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How many relations are there on a set with n elements that are symmetric and a set with n elements that are reflexive and symmetric ?

  • A.\(2^{n(n+1)/2}\) and \(2^{n}\cdot 3^{n(n-1)/2}\)
  • B.\(3^{n(n-1)/2}\) and \(2^{n(n-1)}\)
  • C.\(2^{n(n+1)/2}\) and \(3^{n(n-1)/2}\)
  • D.\(2^{n(n+1)/2}\) and \(2^{n(n-1)/2}\)

Answer: D

A symmetric relation on n elements is fixed by choosing, for each of the n diagonal pairs and the n(n-1)/2 unordered off-diagonal pairs, whether it is in the relation, which gives \(2^{n(n+1)/2}\) relations. A reflexive and symmetric relation must contain all the diagonal pairs, so only the n(n-1)/2 off-diagonal pairs are free, which gives \(2^{n(n-1)/2}\). So the answers are \(2^{n(n+1)/2}\) and \(2^{n(n-1)/2}\), option (D). The other options use factors of 3 or other exponents, which do not count these relations.

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