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Class 12MathematicsRelations and FunctionsEquivalence relationHard
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On the integers, aRb when a-b is divisible by 5, and aSc when a-c is divisible by 10. What is R∪S equal to?

Answer and explanation

Correct answer: Congruence modulo 5

Relation R consists of integer pairs whose difference is a multiple of 5, while S consists of pairs whose difference is a multiple of 10. Every multiple of 10 is also a multiple of 5, so S is a subset of R. The union of a relation with its subset is the larger relation itself: R∪S=R. Therefore the union is congruence modulo 5, making option A correct; modulo 10 would describe S, not the union.

Related tags

Equivalence RelationUnion Of RelationsCongruence Modulo 5
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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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