IF THE GIVEN RELATION IS REFLEXIVE SYMMETRIC OR TRANSITIVE

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:

(iv) Let A be the set consisting of all the female members of a family. The relation R defined by “aRb if a is not a sister of b”.

Solution :

Let A be the relation consisting of 4 female members, a grand mother (a), her two children (b and c) and a grand daughter (d).

Condition for reflexive :

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

a is not a sister of a itself. 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.

aRb that is, a is not a sister of b.

bRa that is, b is not a sister of c.

Note : We should not take b and c, because they are sisters, they are not in the relation.

Hence it is symmetric.

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.

aRc that is, a is not a sister of c.

cRb that is, c is not a sister of b. But a is not a sister of b. Hence it is transitive.

Hence the given relation A is reflexive, symmetric and transitive.

(v) On the set of natural numbers the relation R defined by “xRy if x + 2y = 1”.

Solution :

Condition for reflexive :

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

let x = y

x + 2x = 1

3x = 1 ==> x = 1/3

1/3 is not related to 1/3, because 1/3 is not a natural number and it is not in the relation.R is not symmetric.

The given set R is an empty relation. An empty relation can be considered as symmetric and transitive.

Hence R is not reflexive, symmetric and transitive.