3.2: Defining relations

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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.

We can of course define other relations on the same two sets. Let’s define a relation “likes" to contain { (Harry, Dr. Pepper), (Ron, Dr. Pepper), (Hermione, Dr. Pepper), (Hermione, Mt. Dew) }. This states that while everybody likes Dr. Pepper, Hermione herself has broad tastes and also likes Mt. Dew.

Another relation, “hasFaveDrink," might indicate which drink is each person’s favorite. Maybe the extension is { (Harry, Dr. Pepper), (Ron, Dr. Pepper) }. There’s no ordered pair with Hermione in it, perhaps because she actually prefers iced tea.

Yet another relation, “ownsStockIn," represents which people own stock in which beverage companies. In this case, ownsStockIn \(=\varnothing\) since all of the members of \(X\) are too busy studying potions to be stock owners in anything.

Bottom line is: when we talk about a relation, we’re simply designating certain elements of one set to “go with" or “be associated with" certain elements of another set. Normally this corresponds to something interesting in the real world — like which people have tasted which drinks, or which people own stock in which companies. Even if it doesn’t, though, it still “counts" as a relation, and we can simply list the ordered pairs it contains, one for each association.