt. Dew Hermione hasTasted Dr. Pepper Hermione hasTasted Mt. Dew. Both of these are extensional definitions. But of course the meaning behind the relation “hasTasted” is that if x hasTasted y, then in real life, the person x has given a can of y a try. We’re using this relation to state that although Ron and Hermione have sampled both drinks, Harry (perhaps because of his persecuted childhood at the Dursleys) has not."