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} = S_{1}

(2)_{R} = (4)_{R}= (6)_{R} = {2, 4, 6} = S_{2}

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 = S_{1 }+ S_{2}

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, S_{1} and S_{2} 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 R_{f}.

Any f on any set induces an equivalence relation on that set.