How many equivalence relations on a 4-element set have exactly 2 equivalence classes?
Answer and explanation
Correct answer: 7
There is a one-to-one correspondence between equivalence relations on a set and partitions of that set. Therefore the question asks for the number of partitions of 4 distinct elements into exactly 2 nonempty, unlabeled blocks, namely the Stirling number of the second kind S(4,2). It can be calculated by listing block-size possibilities: a 1+3 split gives C(4,1) / 2 = 2 partitions, and a 2+2 split gives C(4,2) / 2 = 3 partitions. Thus the total is 2 + 3 = 5, not 7. However, the supplied correct option B (7) is mathematically inconsistent; the valid answer 5 is absent. This item must be revised before passing.
Frequently asked questions
What is the correct answer to this question?
7
Why is this the correct answer?
There is a one-to-one correspondence between equivalence relations on a set and partitions of that set. Therefore the question asks for the number of partitions of 4 distinct elements into exactly 2 nonempty, unlabeled blocks, namely the Stirling number of the second kind S(4,2). It can be calculated by listing block-size possibilities: a 1+3 split gives C(4,1) / 2 = 2 partitions, and a 2+2 split gives C(4,2) / 2 = 3 partitions. Thus the total is 2 + 3 = 5, not 7. However, the supplied correct option B (7) is mathematically inconsistent; the valid answer 5 is absent. This item must be revised before passing.
Which subject and chapter does this question cover?
This is a Class 12 Mathematics question. Chapter: Relations and Functions. Topic: Equivalence relation.
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