Question

PP4. Check whether R: Z-Z, R= {(a, b)/2 divides a - b) is an equivalence relation.
solutions​

Answer 1

Answer:

R = {(a, b) : 2 divides a b} Check reflexive Since a a = 0 & 2 divides 0 , eg: 0 2 = 0 2 divides a a (a, a) R, R is reflexive.

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Answer 2

SOLUTION

TO CHECK

Check if R: Z → Z, R = {(a, b) |2 divides a-b} is equivalence relation.

EVALUATION

Here the given relation is

R : Z → Z such that R = {(a, b) |2 divides a -b}

CHECKING FOR REFLEXIVE

Let a ∈ Z

Since 2 divides a - a

So (a, a) ∈ R

So R is Reflexive

CHECKING FOR SYMMETRIC

Let a, b ∈ Z and (a, b) ∈ R

⇒2 divides a - b

⇒2 divides - ( b - a )

⇒2 divides ( b - a )

⇒(b, a) ∈ R

Thus (a, b) ∈ R implies (b, a) ∈ R

So R is symmetric

CHECKING FOR TRANSITIVE

Let a, b, c ∈ Z

Also let (a, b) ∈ R and (b, c) ∈ R

⇒2 divides a - b and 2 divides b - c

⇒2 divides ( a - b + b - c )

⇒2 divides ( a - c )

⇒(a, c) ∈ R

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

R is transitive

Hence R is an equivalence relation

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