HOW TO DISCUSS THE RELATION FOR REFLEXIVE SYMMETRIC OR TRANSITIVE

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Reflexive, Symmetric and transitive Relation

Let S be any non-empty set. Let R be a relation on S. Then

R is said to be reflexive if a is related to a for all a ∈ S.
R is said to be symmetric if a is related to b implies that b is related to a.
R is said to be transitive if “a is related to b and b is related to c” implies that a is related to c.

Question 1 :

Discuss the following relations for reflexivity, symmetricity and transitivity:

(i) The relation R defined on the set of all positive integers by “mRn if m divides n”.

Solution :

Condition for reflexive :

R is said to be reflexive, if a is related to a for a ∈ S.

Every element is divided by itself, for all m ∈ R, m divides m and for all n ∈ R n divides n.

Hence it is reflexive.

Condition for symmetric :

R is said to be symmetric if a is related to b implies that b is related to a.

From the given question, we come to know that m divides n, but the vice versa is not true.

For example, if m = 2 and n = 4 ∈ R, then we may say that 2 divides 4. But we cannot say that 4 divides 2. Hence symmetric is not true.

Condition for transitive :

R is said to be transitive if “a is related to b and b is related to c” implies that a is related to c.

If m divides n, then n = mk ----(1)

If n divide p, then p = nq ----(2)

Here "k" and "q" are constants.

By applying the value of "n" in (2), we get

p = m k q

From this, we come to know that p is the multiple of m. So, it is transitive.

Hence the given relation is reflexive, not symmetric and transitive.