Thread: Order Relation View Single Post
PF Gold
P: 2,330
 Quote by laminatedevildoll Do you mean this relationship? P={(x,y): x P y if y L x} or do I have to jump into the reflexive property?
I mean that P and L are inverses. Remember that binary relations are sets of pairs. You're told that if (x, y) is in L, then (y, x) is in P, and if (y, x) is in P, then (x, y) is in L. In other words, L and P contain the same pairs- just reversed. If (x, x) is in L, then (x, x) is in P. If (x, y) and (y, z) are in L, then (y, x) and (z, y) are in P.
In order to prove that P is an order relation, you need to know that L is an order relation. Otherwise, you could only prove that if L is an order relation, then P is an order relation. But assume you already know L is an order relation. Then the following holds:
For all x, y, and z in R
1) (x, x) is in L.
2) If (x, y) and (y, x) are in L, then x = y.
3) If (x, y) and (y, z) are in L, then (x, z) is in L.
L and P have the same pairs- just reversed, so:
For all x, y, and z in R
4) (x, x) is in P.
Can you see the rest now? To see the rest, highlight:
5) If (y, x) and (x, y) are in P, then y = x.
6) If (y, x) and (z, y) are in P, then (z, x) is in P.
To make it even clearer, (6) can be rewritten as
7) If (z, y) and (y, x) are in P, then (z, x) is in P.
So (4), (5), and (7) show that P is an order relation. Notice that P is usually called "greater than or equal to" :)

Does that make sense?