Q. 14 D5.0( 1 Vote )

Give an example of a relation which is

symmetric but neither reflexive nor transitive.

Answer :

Recall that for any binary relation R on set A. We have,


R is reflexive if for all x A, xRx.


R is symmetric if for all x, y A, if xRy, then yRx.


R is transitive if for all x, y, z A, if xRy and yRz, then xRz.


Let there be a set A.


A = {1, 2, 3, 4}


We need to define a relation which is symmetric but neither reflexive nor transitive.


Let there be a set A.


A = {1, 2, 3, 4}


Symmetric Relation:


{(1, 3), (3, 1)}


This is neither reflexive nor transitive.


(1, 1) R


(3, 3) R


Hence, R is not reflexive.


(1, 3) R and (3, 1) R


Then, (1, 1) R


Hence, R is not transitive.


Thus, the relation which is symmetric but neither nor transitive is:


R = {(1, 3), (3, 1)}


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