Q. 18 B5.0( 2 Votes )

# Each of the following defines a relation on N :x + y = 10, 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 = 10, 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 = 10} on N

Check for Reflexivity:

x N

We should have, (x, x) R

x + x = 10, which is not true everytime.

Take x = 4.

x + x = 10

4 + 4 = 10

8 = 10, which is not true.

That is 8 ≠ 10.

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

R is not reflexive.

Check for Symmetry:

x, y N

If (x, y) R

x + y = 10

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

y + x = 10, which is as same as x + y = 10.

y + x = 10

(y, x) R

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

R is symmetric.

Check for Transitivity:

x, y, z N

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

x + y = 10 and y + z = 10

x + z = 10, may or may not be true.

Let us take x = 6, y = 4 and z = 6

x + y = 10

6 + 4 = 10

10 = 10, which is true.

y + z = 10

4 + 6 = 10

10 = 10, which is true.

x + z = 10

6 + 6 = 10

12 = 10, which is not true

That is, 12 ≠ 10

x + z ≠ 10

(x, z) R

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

x, y, z N

R is not transitive.

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

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