Set Theory: What is a relation?
Relation definition
Let us suppose A and B to be two non-empty sets, then every subset of A 8 B describes a relation from A to B and every relation from A to B is a subset of A * B. Now let us suppose
Then it is said that a is related to b by the relation R and it is written as a R b. If (a, b)
, then it is written as a R b.
· Total number of relations: Let us suppose that A and B are two non-empty finite sets which consists of m and n elements respectively. Then AB has mn ordered pairs. Thus, the total number of subsets of A * B
is Each subset of A * B describes a relation from A * B, so the total number of relations from A to B is
Out of these
relations, the void relation f and the universal relation A * b are small relations from A to B.
· Domain and range of a relation: Let us suppose that R is a relation from a set A to a set B. Then the set consisting of all first components or coordinates of the ordered pairs which belongs to R is known as the domain of R. The set which has all second components or coordinates of the ordered pairs in R is known as the range of R.
Therefore,
Inverse Relation
Let us suppose that A, B are two sets and R has a relation from a set A to a set B. Then the inverse of R, which is depicted by
, has a relation from B to A and is given as
Types of Relations
· Reflexive relation: A relation R on a set A is called reflexive when every element of A is related to itself.
Therefore,
·Symmetric relation: A relation R on a set A is called symmetric relation if
· Anti-symmetric relation: Let us take A to be any set. A relation R on set A is called anti-symmetric relation
Therefore, if a is not related to b, then a may be related to b or b may also be related to a, but it cannot be related to both.
· Transitive relation: Let us take A as any set. A relation R on a set A is called transitive relation
Transitivity does not work when there exists a, b, c such that a Rb, b R c but
· Identity relation: Let us take A to be any set. Then the relation
is known as the identity relation on A. A relation
is known as identity relation when every element of A is related to itself only. Every identity relation is reflexive, symmetric and transitive.
· Equivalence relation: A relation R on a set A is called as an equivalence relation on A
Congruence modulo (m): Let us consider m to be an arbitrary but fixed integer. Two integers a and b are called as congruence modulo m if a — b is divisible by m and it is written as .
Equivalence classes of an equivalence relation: Let us take R as equivalence relation in
Then the equivalence class of a which is denoted by [a] is said to be the set of all those points of A which has a relation to a under the relation R.
Therefore,
Now we know that:
3. Two equivalence classes can be either disjoint or identical.
Composition of relations
Let us suppose R and S to be two relations from sets A to B and B to C respectively. Thus, a relation SoR from A to C can be defined such that
This type of relation is known as the composition of R and S.
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