Applying Complements and Differences in Set Problems

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Reviewed by Cierra Henderson
Cierra Henderson, MBA |
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Cierra is an educational consultant and curriculum developer who has worked with students in K-12 for a variety of subjects including English and Math as well as test prep. She specializes in one-on-one support for students especially those with learning differences. She holds an MBA from the University of Massachusetts Amherst and a certificate in educational consulting from UC Irvine.
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| Attempts: 11 | Questions: 10 | Updated: Jan 21, 2026
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1) If U has n elements and A has k elements, how many elements are in A′?

Explanation

The complement of A contains all elements not in A, so |A′| = n − k.

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About This Quiz
Applying Complements and Differences In Set Problems - Quiz

What happens when sets overlap, exclude, or leave behind certain elements? Complements and differences are the tools mathematicians use to sort it all out. In this quiz, you’ll move beyond basics to solve applied problems, reasoning through more complex examples of set operations. Take this quiz to test your skills... see moreand see how well you can handle complements and differences in action.
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2) If U has 50 elements and A has 20, B has 15, and A∩B has 5, how many elements are in A′∩B′?

Explanation

|A ∪ B| = 20 + 15 − 5 = 30. So |A′∩B′| = 50 − 30 = 20.

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3) For any set A, what is (A′)′?

Explanation

Taking the complement twice returns the original set A.

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4) For any two sets A and B, what is A − B equivalent to?

Explanation

Set difference means elements in A but not in B → A ∩ B′.

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5) For any two sets A and B, what is (A − B)′ equivalent to?

Explanation

By De Morgan’s law: (A − B)′ = (A ∩ B′)′ = A′ ∪ B.

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6) For any sets A and B, simplify (A′ − B′).

Explanation

A′ − B′ = A′ ∩ (B′)′ = A′ ∩ B.

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7) For any sets A and B, what is (A − A′)?

Explanation

A and A′ are disjoint, so A − A′ = A.

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8) For any sets A and B, what is (A′ − A)?

Explanation

A′ already excludes A, so subtracting A doesn’t change it.

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9) For sets A and B in universal set U, what is (A − B) ∪ (B − A)?

Explanation

This is the symmetric difference definition.

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10) For sets A and B in U, what is (A − B) ∩ (B − A)?

Explanation

The two sets have no overlap, so their intersection is the empty set.

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Cierra Henderson |MBA |
K-12 Expert
Cierra is an educational consultant and curriculum developer who has worked with students in K-12 for a variety of subjects including English and Math as well as test prep. She specializes in one-on-one support for students especially those with learning differences. She holds an MBA from the University of Massachusetts Amherst and a certificate in educational consulting from UC Irvine.
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If U has n elements and A has k elements, how many elements are in...
If U has 50 elements and A has 20, B has 15, and A∩B has 5, how many...
For any set A, what is (A′)′?
For any two sets A and B, what is A − B equivalent to?
For any two sets A and B, what is (A − B)′ equivalent to?
For any sets A and B, simplify (A′ − B′).
For any sets A and B, what is (A − A′)?
For any sets A and B, what is (A′ − A)?
For sets A and B in universal set U, what is (A − B) ∪ (B − A)?
For sets A and B in U, what is (A − B) ∩ (B − A)?
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