Q. 14 A5.0( 3 Votes )

Give an example of a relation which is

reflexive and symmetric but not 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 on A which is reflexive and symmetric but not transitive.


Let there be a set A.


A = {1, 2, 3, 4}


Reflexive relation:


R = {(1, 1), (2, 2), (3, 3), (4, 4)} …(1)


Symmetric relation:


R = {(3, 4), (4, 3)} …(2)


Combine results (1) and (2), we get


R = {(1, 1), (2, 2), (3, 3), (4, 4), (3, 4), (4, 3)}


Check for Transitivity:


If (3, 4) R and (4, 3) R


Then, (3, 3) R


3, 4 A [ A = {1, 2, 3, 4}]


So eliminate (3, 3) from R, we get


R = {(1, 1), (2, 2), (4, 4), (3, 4), (4, 3)}


Check for Transitivity:


If (4, 3) R and (3, 4) R


Then, (4, 4) R


3, 4 A


So, eliminate (4, 4) from R, we get


R = {(1, 1), (2, 2), (3, 4), (4, 3)}


Thus, the relation which is reflexive and symmetric but not transitive is:


R = {(1, 1), (2, 2), (3, 4), (4, 3)}


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