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Subjects
Class 12MathematicsRelations and FunctionsTransitive relationHard
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If (R={(a,b):a,b\in\mathbb{R}\text{ and }a\le b+2}), is (R) transitive?

Answer and explanation

Correct answer: No

Step 1: From (a\le b+2) and (b\le c+2), we only get (a\le c+4). Step 2: Transitivity requires (a\le c+2), which need not hold. Take (a=4), (b=2), (c=0). Step 3: When adding inequalities, the bound may change.

Related tags

Inequality CounterexampleNon TransitiveHardTransitive RelationRelations And FunctionsMathematicsClass 12 Mcq
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10 more MCQs
  1. 1Which transitivity requirement fails for R={(1,1),(2,2),(3,3),(4,4),(1,2),(2,3),(3,4),(1,3),(2,4)} on A={1,2,3,4}?
  2. 2On A={1,2,3}, what is the smallest addition to R={(1,2),(2,3)} that makes R transitive?
  3. 3If R and S are transitive relations on the same set, which statement about their union R∪S is always valid?
  4. 4For R={(1,1),(2,2),(3,3),(4,4),(1,2),(2,1),(2,4),(4,2)} on A={1,2,3,4}, which missing ordered pair directly witnesses that R is not transitive?
  5. 5Which relation on A={1,2,3} is transitive but not symmetric?
  6. 6If R and S are transitive relations on A, which implication proves that R∩S is transitive?
  7. 7Which pair of transitive relations gives a union that is not transitive?
  8. 8Which is the correct definition of a transitive relation?
  9. 9On (A={1,2,3}), (R={(1,2),(2,3),(1,3)}) is given. Is (R) transitive?
  10. 10On (A={1,2,3}), (R={(1,2),(2,3)}). Why is (R) not transitive?
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