Question

Which one of the following relations on the set of real numbers R is an equivalence relation?

Options

• aR₁b `⇔` |a| = |b|

• aR₂b `⇔` |a| ≥ |b|

• aR₃b `⇔` a divides b

• aR₄b `⇔` |a| < |b|

Answer 1

aR₁b `⇔` |a| = |b|

Explanation:

The relation R₁ is an equivalence relation

∀a ∈ R, |a| = |a|, i.e. aR₁a ∀a ∈ R

∴ R₁ is reflexive.

Again ∀ a, b ∈ R, |a| = |b| ⇒ |b| = |a|

∴ aR₁b ⇒ bR₁a. Therefore R is symmetric.

Also, ∀ a, b, c ∈ R, |a| = |b| and |b| = |c|

⇒ |a| = |c| ∴ aR₁b and bR₁c ⇒ aR₁c

⇒ R₁ is transitive

R₂ and R₃ are not symmetric.

R₄ is neither reflexive nor symmetric.