Let S be a set of n elements. The number of ordered pairs in the largest and the smallest equivalence relations on S are: a. n and n b. n2 and n c. n and 0 d. n and 1
Question
Let S be a set of n elements. The number of ordered pairs in the largest and the smallest equivalence relations on S are: a. n and n b. n2 and n c. n and 0 d. n and 1
Solution
The largest equivalence relation on a set S of n elements is the universal relation, which includes every possible ordered pair of elements. Since each element can pair with every element including itself, there are n^2 such pairs.
The smallest equivalence relation on a set S of n elements is the identity relation, where each element is only related to itself. Since there are n elements, there are n such pairs.
So, the correct answer is b. n^2 and n.
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