Let S be the set of all real numbers and let
R = {(a, b) : a, b ∈ S and a = ± b}.
Show that R is an equivalence relation on S.
In order to show R is an equivalence relation we need to show R is Reflexive, Symmetric and Transitive.
Given that, ∀ a, b ∈ S, R = {(a, b) : a = ± b }
Now,
R is Reflexive if (a,a) ∈ R ∀ a ∈ S
For any a ∈ S, we have
a = ±a
⇒ (a,a) ∈ R
Thus, R is reflexive.
R is Symmetric if (a,b) ∈ R ⇒ (b,a) ∈ R ∀ a,b ∈ S
(a,b) ∈ R
⇒ a = ± b
⇒ b = ± a
⇒ (b,a) ∈ R
Thus, R is symmetric .
R is Transitive if (a,b) ∈ R and (b,c) ∈ R ⇒ (a,c) ∈ R ∀ a,b,c ∈ S
Let (a,b) ∈ R and (b,c) ∈ R ∀ a, b,c ∈ S
⇒ a = ± b and b = ± c
⇒ a = ± c
⇒ (a, c) ∈ R
Thus, R is transitive.
Hence, R is an equivalence relation.