Let R be the relation on the set Z of all integers defined by R = {(x,y):x y - Maths - Relations and Functions

Question

Let R be the relation on the set Z of all integers defined by R =
{(x,y):x – y is divisible by n} Prove that
a) (x,y) ÎR for all x Î Z
b) (x,y) ÎR Þ (y,x) ÎR for all x,y ÎZ
c) (x,y) ÎR & (y,z) ÎR Þ (x,z) Î R for
all x,y,z ÎR

Rishabh Mittal · Accepted Answer

a
 For any x∈Z, we havex-x=0 , which is divisible by n 
So, x,x∈R for all x∈Zb
 Let x,y∈R⇒x-y is divisible by n⇒-y-x is divisible by n⇒y-x is divisible by n⇒y,x ∈ Rc
Let  x,y∈R and y,z∈R⇒x-y is divisible by n and y-z is divisible by n⇒x-y+y-z is divisible by n⇒x-z is divisible by n⇒x,z ∈ R

. Let R be the relation on the set Z of all integers defined by R = {(x,y):x – y is divisible by n}. Prove that a) (x,y) ÎR for all x Î Z b) (x,y) ÎR Þ (y,x) ÎR for all x,y ÎZ c) (x,y) ÎR & (y,z) ÎR Þ (x,z) Î R for all x,y,z ÎR

. Let R be the relation on the set Z of all integers defined by R = {(x,y):x – y is divisible by n}. Prove that
a) (x,y) ÎR for all x Î Z
b) (x,y) ÎR Þ (y,x) ÎR for all x,y ÎZ
c) (x,y) ÎR & (y,z) ÎR Þ (x,z) Î R for all x,y,z ÎR