Let S be a relation on the set R of all real numbers defined by S = {(a, b) ∈ R × R : a 2 + b 2 = 1}. Prove that S is not an equivalence relation on R.

Question

Philoid · Accepted Answer

S = {(a, b) : a² + b² =1}

Proof :

To prove that relation is not equivalence, we need to prove that it is either not reflexive, or not symmetric or not transitive.

Reflexivity : For Reflexivity, we need to prove that-

(a, a) ∈ R

Let a = 1/2, a ∈ R

Then,

⇒ (a, a) ∉ S

⇒ S is not reflexive

Hence, S is not an equivalence relation on R.

Let S be a relation on the set R of all real numbers defined by S = {(a, b) ∈ R × R : a2 + b2 = 1}. Prove that S is not an equivalence relation on R.

Let S be a relation on the set R of all real numbers defined by S = {(a, b) ∈ R × R : a² + b² = 1}. Prove that S is not an equivalence relation on R.