Inclusion–Exclusion with Three Sets Quiz

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| Questions: 15 | Updated: Dec 1, 2025
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1) For three sets A,B,C, the inclusion–exclusion formula for the union is:

Explanation

Singles added, pairs subtracted, triple added.

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About This Quiz
Inclusionexclusion With Three Sets Quiz - Quiz

Are you ready to see how counting changes when three sets overlap? In this quiz, you’ll explore the full three-set inclusion–exclusion formula and learn how each term — singles, pairwise intersections, and the triple intersection — contributes to the final union. You’ll work through problems involving sports, subjects, and numbe... see moredivisibility to practice computing how many elements are in at least one, in none, or in exactly one of the sets. As you solve each question, you’ll get comfortable with the classic pattern “+ singles, – pairs, + triple” and understand why each correction is needed. see less

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2) In the three-set formula, the term with pairwise intersections is:

Explanation

These are exactly the pairwise overlaps.

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3) For three sets, the signs in inclusion–exclusion are: + (singles), – (pairs), + (triple).

Explanation

Signs alternate with intersection size.

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4) Given |A|=20, |B|=18, |C|=15; |A∩B|=5, |A∩C|=4, |B∩C|=3; |A∩B∩C|=2. Then |A∪B∪C| equals:

Explanation

20+18+15 − (5+4+3) + 2 = 43.

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5) Suppose |A|=18, |B|=16, |C|=14; overlaps 5,4,3; and |A∪B∪C|=40. What is |A∩B∩C|?

Explanation

Solve: 40 = 48 − 12 + x → x = 4.

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6) 100 students: 40 football, 35 basketball, 30 baseball, overlaps 10,8,6, triple 4. At least one?

Explanation

40+35+30 − (10+8+6) + 4 = 85.

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7) How many play none of the sports (from Q6)?

Explanation

100 − 85 = 15.

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8) To compute |A∪B∪C| you must know singles, pairwise intersections, triple.

Explanation

All needed in formula.

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9) Quantities always needed for 3-set inclusion–exclusion:

Explanation

These appear directly in the formula.

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10) Match: singles / subtract pairs / add triple

Explanation

Correct roles.

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11) Integers 1..60 divisible by 2,3,5:

Explanation

30+20+12 − (10+6+4) + 2 = 44.

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12) If three sets pairwise disjoint and triple empty, union is sum.

Explanation

All intersections zero.

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13) 22 Algebra,18 Geometry,16 Stats; overlaps 6,5,4; triple 3. At least one?

Explanation

22+18+16 − (6+5+4) + 3 = 44.

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14) Exactly one of A,B,C:

Explanation

Remove pair overlaps twice, add triple three times.

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15) Union never exceeds sum of |A|+|B|+|C|.

Explanation

Disjoint case gives max; overlaps reduce union.

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For three sets A,B,C, the inclusion–exclusion formula for the union...
In the three-set formula, the term with pairwise intersections is:
For three sets, the signs in inclusion–exclusion are: + (singles),...
Given |A|=20, |B|=18, |C|=15; |A∩B|=5, |A∩C|=4, |B∩C|=3;...
Suppose |A|=18, |B|=16, |C|=14; overlaps 5,4,3; and |A∪B∪C|=40....
100 students: 40 football, 35 basketball, 30 baseball, overlaps...
How many play none of the sports (from Q6)?
To compute |A∪B∪C| you must know singles, pairwise intersections,...
Quantities always needed for 3-set inclusion–exclusion:
Match: singles / subtract pairs / add triple
Integers 1..60 divisible by 2,3,5:
If three sets pairwise disjoint and triple empty, union is sum.
22 Algebra,18 Geometry,16 Stats; overlaps 6,5,4; triple 3. At least...
Exactly one of A,B,C:
Union never exceeds sum of |A|+|B|+|C|.
Alert!

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