Show that the following are equivalence relation:R in A = (x ∈ N/x �

1.	Show that the following are equivalence relation:R in A = (x ∈ N/x ≤ 10} given by R = {(a, b) \| a = b}
Answer» a. Since a = a ∴ (a, a) ∈ R ∴ R is reflexive. b. Let (a, b) ∈ R Then a = b ∴ b = a ∴ (b, a) ∈ R ∴ R is symmetric. c. Let (a, b), (b, c) ∈ R Then, a = b, b = c ∴ a = c ∴ (a, c) ∈ R ∴ R is transitive. Thus, R is an equivalence relation.

Show that the following are equivalence relation:R in A = (x ∈ N/x ≤ 10} given by R = {(a, b) | a = b}

Answer»

a. Since a = a

∴ (a, a) ∈ R

∴ R is reflexive.

b. Let (a, b) ∈ R

Then a = b

∴ b = a

∴ (b, a) ∈ R

∴ R is symmetric.

c. Let (a, b), (b, c) ∈ R

Then, a = b, b = c

∴ a = c

∴ (a, c) ∈ R

∴ R is transitive.

Thus, R is an equivalence relation.