Transitive Relation on Set

What is transitive relation on set?

Let A be a set in which the relation R defined.

R is said to be transitive, if

(a, b) ∈ R and (b, a) ∈ R ⇒ (a, c) ∈ R,

That is aRb and bRc ⇒ aRc where a, b, c ∈ A.

The relation is said to be non-transitive, if

(a, b) ∈ R and (b, c) ∈ R do not imply (a, c ) ∈ R.

For example, in the set A of natural numbers if the relation R be defined by ‘x less than y’ then

a < b and b < c imply a < c, that is, aRb and bRc ⇒ aRc.

Hence this relation is transitive.


Solved example of transitive relation on set:

1. Let k be given fixed positive integer.

Let R = {(a, a) : a, b  ∈ Z and (a – b) is divisible by k}.

Show that R is transitive relation.

Solution:

Given R = {(a, b) : a, b ∈ Z, and (a – b) is divisible by k}.

Let (a, b) ∈ R and (b, c) ∈ R. Then

      (a, b) ∈ R and (b, c) ∈ R

   ⇒ (a – b) is divisible by k and (b – c) is divisible by k.

   ⇒ {(a – b) + (b – c)} is divisible by k.

   ⇒ (a – c) is divisible by k.

   ⇒ (a, c) ∈ R.

Therefore, (a, b) ∈ R and (b, c) ∈ R    (a, c) ∈ R.

So, R is transitive relation.


2. A relation ρ on the set N is given by “ρ = {(a, b) ∈ N × N : a is divisor of b}”. Examine whether ρ is transitive or not transitive relation on set N.

Solution:

Given ρ = {(a, b) ∈ N × N : a is divisor of b}.

Let m, n, p ∈ N and (m, n) ∈ ρ and  (n, p ) ∈ ρ. Then

                                                 (m, n) ∈ ρ and  (n, p ) ∈ ρ

                                              ⇒ m is divisor of n and n is divisor of p

                                              ⇒ m is divisor of p

                                              ⇒ (m, p) ∈ ρ

Therefore, (m, n) ∈ ρ and (n, p) ∈ ρ ⇒ (m, p) ∈ ρ.

So, R is transitive relation.

Set Theory

Sets

Representation of a Set

Types of Sets

Pairs of Sets

Subset

Practice Test on Sets and Subsets

Complement of a Set

Problems on Operation on Sets

Operations on Sets

Practice Test on Operations on Sets

Word Problems on Sets

Venn Diagrams

Venn Diagrams in Different Situations

Relationship in Sets using Venn Diagram

Examples on Venn Diagram

Practice Test on Venn Diagrams

Cardinal Properties of Sets



7th Grade Math Problems

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