Q. 9 B4.2( 6 Votes )

If A = {1, 2, 3,

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 symmetric but neither reflexive nor transitive.


The relation R must be defined on A.


Symmetric relation:


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


Note that, the relation R here is neither reflexive nor transitive, and it is the shortest relation that can be form.


Similarly, we can also write:


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


Or R = {(3, 4), (4, 3)}


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


And so on…


All of these are right answers.


Thus, we have got the relation which is symmetric but neither reflexive nor transitive.


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