3.3: Relations between a set and itself
In the above example, the two sets contained different kinds of things: people, and drinks. But many relations are defined in which the left and right elements are actually drawn from the same set. Such a relation is called (don’t laugh) an endorelation .
Consider the relation “hasACrushOn" between \(X\) and \(X\) , whose intensional meaning is that if \((x,y)\in \text{hasACrushOn}\) , then in real life \(x\) is romantically attracted to \(y\) . The extension is probably only { (Ron, Hermione), (Hermione, Ron) }, although who knows what goes through teenagers’ minds.
Another example would be the relation “hasMoreCaloriesThan" between \(Y\) and \(Y\) : this relation’s extension is { (Mt. Dew, Dr. Pepper) }. (Fun fact: Dr. Pepper has only 150 calories per can, whereas Mt. Dew has 170.)
Note that just because a relation’s two sets are the same, that doesn’t necessarily imply that the two elements are the same for any of its ordered pairs. Harry clearly doesn’t have a crush on himself, nor does anyone else have a self-crush. And no soda has more calories than itself, either — that’s impossible. That being said, though, an ordered pair can have the same two elements. Consider the relation “hasSeen" between \(X\) and \(X\) . Surely all three wizards have looked in a mirror at some point in their lives, so in addition to ordered pairs like (Ron, Harry) the hasSeen relation also contains ordered pairs like (Ron, Ron) and (Hermione, Hermione).