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CA Foundation · Quantitative Aptitude · Sets, Relations and Functions, Limits and Continuity

On the set of integers, a relation R is defined by a R b if (a − b) is divisible by 5. Which of the following is correct?

R is an equivalence relation. Zero is divisible by 5 so it is reflexive, the negative of a multiple of 5 is a multiple of 5 so it is symmetric, and the sum of two multiples of 5 is a multiple of 5 so it is transitive.

  1. AR is reflexive and symmetric but not transitive
  2. BR is symmetric only
  3. CR is an equivalence relationCorrect
  4. DR is reflexive and transitive but not symmetric

Explanation

a − a = 0 is divisible by 5, so R is reflexive. If a − b is divisible by 5, so is b − a, so R is symmetric. If a − b and b − c are multiples of 5, their sum a − c is also a multiple of 5, so R is transitive. All three hold, so R is an equivalence relation.

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