CHECKING IF THIS RELATION IS REFLEXIVE SYMMETRIC AND TRANSITIVE

Checking If this Relation is Reflexive symmetric and Transitive :

Here we are going to see how to check if the given relation is reflexive, symmetric and 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.

Checking If this Relation is Reflexive symmetric and Transitive - Practice Questions

Question 1 :

Discuss the following relations for reflexivity, symmetricity and transitivity:

Let P denote the set of all straight lines in a plane. The relation R defined by “lRm if l is perpendicular to m”.

Solution :

Condition for reflexive :

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

A line will not be perpendicular to itself. Hence it is not  reflexive.

Condition for symmetric :

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

lRm that is, l perpendicular to m. 

mRl, m is perpendicular to l, both are true. 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.

Let l, m and n be the set of lines in P.

If  “l is related to m and m is related to n” implies that l is not related to n, because they l and n are parallel lines.

So, is transitive is not true. 

Hence P is relation which is reflexive but not symmetric and not transitive.

(iii) Let A be the set consisting of all the 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 elements mother (a),  father (b), a son (c) and a daughter (d).

Condition for reflexive :

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

Let "a" be a member of a relation A, a will be not a sister of a. Hence it is not 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.

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

dRc that is, d is a sister of c.

Hence it is not 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.

dRa that is, d is not a sister of a.

aRc that is, a is not a sister of c.  But a is a sister of c, this is not in the relation. Hence it is not transitive.

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

Related Topics

Reflexive relation

Symmetric relation

Transitive relation

Equivalence relation

Identity relation

Inverse relation

Difference between reflexive and identity relation

After having gone through the stuff given above, we hope that the students would have understood, how to check whether a relation is reflexive, symmetric and transitive.

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