Q. 18 A5.0( 1 Vote )

# Each of the following defines a relation on N :x > y, x, y ∈ NDetermine which of the above relations are reflexive, symmetric and transitive.

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.

We have

x > y, x, y N

This relation is defined on N (set of Natural Numbers)

The relation can also be defined as

R = {(x, y): x > y} on N

Check for Reflexivity:

x N

We should have, (x, x) R

x > x, which is not true.

1 can’t be greater than 1.

2 can’t be greater than 2.

16 can’t be greater than 16.

Similarly, x can’t be greater than x.

So, x N, then (x, x) R

R is not reflexive.

Check for Symmetry:

x, y N

If (x, y) R

x > y

Now, replace x by y and y by x. We get

y > x, which may or not be true.

Let us take x = 5 and y = 2.

x > y

5 > 2, which is true.

y > x

2 > 5, which is not true.

y > x, is not true as x > y

(y, x) R

So, if (x, y) R, but (y, x) R x, y N

R is not symmetric.

Check for Transitivity:

x, y, z N

If (x, y) R and (y, z) R

x > y and y > z

x > y > z

x > z

(x, z) R

So, if (x, y) R and (y, z) R, and then (x, z) R

x, y, z N

R is transitive.

Hence, the relation is transitive but neither reflexive nor symmetric.

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