If A has n elements, what is the number of symmetric relations on A?
Answer and explanation
Correct answer: 2^{n(n+1)/2}
A symmetric relation is determined by its n diagonal pairs and by the choices for each unordered pair of distinct elements. There are n(n−1)/2 such unordered pairs, and each may be included in both directions or omitted, giving two choices. The diagonal pairs also have two independent choices. Therefore the total is 2^n·2^{n(n−1)/2}=2^{n(n+1)/2}, so A is correct.
Frequently asked questions
What is the correct answer to this question?
2^{n(n+1)/2}
Why is this the correct answer?
A symmetric relation is determined by its n diagonal pairs and by the choices for each unordered pair of distinct elements. There are n(n−1)/2 such unordered pairs, and each may be included in both directions or omitted, giving two choices. The diagonal pairs also have two independent choices. Therefore the total is 2^n·2^{n(n−1)/2}=2^{n(n+1)/2}, so A is correct.
Which subject and chapter does this question cover?
This is a Class 12 Mathematics question. Chapter: Relations and Functions. Topic: Types of relations.
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