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

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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