Consider the relations R_{1} and R_{2} defined as aR_{1... | Relation (Mathematics)

Exam ( $1^{st\ }$ Feb $2^{nd\ }$ Shift 2024) Shift 2

Mathematics

Consider the relations

R_{1}

and

R_{2}

defined as

aR_{1}b \Leftrightarrow a^{2} + b^{2} = 1

for all

a,b \in R

and

(a,b)R_{2}(c,d)

\Leftrightarrow a + d = b + c

for all

(a,b),(c,d) \in N \times N

. Then

R_{1}

and

R_{2}

both are equivalence relations

Only

R_{1}

is an equivalence relation

Only

R_{2}

is an equivalence relation

Neither

R_{1}

nor

R_{2}

is an equivalence relation

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Correct Answer:

Explanation

$a_1 b \Leftrightarrow a^2+b^2=1 ; a, b \in R \\$

$(a, b) R_2(c, d) \Leftrightarrow a+d=b+c ;(a, b),(c, d) \in N$

for $\mathbf{R}_{\mathbf{1}}$ : Not reflexive symmetric not transitive for $\mathbf{R}_2$ : $\mathbf{R}_2$ is reflexive, symmetric and transitive Hence only $\mathrm{R}_2$ is equivalence relation.

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