0. PRELIMINARIES
19
Theorem (0.7 — Equivalence Classes Partition).
The equivalence classes of an equivalence relation on a set S constitute a
partition of S. Conversely, for any patition P of S, there is an equivalence
relation on S whose equivalence classes are the elements of P .
Proof. Let ⇠ be an equivalence relation on S. For any a 2 S, a ⇠ a, so
a 2 [a] and S = [ [a]. Now suppose [a] 6= [b]. [To show [a] \ [b] = ;.]
a2S
By way of contradiction, suppose c 2 [a] \ [b]. [We show [a] = [b]. Recall
A ✓ B and B ✓ A =) A = B.]
Let x 2 [a]. Then c ⇠ a, c ⇠ b, and x ⇠ a, so a ⇠ c and thus x ⇠ c and
x ⇠ b. We now have x 2 [b] =) [a] ✓ [b].
Analagously, [b] ✓ [a] =) [a] = [b], a contradiction. Thus [a] \ [b] = ;, giving
us a partition of S.
For the second part, let P be a partition of S. Define a ⇠ b if a and b belong to
the same subset in the partition. To finish the proof, show ⇠ is an equivalence
relation.
⇤
Functions (Mappings)
Definition (Function (Mapping)).
A function (or mapping) from a set A (the domain of ) to a set B (the
range of ) is a rule that assigns to each element a of A exactly one element b
of B. : A ! B means is a mapping from A to B.
If assigns b 2 B to a 2 A, we write (a) = b or
image of A under .
: a ! b, and say b is the
(A) = {b 2 B| (a) = b for some a 2 A} is the image of A under .
(A) ✓ B.