Question

Define Transitive Relation. Given one example.

Answer 1

In mathematics, a binary relation R over a set X is transitive if wheneveran element a is related to an element band b is related to an element c then a is also related to c. Transitivity (or transitiveness) is a key property of both partial order relations andequivalence relations.





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.

Answer 2

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