Let R be the set of all binary relations on the set {1, 2, 3}. Suppose a relation is chosen from R at random. The probab...

Question

Let R be the set of all binary relations on the set {1, 2, 3}. Suppose a relation is chosen from R at random. The probability that the chosen relation is reflexive (round off to 3 decimal places) is ______.

Accepted Answer

0.125 A binary relation on a set with n elements is a subset of S x S (which has n^2 pairs). A reflexive relation must include all n diagonal pairs (1,1), (2,2), ..., (n,n). The remaining n^2 - n pairs can be freely included or excluded. So the probability = 2^(n^2 - n) / 2^(n^2) = 1 / 2^n. For n=3, this is 1/8 = 0.125.