Question

Answer the following:

Show that the following is an equivalence relation

R in A = {x ∈ Z | 0 ≤ x ≤ 12} given by R = {(a, b)/|a − b| is a multiple of 4}

Answer 1

A = {x ∈ Z | 0 ≤ x ≤ 12} 

R = {(a, b)/|a − b| is a multiple of 4; a, b ∈ A} 

|a − a| = 0 is a multiple of 4

∴ aRa ∀ a∈A 

∴ R is reflexive

Let aRb

∴ |a − b| is a multiple of 4

∴ |b − a| = |a − b|

∴ |b − a| is a multiple of 4

∴ aRb ⇒ bRa ∀a, b ∈ A 

∴ R is symmetric

Let aRb and bRc

∴ |a − b| and |b − c| are multiples of 4

∴ a − b = 4m, b − c = 4n; m, n ∈ Z

a − c = (a − b) + (b − c) = 4m + 4n 

= 4(m + n); (m + n) ∈ Z

∴ |a − c| is a multiple of 4

∴ aRc

∴ aRb, bRc ⇒ aRc ∀a, b, c ∈ A

∴ R is transitive

∵ R is reflexive, symmetric and transitive

∴ R is an equivalence relation.