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# 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. - CBSE (Arts) Class 12 - Mathematics

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#### Question

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.

#### Solution

We observe the following properties of S.

Reflexivity :

Let a be an arbitrary element of R. Then,

∈ R

⇒  a2+a2 ≠ 1∀ R

⇒ (a, a∉ S

So, S is not reflexive on R.

Symmetry: Let (a, b∈ R

⇒ a2 b21

⇒ b2a21

⇒ (b, a∈ S for all a, ∈ R

So, S is symmetric on R.

Transitivity :

Let (a, b) and (b, c∈ S

⇒ a2+b2=1 and b2+c2=1

Adding the above two, we get

a2+c2=22b21 for all a, b,  ∈ R

So, S is not transitive on R.

Hence, S is not an equivalence relation on R.

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Solution 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. Concept: Types of Relations.
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