Set Theory → Union and Intersection (Advanced)

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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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Quizzes Created: 8156 | Total Attempts: 9,588,805
| Attempts: 15 | Questions: 10 | Updated: Jan 20, 2026
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Question 1 / 11
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1) Simplify (A ∪ B ∪ C) ∩ (A ∪ B′ ∪ C).

Explanation

Both terms contain A ∪ C; the variation in B cancels.

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About This Quiz
Set Theory Union and Intersection (Advanced) - Quiz

Take your set reasoning further! In this quiz, you’ll apply unions and intersections to advanced problems, combining logic and calculation. Take this quiz to master higher-level set theory challenges.

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2) Simplify (A ∪ B) ∩ (B ∪ C).

Explanation

Distributive law: (A ∪ B) ∩ (B ∪ C) = B ∪ (A ∩ C).

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3) Simplify (A ∩ B) ∪ (A ∩ B′) ∪ B.

Explanation

(A ∩ B) ∪ (A ∩ B′) = A; then A ∪ B remains.

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4) Simplify (A ∪ B′) ∩ (A ∪ B).

Explanation

Distribute: (A ∪ B′) ∩ (A ∪ B) = A ∪ (B ∩ B′) = A ∪ ∅ = A.

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5) Simplify ((A ∪ B)′ ∪ C)′ using De Morgan’s Laws.

Explanation

Outer complement turns union into intersection; inner De Morgan gives (A ∪ B); result: (A ∪ B) ∩ C′.

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6) Simplify (A ∩ B′) ∪ (A′ ∩ B) ∪ (A ∩ B).

Explanation

All disjoint regions of A and B are included; together they form A ∪ B.

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7) Simplify (A ∩ (B ∪ C)) ∪ (A′ ∩ B).

Explanation

This expression is already simplified; nothing cancels further.

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8) Simplify ((A ∩ B)′ ∩ A).

Explanation

(A ∩ B)′ = A′ ∪ B′; intersecting with A gives (A ∩ B′).

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9) Simplify A ∪ (B ∩ (A ∪ C)).

Explanation

B ∩ (A ∪ C) = (B ∩ A) ∪ (B ∩ C); union with A absorbs (B ∩ A), leaving A ∪ (B ∩ C).

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10) Simplify (A′ ∪ B′)′.

Explanation

By De Morgan: (A′ ∪ B′)′ = A ∩ B.

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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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Simplify (A ∪ B ∪ C) ∩ (A ∪ B′ ∪ C).
Simplify (A ∪ B) ∩ (B ∪ C).
Simplify (A ∩ B) ∪ (A ∩ B′) ∪ B.
Simplify (A ∪ B′) ∩ (A ∪ B).
Simplify ((A ∪ B)′ ∪ C)′ using De Morgan’s Laws.
Simplify (A ∩ B′) ∪ (A′ ∩ B) ∪ (A ∩ B).
Simplify (A ∩ (B ∪ C)) ∪ (A′ ∩ B).
Simplify ((A ∩ B)′ ∩ A).
Simplify A ∪ (B ∩ (A ∪ C)).
Simplify (A′ ∪ B′)′.
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