Types of Relations in Discrete Mathematics

In this tutorial, we will learn about the different types of relations in discrete mathematics. By Prerana Jain Last updated : May 09, 2023

Types of Relations in Discrete Mathematics

There are many types of relation which is exist between the sets,

1. Universal Relation

A relation r from set a to B is said to be universal if: R = A * B

Example:

A = {1,2} B = {a, b}

R = { (1, a), (1, b), (2, a), (2, b) is a universal relation.

2. Compliment Relation

Compliment of a relation will contain all the pairs where pair do not belong to relation but belongs to Cartesian product.

R = A * B – X

Example:

A = { 1, 2}   B = { 3, 4}
R = { (1, 3) (2, 4) }
Then the complement of R
Rc = { (1, 4) (2, 3) }

3. Empty Relation

A null set phie is subset of A * B.

R = phie is empty relation

4. Inverse of Relation

An inverse of a relation is denoted by R^-1 which is the same set of pairs just written in different or reverse order. Let R be any relation from A to B. The inverse of R denoted by R^-1 is the relation from B to A defined by:

R^-1 = { (y, x) : yEB, xEA, (x, y) E R}

5. Composite Relation

Let A, B, and C be any three sets. Let consider a relation R from A to B and another relation from B to C. The composition relation of the two relation R and S be a Relation from the set A to the set C, and is denoted by RoS and is defined as follows:

Ros = { (a, c) : an element of B such that (a, b) E R and (b, c) E s, when a E A , c E C}
Hence, (a, b) E R (b, c) E S => (a, c) E RoS.

6. Equivalence Relation

The relation R is called equivalence relation when it satisfies three properties if it is reflexive, symmetric, and transitive in a set x. If R is an equivalence relation in a set X then D(R) the domain of R is X itself. Therefore, R will be called a relation on X.

The following are some examples of the equivalence relation:

Equality of numbers on a set of real numbers.
Equality of subsets of a universal set.
Similarities of triangles on the set of triangles.
Relation of lines being a parallel onset of lines in a plane.
Relation of living in the same town on the set of persons living in Canada.

7. Partial Order Relation

Let, R be a relation in a set A then, R is called partial order Relation if,

R is reflexive
i.e. aRa ,a belongs to A
R is anti- symmetric
i.e. aRb, bRa => a = b, a, b belongs to a
R is transitive
aRb, bRc => aRc, a, b, c belongs to A

8. Antisymmetric Relation

A relation R on a set a is called on antisymmetric relation if for x, y if for x, y =>

If (x, y) and (y, x) E R then x = y

Example: { (1, 2) (2, 3), (2, 2) } is antisymmetric relation.

A relation that is antisymmetric is not the same as not symmetric. A relation can be antisymmetric and symmetric at the same time.

9. Irreflective Relation

A relation R is said to be on irreflective relation if x E a (x ,x) does not belong to R.

Example:

a = {1, 2, 3}
R = { (1, 2), (1, 3) if is an irreflexive relation

10. Not Reflective Relation

A relation R is said to be not reflective if neither R is reflexive nor irreflexive.

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