Question

examine whether the relation R= {a, b ∈ Z×Z :|a-b|<=3} is an equivalence relation.​

Answer 1

Answer:

Given:

R={(a,b):a,b∈Z and (a−b) is divisible by 5}.

R=(a,b)

(a−b) is divisible by 5

Reflexive

(a,a)⇒(a−a) is divisible by 5

Symmetric

(a,b)⇔(b,a)

(a−b) is divisible by 5

(b−a) is divisible by 5

Transitive

(a,b),(b,c)⇔(a,c)

(a−b) is divisible by 5

(b−c) is divisible by 5

(a−c) is divisible by 5

R is an equivalent relation on Z