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Class 12MathematicsRelations and FunctionsEquivalence relationHard
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On non-zero real numbers, (aRb) holds when (\frac{a}{b}\in\mathbb{Q}). If (aRb) and (bRc), which expression is useful to prove transitivity?

Answer and explanation

Correct answer: (\frac{a}{c}=\frac{a}{b}\cdot\frac{b}{c})

Step 1: From (aRb), (\frac{a}{b}) is rational, and from (bRc), (\frac{b}{c}) is rational. Step 2: Their product is (\frac{a}{c}). Step 3: A product of rational numbers is rational, proving transitivity.

Related tags

Rational RatioTransitivityProof IdeaEquivalence RelationRelations And FunctionsMathematicsClass 12 Mcq
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10 more MCQs
  1. 1What is the smallest change needed to make R={(1,1),(2,2),(3,3),(1,2),(2,1),(2,3),(3,2)} an equivalence relation on A={1,2,3}?
  2. 2For the relation on A={1,2,3,4} whose equivalence classes are {1,2} and {3,4}, which pair is definitely absent?
  3. 3How many ordered pairs does the equivalence relation with classes {1,2} and {3,4} contain?
  4. 4Which pair must belong to the smallest equivalence relation on A={1,2,3,4} containing R={(1,2),(2,3),(4,4)}?
  5. 5Which pair is not present in the smallest equivalence relation on A={1,2,3,4} containing R={(1,2),(2,3),(3,1)}?
  6. 6Which single pair should be added to R={(1,1),(2,2),(3,3),(1,2)} so that it becomes an equivalence relation on A={1,2,3}?
  7. 7On R, define aRb if and only if a−b is rational. What is the equivalence class of √2?
  8. 8Let R be an equivalence relation on A and let [a] and [b] denote equivalence classes. If aRb, which statement must be true?
  9. 9For the relation on A={1,2,3,4,5,6} defined by “a and b are both multiples of 3 or both are not multiples of 3”, which statement is true?
  10. 10For the relation defined by aRb iff a=b or both a and b are greater than 2 on A={1,2,3,4}, what is the equivalence class of 3?
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