Q. 115.0( 2 Votes )

Show that the relation R = {(a, b) : a > b} on N is transitive but neither reflexive nor symmetric.

Answer :

We have, R = {(a, b) : a > b} relation defined on N.


Now,


We observe that, any element a N cannot be greater than itself.


(a,a) R a N


R is not reflexive.


Let (a,b) R a, b N


a is greater than b


But b cannot be greater than a if a is greater than b.


(b,a) R


For e.g., we observe that (5,2) R i.e 5 > 2 but 2 5 (2,5) R


R is not symmetric


Let (a,b) R and (b,c) R a, b,c N


a > b and b > c


a > c


(a,c) R


For e.g., we observe that


(5,4) R 5 > 4 and (4,3) R 4 > 3


And we know that 5 > 3 (5,3) R


R is transitive.


Thus, R is transitive but not reflexive not symmetric.


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