Q. 14 B5.0( 2 Votes )

Give an example of a relation which is

reflexive and transitive but not symmetric.

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 transitive but not symmetric.


Let there be a set A.


A = {1, 2, 3, 4}


Reflexive relation:


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


Transitive relation:


R = {(3, 4), (4, 1), (3, 1)} …(2)


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


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


Check for Symmetry:


If (3, 4) R


Then, (4, 3) R


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


One example is enough to prove that R is not symmetric.


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


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


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