Definition 1.1.3: The Equivalence Relation above gives us two Equivalence Classes
For example, pick any s in S, say 3:
3R1, 3R3, and 3R5
therefore, the equivalence class of 3 under R, denoted by (3)R or (3), is the set
{1, 3, 5}.
If we determine equivalence classes for each of the six elements in S, we will find 2 different equivalence classes:
(1) R = (3)R= (5)R = {1, 3, 5} = S1
(2)R = (4)R= (6)R = {2, 4, 6} = S2
Notice, as Rosen points out, that picking an s and an s' from S leads to either
(s)R = (s')R, or
(s)R Ç (s')R = Æ
Thus S can be written in terms of the sum of these subsets
S = S1 + S2
Definition 1.1.4
Breaking S down this way makes each element s belong to only one of the subsets. Thus the equivalence relation R has defined a partition of S, S1 and S2 being blocks of the partition.
From this it is easy to see that every partition of S defines some equivalence relation on S. Conversely, every equivalence relation on S defines a partition of S. (See lemma 1.1.1)
Example 1.1.1
Let S = {1, 2, 3, 4, 5, 6} and define a mapping f defined as:
Thus the elements of S, 3 and 5 are f related because
f(3) = 1 and f(5) = 1. In other words, f(3) = f(5).
This relation, which turns out to be the equivalence relation above, is Rf.
Any f on any set induces an equivalence relation on that set.