Thread: Order Relation
View Single Post
laminatedevildoll
laminatedevildoll is offline
#11
May5-05, 01:35 PM
P: 233
Quote Quote by honestrosewater
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?
Yes, it makes sense. I got confused with the "precedes" symbol, when in reality it somewhat refers closely to the "less than or equal to" sign. Thank you.