<p><strong>Let <math><mrow><mi>A</mi><mo>=</mo><mrow><mo>{</mo><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mo>…</mo><mo>,</mo><mn>9</mn></mrow><mo>}</mo></mrow></mrow></math> and <math><mrow><mi>R</mi><mo> </mo></mrow></math> be the relation in <math><mrow><mi>A</mi><mo>×</mo><mi>A</mi></mrow></math> defined by <math><mrow><mrow><mo>(</mo><mrow><mi>a</mi><mo>,</mo><mi>b</mi></mrow><mo>)</mo></mrow><mi>R</mi><mrow><mo>(</mo><mrow><mi>c</mi><mo>,</mo><mi>d</mi></mrow><mo>)</mo></mrow></mrow></math> if <math><mrow><mo> </mo><mo> </mo><mi>a</mi><mo>+</mo><mi>d</mi><mo>=</mo><mi>b</mi><mo>+</mo><mi>c</mi></mrow></math> for <math><mrow><mrow><mo>(</mo><mrow><mi>a</mi><mo>,</mo><mi>b</mi></mrow><mo>)</mo></mrow><mo>,</mo><mrow><mo>(</mo><mrow><mi>c</mi><mo>,</mo><mi>d</mi></mrow><mo>)</mo></mrow><mo> </mo><mi>i</mi><mi>n</mi><mo> </mo><mi>A</mi><mo>×</mo><mi>A</mi></mrow></math> . Prove that <math><mrow><mo> </mo><mi>R</mi></mrow></math> is an equivalence relation and also obtain the equivalent class <math><mrow><mrow><mo>[</mo><mrow><mrow><mo>(</mo><mrow><mn>2</mn><mo>,</mo><mn>5</mn></mrow><mo>)</mo></mrow></mrow><mo>]</mo></mrow></mrow></math> .</strong></p>

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Ncert Solutions Maths class 12th Maths Class 12th Maths NCERT Exemplar Solutions Class 12th Chapter One Relations and Functions

Let $A = {1, 2, 3, \dots, 9}$ and $R$ be the relation in $A \times A$ defined by $(a, b) R (c, d)$ if $a + d = b + c$ for $(a, b), (c, d) i n A \times A$ . Prove that $R$ is an equivalence relation and also obtain the equivalent class $[(2, 5)]$ .

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alok kumar singh | Contributor-Level 10

3 months ago

This is a Long Answer Type Question as classified in NCERT Exemplar

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$\begin{array}{l} H e r e, A = {1, 2, 3, \dots, 9} \\ a n d R \to A \times A d e f i n e d b y (a, b) R (c, d) \Rightarrow a + d = b + c \\ \forall (a, b), (c, d) \in A \times A \\ F o r r e f l e x i v e : (a, b) R (a, b) = a + b = b + a \forall a, b \in A w h i c h i s t r u e . S o, R i s r e f l e x i v e . \\ F o r s y m m e t r i c : (a, b) R (c, d) = (c, d) R (a, b) \\ L . H . S . a + d = b + c \\ R . H . S . c + b = d + a \\ L . H . S . = R . H . S . S o, R i s s y m m e t r i c . \\ F o r t r a n s i t i v e : (a, b) R (c, d) = (c, d) R (e, f) \Leftrightarrow (a, b) R (e, f) \\ \Rightarrow a + d = b + c a n d c + f = d + e \\ \Rightarrow a + d = b + c a n d d + e = c + f \\ \Rightarrow (a + d) - (d + e) = (b + c) - (c + f) \\ \Rightarrow a - e = b - f \\ \Rightarrow a + f = b + e \\ \Rightarrow (a, b) R (e, f) \\ S o, R i s t r a n s i t i v e . \\ H e n c e, R i s a n e q u i v a l e n c e r e l a t i o n . \\ E q u i v a l e n t c l a s s o f {(2, 5)} i s {(1, 4), (2, 5), (3, 6), (4, 7), (5, 8), (6, 9)} \end{array}$

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Let $A = {1, 2, 3, \dots, 9}$ and $R$ be the relation in $A \times A$ defined by $(a, b) R (c, d)$ if $a + d = b + c$ for $(a, b), (c, d) i n A \times A$ . Prove that $R$ is an equivalence relation and also obtain the equivalent class $[(2, 5)]$ .

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Let A={1,2,3,…,9} and R be the relation in A×A defined by (a,b)R(c,d) if a+d=b+c for (a,b),(c,d) in A×A . Prove that R is an equivalence relation and also obtain the equivalent class [(2,5)] .

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Let $A = {1, 2, 3, \dots, 9}$ and $R$ be the relation in $A \times A$ defined by $(a, b) R (c, d)$ if $a + d = b + c$ for $(a, b), (c, d) i n A \times A$ . Prove that $R$ is an equivalence relation and also obtain the equivalent class $[(2, 5)]$ .