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Class 12MathematicsRelations and FunctionsSymmetric relationHard
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For every symmetric relation (R), will (R\circ R) also be symmetric?

Answer and explanation

Correct answer: Yes, always

Step 1: Suppose ((a,c)\in R\circ R). Then there is some (b) such that (aRb) and (bRc). Step 2: Since (R) is symmetric, (cRb) and (bRa), so ((c,a)\in R\circ R). Step 3: In such proofs, reuse the intermediate element in the reverse direction.

Related tags

CompositionSymmetric RelationProof Based McqRelations And FunctionsMathematicsClass 12 Mcq
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10 more MCQs
  1. 1Let A={1,2,3}. If a symmetric relation R on A must contain (1,2), how many such relations are possible?
  2. 2On A={1,2}, which property is satisfied by R={(1,1),(1,2),(2,1)}?
  3. 3If A has n elements, what is the number of symmetric relations on A?
  4. 4Which relation on A={1,2,3} is symmetric but not transitive?
  5. 5For a five-element set A, how many symmetric relations on A contain no diagonal ordered pair (a,a)?
  6. 6On the set (A={1,2,3}), the relation (R={(1,2),(2,1),(3,3)}) is given. What type of relation is it?
  7. 7If a relation (R) contains ((4,7)) and (R) is symmetric, which ordered pair must definitely be in (R)?
  8. 8On (A={1,2}), (R={(1,1),(1,2)}) is given. Which pair must be added to make it symmetric?
  9. 9Which statement correctly identifies a symmetric relation?
  10. 10On the set (A={1,2,3}), which relation is not symmetric?
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