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NDA 2023 Maths · Mathematics · Sets, Relations and Functions · medium

The number of relations on the set A=\1,2,3\ that are both reflexive and symmetric is

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The number of relations on the set A={1,2,3}A=\{1,2,3\} that are both reflexive and symmetric is
  1. A. 66
  2. B. 88
  3. C. 1616
  4. D. 6464
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Answer: (B) 88

Reflexive forces (1,1),(2,2),(3,3)(1,1),(2,2),(3,3). Symmetry then only allows choosing, for each of the three unordered pairs {1,2},{1,3},{2,3}\{1,2\},\{1,3\},\{2,3\}, whether both ordered pairs are in or out: 23=82^3=8 relations. The count 64=2664=2^6 is for symmetric relations alone and ignores the reflexivity condition.
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