Q. 3 A4.2( 10 Votes )

Test whether the following relations R1, R2 and R3 are (i) reflexive (ii) symmetric and (iii) transitive :

R1 on Q0 defined by (a, b) ϵ R1 a = 1/b

Answer :

Here, R1, R2, R3, and R4 are the binary relations.


So, 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.


So, using these results let us start determining given relations.


We have


R1 on Q0 defined by (a, b) R1


Check for Reflexivity:


a, b Q0,


(a, a), (b, b) R1 needs to be proved for reflexivity.


If (a, b) R1


Then, …(1)


So, for (a, a) R1


Replace b by a in equation (1), we get



But, we know



(a, a) R1


So, a Q0, then (a, a) R1


R1 is not reflexive.


Check for Symmetry:


If (a, b) R1


Then, (b, a) R1


a, b Q0


If (a, b) R1


We have, …(2)


Now, for (b, a) R1


Replace a by b & b by a in equation (2), we get



(b, a) R2


So, if (a, b) R1, then (b, a) R1


a, b Q0


R1 is symmetric.


Check for Transitivity:


If (a, b) R1 and (b, c) R1



We need to eliminate b.


We have




Putting in , we get




But,


(a, c) R1


So, if (a, b) R1 and (b, c) R1, then (a, c) R1


a, b, c Q0


R1 is not transitive.


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