Let \(\mathcal{R}\) be the set of all binary relations on the set…
2020
Let \(\mathcal{R}\) 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 ________ .
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Correct answer: 0.125
Key facts: a binary relation on a 3-element set is any subset of the 3×3 = 9 ordered pairs.
Total number of relations = 2^9 = 512.
A reflexive relation must include (1,1), (2,2), (3,3); the remaining 6 ordered pairs can be chosen arbitrarily, giving 2^6 = 64 reflexive relations.
Probability = 64 / 512 = 1/8 = 0.125. Rounded to three decimal places: 0.125.
Answer: 0.125