Q. 18 B5.0( 2 Votes )

Each of the following defines a relation on N :

x + y = 10, x, y N

Determine which of the above relations are reflexive, symmetric and transitive.

Answer :

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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