Q11 of 109 Page 1

Is it true that every relation which is symmetric and transitive is also reflexive? Give reasons.

It is not true that every relation which is symmetric and transitive is also reflexive.

Take for example:


Take a set A = {1, 2, 3, 4}


And define a relation R on A.


Symmetric relation:


R = {(1, 2), (2, 1)}, is symmetric on set A.


Transitive relation:


R = {(1, 2), (2, 1), (1, 1)}, is the simplest transitive relation on set A.


⇒ R = {(1, 2), (2, 1), (1, 1)} is symmetric as well as transitive relation.


But R is not reflexive here.


If only (2, 2) ∈ R, had it been reflexive.


Thus, it is not true that every relation which is symmetric and transitive is also reflexive.


More from this chapter

All 109 →
9

If A = {1, 2, 3, 4}, define relations on A which have properties of being

reflexive, symmetric and transitive.

 10

Let R be a relation defined on the set of natural numbers N as

R = {(x, y) : x, y ∈ N, 2x + y = 41}


Find the domain and range of R. Also, verify whether R is


(i) reflexive,


(ii) symmetric


(iii) transitive.

 12

An integer m is said to be related to another integer n if m is a multiple of n. Check if the relation is symmetric, reflexive and transitive.

 13

Show that the relation “≥” on the set R of all real numbers is reflexive and transitive but not symmetric.