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Relations and Functions
NCERT Class 12
CBSE
Grade 12
Question:
Let $A = \mathbb{N} \times \mathbb{N}$ and $R$ be the relation on $A$ defined by $(a, b) R (c, d) \iff ad(b + c) = bc(a + d)$. Show that $R$ is an equivalence relation.
Step-by-Step Solution
Equivalent condition: $\dfrac{1}{a} + \dfrac{1}{d} = \dfrac{1}{b} + \dfrac{1}{c}$. [1.0 Mark] Reflexive: $\dfrac{1}{a} + \dfrac{1}{b} = \dfrac{1}{b} + \dfrac{1}{a} \Rightarrow (a,b) R (a,b)$. [1.0 Mark] Symmetric: $\dfrac{1}{a} + \dfrac{1}{d} = \dfrac{1}{b} + \dfrac{1}{c} \Rightarrow (c,d) R (a,b)$. [1.5 Marks] Transitive: Adding the two system equations gives $\dfrac{1}{a} + \dfrac{1}{f} = \dfrac{1}{b} + \dfrac{1}{e} \Rightarrow (a,b) R (e,f)$. [1.5 Marks]
--- 🎯 Official CBSE Marking Scheme: Simplifying relation to reciprocal sum form: 1.0 Mark Proving Reflexivity: 1.0 Mark Proving Symmetry: 1.5 Marks Proving Transitivity to conclude Equivalence Relation: 1.5 Marks
Correct Answer:
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