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Subjects
Class 12MathematicsRelations and FunctionsSymmetric relationHardLevel 10
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If A has n elements, how many symmetric relations on A contain no diagonal pair (a,a)?

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Answer and explanation

Correct answer: 2^(n(n−1)/2)

Option A is correct. In a symmetric relation, each off-diagonal pair {a,b} must be selected as a complete two-way choice: either both (a,b) and (b,a) are included or neither is included. There are n(n−1)/2 such unordered pairs. Since diagonal pairs are forbidden and each off-diagonal pair can be chosen independently, the total number is 2^(n(n−1)/2).

Related tags

Symmetric RelationRelationsCombinatoricsDiagonal Pairs

Frequently asked questions

What is the correct answer to this question?

2^(n(n−1)/2)

Why is this the correct answer?

Option A is correct. In a symmetric relation, each off-diagonal pair {a,b} must be selected as a complete two-way choice: either both (a,b) and (b,a) are included or neither is included. There are n(n−1)/2 such unordered pairs. Since diagonal pairs are forbidden and each off-diagonal pair can be chosen independently, the total number is 2^(n(n−1)/2).

Which subject and chapter does this question cover?

This is a Class 12 Mathematics question. Chapter: Relations and Functions. Topic: Symmetric relation.

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10 more MCQs
  1. 1On the set (A={1,2,3}), the relation (R={(1,2),(2,1),(3,3)}) is given. What type of relation is it?
  2. 2If a relation (R) contains ((4,7)) and (R) is symmetric, which ordered pair must definitely be in (R)?
  3. 3On (A={1,2}), (R={(1,1),(1,2)}) is given. Which pair must be added to make it symmetric?
  4. 4Which statement correctly identifies a symmetric relation?
  5. 5On the set (A={1,2,3}), which relation is not symmetric?
  6. 6If (R={(2,2),(3,5),(5,3)}), which statement about (R) is correct?
  7. 7On (A={1,2,3}), (R={(1,2),(2,1),(2,3)}). Why is (R) not symmetric?
  8. 8Is the empty relation (R=\varnothing) on a non-empty set (A) symmetric or not?
  9. 9Is the universal relation (R=A \times A) on a set (A) symmetric or not?
  10. 10On (A={1,2,3}), (R={(1,1),(2,2),(3,3)}). Why is this relation symmetric?
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