Category: 02. Relations

A relation R on a set A is called a partial order relation if it satisfies the following three properties: Example1: Show whether the relation (x, y) ∈ R, if, x ≥ y defined on the set of +ve integers is a partial order relation. Solution: Consider the set A = {1, 2, 3, 4} containing four…

A relation R on a set A is called an equivalence relation if it satisfies following three properties: Example: Let A = {1, 2, 3, 4} and R = {(1, 1), (1, 3), (2, 2), (2, 4), (3, 1), (3, 3), (4, 2), (4, 4)}. Show that R is an Equivalence Relation. Solution: Reflexive: Relation R is…

Consider a given set A, and the collection of all relations on A. Let P be a property of such relations, such as being symmetric or being transitive. A relation with property P will be called a P-relation. The P-closure of an arbitrary relation R on A, indicated P (R), is a P-relation such that…

1. Reflexive Relation: A relation R on set A is said to be a reflexive if (a, a) ∈ R for every a ∈ A. Example: If A = {1, 2, 3, 4} then R = {(1, 1) (2, 2), (1, 3), (2, 4), (3, 3), (3, 4), (4, 4)}. Is a relation reflexive? Solution: The relation is reflexive as for every…

Let A, B, and C be sets, and let R be a relation from A to B and let S be a relation from B to C. That is, R is a subset of A × B and S is a subset of B × C. Then R and S give rise to a relation…

Relations can be represented in many ways. Some of which are as follows: 1. Relation as a Matrix: Let P = [a1,a2,a3,…….am] and Q = [b1,b2,b3……bn] are finite sets, containing m and n number of elements respectively. R is a relation from P to Q. The relation R can be represented by m x n matrix…

Let P and Q be two non- empty sets. A binary relation R is defined to be a subset of P x Q from a set P to Q. If (a, b) ∈ R and R ⊆ P x Q then a is related to b by R i.e., aRb. If sets P and Q…