Q. 9 C4.2( 5 Votes )

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

reflexive, symmetric and 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.


Using these properties, we can define R on A.


A = {1, 2, 3, 4}


We need to define a relation (say, R) which is reflexive, symmetric and transitive.


The relation must be defined on A.


Reflexive Relation:


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


Or simply shorten it and write,


R = {(1, 1), (2, 2)} …(1)


Symmetric Relation:


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


Or simply shorten it and write,


R = {(1, 2), (2, 1)} …(2)


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


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


It is reflexive, symmetric as well as transitive as per definition.


Similarly, we can find other combinations too.


Thus, we have got the relation which is reflexive, symmetric as well as transitive.


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