Binary Relations and Equivalence Relations and Partitions
Relations:
Relation, a mathematical concept, is a set of ordered pairs. More precisely, the
concept is called binary relation, but usually “binary” is omitted.
The Definition of Relation
Let X and Y be sets. A relation, R, from X to Y is a subset of the Cartesian
product X × Y. The elements of X × Y are called ordered pairs.
Let x be an element of X and y be an element of Y. The notations (x, y) is an
element of R and x R y (say x is in relation R to y) are equivalent.
If X = Y, then R is called a relation on X and is, of course, a subset of X × X.
Example of Relation
Let X and Y be sets. The trivial relation is the empty set, which is, of course, a
subset of every set including X × Y. Because the empty set is a subset of every
subset of X × Y, the trivial relation is a subset of every relation from X to Y. Also,
the domain and range are both equal to the empty set.
The Cartesian product, X × Y, is also a relation. It is obviously the largest relation
from X to Y since, by definition of relation, it contains every relation from X to Y.
Also, the domain is X and the range is Y.
The Definition of Binary Relation
Given a set of objects , a binary relation is a subset of the Cartesian Product .
A binary relation from a set A to a set B is a set of ordered pairs <a, b> where a is an
element of A and b is an element of B.
When an ordered pair <a, b> is in a relation R, we write a R b, or <a, b> R. It means
that element a is related to element b in relation R.
When A = B, we call a relation from A to B a (binary) relation on A.
Definition of Cartesian product
The set of all ordered pairs <a, b>, where a is an element of A and b is an element of B,
is
called the Cartesian product of A and B and is denoted by A B.
Thus a binary relation from A to B is a subset of Cartesian product A B.
The Definition of Equivalence Relation
An Equivalence Relation is a binary relation between two elements of a set which
groups them together as being "equivalent" in some way. That a is equivalent to
b
is denoted as "a ~ b" or "a ≡ b".
An equivalence relation is reflexive, symmetric, and transitive. In other words, for
all elements a, b, and c of the set X, the following must hold for "~" to be an
equivalence relation:
Reflexivity: a ~ a
Symmetry: if a ~ b then b ~ a
Transitivity: if a ~ b and b ~ c then a ~ c.
Reflexivity:
A binary relation can have, among other properties, reflexivity or irreflexivity.
At least in this context, (binary) relation (on X) always means a relation on X×X,
or in other words from a set X into itself.
A reflexive relation R on set X is one where for all a in X, a is R-related to itself.
In mathematical notation, this is:
.
.
The reflexive closure R = is defined as R = = {(x, x) | x X} ∈ ∪ R, i.e., the smallest
reflexive relation over X containing R. This can be seen to be equal to the
intersection of all reflexive relations containing R.
Symmetry:
a binary relation R over a set X is symmetric if it holds for all a and b in X that if a
is related to b then b is related to a.
In mathematical notation, this is:
Transitivity:
A binary relation R over a set X is transitive if it holds for all a, b, and c in X, that
if a is related to b and b is related to c, then a is related to c.
To write this in predicate logic:
For instance, the "greater than" relation is transitive:
If A > B, and B > C, then A > C.
For example, "is greater than," "is at least as great as," and "is equal to" (
equality)
are transitive relations:
whenever A > B and B > C, then also A > C
whenever A ≥ B and B ≥ C, then also A ≥ C
whenever A = B and B = C, then also A = C
Partition: Definition:
A partition of a set X is a division of X into non-overlapping "parts" or "blocks" or
"cells" that cover all of X. More formally, these "cells" are both collectively
exhaustive and mutually exclusive with respect to the set being partitioned.
A partition of a set X is a set of nonempty subsets of X such that every element x
in X is in exactly one of these subsets.
Equivalently, a set P of subsets of X, is a partition of X if
No element of P is empty.
The union of the elements of P is equal to X.
The intersection of any two elements of P is empty.
The elements of P are sometimes called the blocks or parts of the partition.
Every singleton set {x} has exactly one partition, namely { {x} }.
For any nonempty set X, P = {X} is a partition of X.
The empty set has exactly one partition, namely one with no blocks.
For any non-empty proper subset A of a set U, this A together with its
complement
is a partition of U.
If we do not use axiom 1, then the above example generalizes so that any subset
(empty or not) together with its complement is a partition.
The set { 1, 2, 3 } has these five partitions.
{ {1}, {2}, {3} }, sometimes denoted by 1/2/3.
{ {1, 2}, {3} }, sometimes denoted by 12/3.
{ {1, 3}, {2} }, sometimes denoted by 13/2.
{ {1}, {2, 3} }, sometimes denoted by 1/23.
{ {1, 2, 3} }, sometimes denoted by 123.
Note that
{ {}, {1,3}, {2} } is not a partition if we are using axiom 1 (because it contains the
empty set); otherwise it is a partition of {1, 2, 3}.
{ {1,2}, {2, 3} } is not a partition (of any set) because the element 2 is contained
in more than one distinct subset.
{ {1}, {2} } is not a partition of {1, 2, 3} because none of its blocks contains 3;
however, it is a partition of {1, 2}.
[edit ] Partitions and equivalence relations
If an equivalence relation is given on the set X, then the set of all equivalence
classes forms a partition of X. Conversely, if a partition P is given on X, we can
define an equivalence relation on X by writing x ~ y if there exists a member of P
which contains both x and y. The notions of "equivalence relation" and "partition"
are thus essentially equivalent.