Question

a set of integers, a relation R is defined by xRy if and only if x-y is divisible by 4,then verify R is equivalence relation​

Answer 1

Answer:

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Step-by-step explanation:

Answer 2

The relation R is equivalence relation

Given :

The relation R on set of integers is defined by xRy if and only if x - y is divisible by 4

To find :

To prove R is equivalence relation

Solution :

Step 1 of 5 :

Write down the given relation

Here the given relation is R on set of integers is defined by xRy if and only if x - y is divisible by 4

So R = { (x, y) ∈ Z : x - y is divisible by 4 }

Step 2 of 5 :

Check for Reflexive

Let a ∈ Z

Since 4 divides a - a

So (a, a) ∈ R

So R is Reflexive

Step 3 of 5 :

Check for Symmetric

Let a, b ∈ Z and (a, b) ∈ R

⇒ 4 divides a - b

⇒ 4 divides - ( b - a )

⇒ 4 divides ( b - a )

⇒ (b, a) ∈ R

Thus (a, b) ∈ R implies (b, a) ∈ R

So R is symmetric

Step 4 of 5 :

Check for transitive

Let a, b, c ∈ Z

Also let (a, b) ∈ R and (b, c) ∈ R

⇒ 4 divides a - b and 4 divides b - c

⇒ 4 divides ( a - b + b - c )

⇒ 4 divides ( a - c )

⇒ (a, c) ∈ R

Thus (a, b) ∈ R and (b, c) ∈ R implies (a, c) ∈ R

R is transitive

Step 5 of 5 :

Check for equivalence relation

Since R is reflexive , symmetric and transitive

Hence R is an equivalence relation

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