ONT Re: Set Theory
¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤
Note 15
¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤
| Elementary Set Theory
|
| Ordered Pairs: Relations (cont.)
|
| 56. Definition.
|
| r is a 'relation' if and only if
| for each member z of r there is
| x and y such that z = (x, y).
|
| A 'relation' is a class whose members are ordered pairs.
|
| 57. Definition.
|
| r o s =
|
| {u : for some x, some y, some z, u = (x, z), (x, y) in s, (y, z) in r}.
|
| The class r o s is the 'composition' of r and s.
|
| To avoid excessive notation we agree that {(x, z) : ···} is to
| be identical with {u : for some x, some z, u = (x, z) and ···}.
| Thus r o s = {(x, z) : for some y, (x, y) in s and (y, z) in r}.
|
| 58. Theorem.
|
| (r o s) o t = r o (s o t).
|
| 59. Theorem.
|
| r o (s |_| t) = (r o s) |_| (r o t),
|
| r o (s |^| t) c (r o s) |^| (r o t).
|
| 60. Definition.
|
| r^-1 = {(x, y) : (y, x) in r}.
|
| If r is a relation, r^-1 is the 'relation inverse to' r.
|
| 61. Theorem.
|
| (r^-1)^-1 = r.
|
| 62. Theorem.
|
| (r o s)^-1 = (s^-1) o (r^-1).
|
| JLK, Gen Top, page 260.
|
| John L. Kelley, 'General Topology',
| Van Nostrand Reinhold, New York, NY, 1955.
¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤