Relations and types of Relations

Let R be a relation on a set A. Then R is said to beLet R be a relation on a set A. Then R is said to be

Equivalence Relation

A relation which is reflexive, symmetric and transitive is called the Equivalence relation.

i.e., A relation R in a set A is called equivalence if it satisfies the following conditions.

Remarks:

• The smallest equivalence relation in set A is the relation of equality in A.

• The largest equivalence relation in A is A x A.

Inverse Relation

Let R Í A x B be a relation from A to B. Then the inverse of R, denoted by R^-1 is a relation from B to A, defined as 

Clearly, Domain of R^-1= Range of R and Range of R^-1= Domain of R.

Example 1:

If R = {(1,2), (1,5),(2,4),(3,5)}

Domain of R^-1= {2, 5, 4} = Range of R

Range of R^-1= {1, 2, 3} = Domain of R

Example 2:

If f is a relation from a set A to set B such that f Í f^-1 then, prove that f = f^-1.

Suggested answer:

From (i) and (ii),

Some important relations

• On the set of positive integers Z⁺, the relation “a divides b” is

• reflexive

• not symmetric

• transitive

• On the set of positive integers Z⁺, the relation "a

• not reflexive

• not symmetric

• transitive

• On the set of triangles, the relation “similar to” is

• reflexive

• symmetric

• transitive

• On the set of lines in a plane, the relation “parallel to” is

• reflexive

• symmetric

• transitive

• On the set of lines in a plane, the relation “perpendicular to” is

• not reflexive

• symmetric

• not transitive

• On the universal set of sets, the relation “subset of” is

• reflexive

• not symmetric

• transitive

• On the universal set of sets, the relation “subset of” is an

antisymmetric relation.

• On the set of integers, the relation “congruence” is an equivalence relation.