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Types of Relation | Discrete Mathematics

In this article, we will learn about the relations and the different types of relation in the discrete mathematics.
Submitted by Prerana Jain, on August 17, 2018

Types of Relation

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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