Q. 4

# Let R = {(a, b) : a, b ∈ Z and (a - b) is divisible by 5}.Show that R is an equivalence relation on Z.

In order to show R is an equivalence relation, we need to show R is Reflexive, Symmetric and Transitive.

Given that, a, b Z, aRb if and only if a b is divisible by 5.

Now,

R is Reflexive if (a,a) R a Z

aRa (a-a) is divisible by 5.

a-a = 0 = 0 × 5 [since 0 is multiple of 5 it is divisible by 5]

a-a is divisible by 5

(a,a) R

Thus, R is reflexive on Z.

R is Symmetric if (a,b) R (b,a) R a,b Z

(a,b) R (a-b) is divisible by 5

(a-b) = 5z for some z Z

-(b-a) = 5z

b-a = 5(-z) [ z Z -z Z ]

(b-a) is divisible by 5

(b,a) R

Thus, R is symmetric on Z.

R is Transitive if (a,b) R and (b,c) R (a,c) R a,b,c Z

(a,b) R (a-b) is divisible by 5

a-b = 5z1 for some z1 Z

(b,c) R (b-c) is divisible by 5

b-c = 5z2 for some z2 Z

Now,

a-b = 5z1 and b-c = 5z2

(a-b) + (b-c) = 5z1 + 5z2

a-c = 5(z1 + z2 ) = 5z3 where z1 + z2 = z3

a-c = 5z3 [ z1,z2 Z z3 Z]

(a-c) is divisible by 5.

(a, c) R

Thus, R is transitive on Z.

Since R is reflexive, symmetric and transitive it is an equivalence relation on Z.

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