EQUIVALENCE RELATION
Maªgorzata Murat
CARTESIAN PRODUCT
Let U be a universe. Let A and B be sets of elements of U .
A × B = {(a, b) : a ∈ A ∧ b ∈ B} -
- the set of all ordered pairs
examples:
N × −1, 1,
h−2, 2) × Z,
(2, 4i × (−4, 2),
{1, 2, 3, 4} × {1, 2, 3, 4}
RELATION
R ⊆ A × B - relation on A and B
R ⊆ A × A - relation on A
examples:
Let A = {1, 2, 3, 4}. Dene a relation R on A by writing
(x, y) ∈ R if x < y .
R=∅
Let A = {1, 2}. Dene a relation R on A by writing
(P, Q) ∈ R if P ⊂ Q.
PROPERTIES OF
R
Let R ⊆ A × A.
R is called
reexive if
^
(a, a) ∈ R
a∈A
R is called
symmetric if
^ (a, b) ∈ R ⇒ (b, a) ∈ R
a,b∈A
R is called ^
transitive
if
h
i
(a, b) ∈ R ∧ (b, c) ∈ R ⇒ (a, c) ∈ R
a,b,c∈A
EXAMPLE
Let
A = {1, 2, 3}.
R = {(1, 2); (2, 1)} is symmetric but not reexive and
transitive
R = {(1, 1); (2, 2); (3, 3); (1, 2)} is reexive and
transitive but not symmetric
R = {(1, 1); (2, 2); (3, 3)} is symmetric, reexive and
transitive
R = {(1, 2); (2, 1); (1, 1); (2, 2)} is symmetric and
transitive but not reexive
R ⊂ A × A is called
an equivalence relation if it is
reexive, symmetric and transitive
example: R ⊂ Z: (a, b) ∈ R if a − b is a multiple of 3
EQUIVALENCE CLASSES
Let R ⊂ A × A is an equivalence relation.
[a] = {b ∈ A : (a, b) ∈ R} =
equivalence class of a
b ∈ [a] ⇔ [a] = [b]
[a] ∩ [b] = ∅ or [a] = [b]
A is the disjoint union of its distinct equivalence classes