Consider the relations R1 and R2 defined as a R1 b ⇔ a^2 + b^2 = 1

Question

Consider the relations \(R_{1}\) and \(R_{2}\) defined as \(a R_{1} b\Leftrightarrow a^{2}+b^{2}=1 \) for all \(a, b, \in R\) and \((a, b) R_{2}(c, d)\Leftrightarrow \mathrm{a}+\mathrm{d}=\mathrm{b}+\mathrm{c}\) for all \((\mathrm{a}, \mathrm{b}),(\mathrm{c}, \mathrm{d}) \in \mathrm{N} \times \mathrm{N}\). Then

(1) Only \(R_{1}\) is an equivalence relation

(2) Only \(R_{2}\) is an equivalence relation

(3) \(R_{1}\) and \(R_{2}\) both are equivalence relations

(4) Neither \(R_{1}\) nor \(R_{2}\) is an equivalence relation

Answer 1

Correct option is (2) Only \(\mathrm R_{2}\) is an equivalence relation

\(\mathrm{aR}_{1} \mathrm{~b} \Leftrightarrow \mathrm{a}^{2}+\mathrm{b}^{2}=1 ; \mathrm{a}, \mathrm{b} \in \mathrm{R}\)

\((\mathrm{a}, \mathrm{b}) \mathrm{R}_{2}(\mathrm{c}, \mathrm{d}) \Leftrightarrow \mathrm{a}+\mathrm{d}=\mathrm{b}+\mathrm{c} ;(\mathrm{a}, \mathrm{b}),(\mathrm{c}, \mathrm{d}) \in \mathrm{N}\)

for \(\mathrm R_{1}\): Not reflexive symmetric not transitive

for \(\mathrm R_{2}\): \(\mathrm R_{2}\) is reflexive, symmetric and transitive

Hence only \(\mathrm R_{2}\) is equivalence relation.