Question

In the set of all natural number, let R be defined by R= {(x, y):
xε N, x-y is divisible by 5, then Prove that R is an equivalence relation. Thanks

Answer 1

Answer:

Step-by-step explanation:

We check one by one all the relations reflexive, transative and symmetric

For reflexive:

We know that (x,x) is reflexive

Put X in place of y

x-x=0 so,it is divisible by 5

Hence,it is reflexive.

For symmetric:

If (X,y)E R then (y,x) E R

X-y is divisible by 5

It implies that y-x is divisible by 5

Hence,it is symmetric

For transative:

If (X,y)E R and (y,z)E R then (X,z) E R

x-y=5I1

Y-z=5I2

When we add them then we get

x-z=(I1+I2)

Hence,it is transative

So,R is an equivalance relation