Question

Each of the following defines a relation on N:

(i) x > y, x, y $\in$ N
(ii) x + y = 10, x, y $\in$ N
(iii) xy is square of an integer, x, y $\in$ N
(iv) x + 4y = 10, x, y $\in$ N

Determine which of the above relations are reflexive, symmetric and transitive. [NCERT EXEMPLAR]

Answer 1

(i) We have,
R = {(x, y) : x > y, x, y ∈ N}
As, x=x ∀x∈N⇒x,x∉RSo, R is not a reflexive relationLet x,y∈R⇒x>ybut y<x⇒y,x∉RSo, R is not a symmeteric relationLet x,y∈R and y,z∈R⇒x>y and y>z⇒x>z⇒x,z∈RSo, R is a transitive relation
(ii) We have,
R = {(x, y) : x + y = 10, x, y ∈ N}
R=1,9,2,8,3,7,4,6,5,5,6,4,7,3,8,2,9,1
As, (1,1)∉R So, R is not a reflexive relation Let x,y∈R⇒x+y=10⇒y+x=10⇒(y,x)∈R So, R is a symmeteric relation As, (1,9)∈R and (9,1)∈R but (1,1)∉R So, R is not a transitive relation
(iii) We have,
R = {(x, y) : xy is square of an integer, x, y ∈ N}
As, $x\times x=x^2$which is a square of an integer x⇒(x,x)∈R So, R is a reflexive relation
Let (x,y)∈R⇒xy is square of an integer⇒yx is also a square of an integer⇒(y,x)∈R So, R is a symmeteric relationLet (x,y)∈R and (y,z)∈R⇒xy is square of an integer and yz is also a square of an interger⇒xz must be a square of an integer⇒(x,z)∈R
So, R is a transitive relation
(iv) We have,
R = {(x, y) : x + 4y = 10, x, y ∈ N}
​R=2,2,6,1
As, (2,2)∉R So, R is not a reflexive relation As, 2,4∈R but 4,2∉RSo, R is not a symmeteric relation
As, (2, 4)∈R but 4 is not related to any natural number
So, R is a transitive relation