Question

Q. 15 Let n be a fixed positive integer. Define a relation Rin Z as follows V a, bez, aRb if and only if a -b is divisible by n. Show that R is an equivalence relation.​

Answer 1

Answer:

Given that,∀ a,b ∈ Z, aRb if and only if a - b is divisible by n.

Now,

aRa ⇒ (a-a) is divisible by n, which is true for any integer a as '0' is divisible by n.

Hence, R is reflexive

aRb

⇒ a-b is divisible by n.

⇒ -(b-a) is divisible by n.

⇒ (b-a) is divisible by n.,

⇒ aRc

Hence , R is transitive .

So, R is an equivalence relation

Step-by-step explanation:

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Answer 2

Given that ∀x,b∈Z,aRb if an only a−b divisible by n

Now

Reflexive

aRa⇒ (a−a) divisible by n, which is true for any integer

′

a

′

as 0 divisible by n

Symmetric

aRb

⇒ z−b is divisible by n

⇒ −b+a is divisible by n

⇒ −(b−s) is divisible by n

⇒ (b−a) is divisible by n

⇒ bRa

Hence R is symmetric

Transitive

Let aRb and bRa

⇒ (a−b) is divisible by n and (b−c) is divisible by n

⇒ (a−b)+(b−c) is divisible by n

⇒ (a−c) is divisible by n

⇒ aRc

Hence, R is transitive.

So, R an equivalence relation.