Show that the following are equivalence relation:R in A = {x ∈ Z

1.	Show that the following are equivalence relation:R in A = {x ∈ Z \| 0 ≤ x ≤ 12} given by R = {(a, b) / \|a – b\| is a multiple of 4}
Answer» a. Since \|a – a\| is a multiple of 4, (a, a) ∈ R ∴ R is reflexive. b. Let (a, b) ∈ R Then a – b = ±4m, ∴ b – a = ±4m, where m is an integer ∴ (b, a) ∈ R ∴ R is symmetric. c. Let (a, b), (b, c) ∈ R a – b = ± 4m, b – c = ± 4n, ∴ a – c = ±4(m + n), where m, n are integers ∴ (a, c) ∈ R ∴ R is transitive Thus, R is an equivalence relation.

Show that the following are equivalence relation:R in A = {x ∈ Z | 0 ≤ x ≤ 12} given by R = {(a, b) / |a – b| is a multiple of 4}

Answer»

a. Since |a – a| is a multiple of 4,

(a, a) ∈ R

∴ R is reflexive.

b. Let (a, b) ∈ R

Then a – b = ±4m,

∴ b – a = ±4m, where m is an integer

∴ (b, a) ∈ R

∴ R is symmetric.

c. Let (a, b), (b, c) ∈ R

a – b = ± 4m, b – c = ± 4n,

∴ a – c = ±4(m + n), where m, n are integers

∴ (a, c) ∈ R

∴ R is transitive

Thus, R is an equivalence relation.