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Class 12MathematicsRelations and FunctionsSymmetric relationMediumLevel 12
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If A has 5 elements, what is the number of symmetric relations on A?

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Answer and explanation

Correct answer: 2¹⁵

A symmetric relation is determined by independent choices on the diagonal and on unordered pairs of distinct elements. For n elements, the number of such choices is n + n(n−1)/2 = n(n+1)/2. With n = 5, the exponent is 5×6/2 = 15. Each choice is binary: include the corresponding diagonal pair, or include both members of an off-diagonal pair; alternatively, omit it. Therefore the number of symmetric relations is 2¹⁵, so option A is correct. The exponents 20 and 25 do not count the required paired positions.

Related tags

Symmetric RelationFinite Set CountingRelations And Functions

Frequently asked questions

What is the correct answer to this question?

2¹⁵

Why is this the correct answer?

A symmetric relation is determined by independent choices on the diagonal and on unordered pairs of distinct elements. For n elements, the number of such choices is n + n(n−1)/2 = n(n+1)/2. With n = 5, the exponent is 5×6/2 = 15. Each choice is binary: include the corresponding diagonal pair, or include both members of an off-diagonal pair; alternatively, omit it. Therefore the number of symmetric relations is 2¹⁵, so option A is correct. The exponents 20 and 25 do not count the required paired positions.

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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