Counting with Three Overlapping Sets Quiz

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| Questions: 15 | Updated: Dec 1, 2025
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1) In the inclusion–exclusion formula for three sets A,B,C, which term has a positive sign?

Explanation

Only the triple intersection is added.

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About This Quiz
Counting With Three Overlapping Sets Quiz - Quiz

Think you can keep track of all the overlaps when three sets intersect? This graduate-level quiz challenges you to apply the three-set inclusion–exclusion formula in more involved situations. You’ll analyze language groups, course enrollments, and divisibility questions where elements can belong to one, two, or all three sets at once.... see moreAlong the way, you’ll practice distinguishing between “at least one,” “none,” “exactly two,” and full triple intersections, and you’ll reason about what must be true when certain equalities hold. Step by step, you’ll sharpen your ability to translate word problems into set expressions — and use inclusion–exclusion to compute the right counts every time. see less

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2) Given ∣A∣=25, ∣B∣=22, ∣C∣=20; ∣A∩B∣=7, ∣A∩C∣=6, ∣B∩C∣=5; and ∣A∩B∩C∣=3, what is ∣A∪B∪C∣?

Explanation

25+22+20 − (7+6+5) + 3 = 52.

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3) For three sets, the inclusion–exclusion formula can be viewed as “singles – pairs + triple”.

Explanation

Signs alternate by intersection size.

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4) Three sets satisfy ∣A∣=18, ∣B∣=16, ∣C∣=14... What is ∣A∩B∩C∣?

Explanation

Solving gives x = 1.

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5) In words, ∣A∪B∪C∣ counts the number of elements that are:

Explanation

Union means at least one.

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6) The triple intersection ∣A∩B∩C∣ can never exceed any of the pairwise intersections.

Explanation

Triple is contained in each pairwise.

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7) How many speak at least one of the three languages?

Explanation

Computed as 95.

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8) How many speak none?

Explanation

120−95=15.

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9) To compute ∣A∪B∪C∣ by inclusion–exclusion, which must you know?

Explanation

All appear in formula.

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10) How many integers from 1 to 90 are divisible by 2, 3, or 7?

Explanation

Result is 64.

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11) If three sets are pairwise disjoint but have a nonempty triple intersection, this is impossible.

Explanation

Triple must be empty.

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12) If ∣A∪B∪C∣=∣A∣+∣B∣+∣C∣, what must be true?

Explanation

Means no overlaps.

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13) For three sets A,B,C, the number of elements in exactly two sets is:

Explanation

Subtract triple thrice.

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14) In a probability setting, P(A∪B∪C) equals:

Explanation

Matches formula.

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15) Knowing only union and singles is sufficient to recover all intersection sizes.

Explanation

Need intersections.

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In the inclusion–exclusion formula for three sets A,B,C, which term...
Given ∣A∣=25, ∣B∣=22, ∣C∣=20; ∣A∩B∣=7,...
For three sets, the inclusion–exclusion formula can be viewed as...
Three sets satisfy ∣A∣=18, ∣B∣=16, ∣C∣=14... What is...
In words, ∣A∪B∪C∣ counts the number of elements that are:
The triple intersection ∣A∩B∩C∣ can never exceed any of the...
How many speak at least one of the three languages?
How many speak none?
To compute ∣A∪B∪C∣ by inclusion–exclusion, which must you...
How many integers from 1 to 90 are divisible by 2, 3, or 7?
If three sets are pairwise disjoint but have a nonempty triple...
If ∣A∪B∪C∣=∣A∣+∣B∣+∣C∣, what must be true?
For three sets A,B,C, the number of elements in exactly two sets is:
In a probability setting, P(A∪B∪C) equals:
Knowing only union and singles is sufficient to recover all...
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