Inclusion–Exclusion Principle Essentials Quiz

1) The Inclusion–Exclusion Principle is primarily used to:

Count derivatives

Determine sizes of overlapping sets

Evaluate integrals

Compute eigenvalues

It corrects for double-counting when sets overlap.

Explanation

It corrects for double-counting when sets overlap.

2) For sets A and B, Inclusion–Exclusion states:

|A∪B|=|A|+|B|

|A∪B|=|A|+|B|−|A∩B|

|A∪B|=|A|−|B|

|A∪B|=|A∩B|

The overlap must be subtracted once to avoid double-counting.

Explanation

The overlap must be subtracted once to avoid double-counting.

3) If |A|=50, |B|=40, and |A∩B|=10, then |A∪B| equals:

80

100

50

40

50 + 40 − 10 = 80.

Explanation

50 + 40 − 10 = 80.

4) For three sets A, B, C, the term |A∩B∩C| is:

Added

Subtracted

Ignored

Squared

Triple intersections are added back.

Explanation

Triple intersections are added back.

5) The Inclusion–Exclusion Principle is most helpful when:

Sets are disjoint

Intersections are empty

Sets overlap

Only one set is involved

Overlap requires adjusting counts.

Explanation

Overlap requires adjusting counts.

6) Which are components of the three-set Inclusion–Exclusion formula?

|A|+|B|+|C|

-(|A∩B| + |A∩C| + |B∩C|)

+|A∩B∩C|

-|A∩B∩C|

Singles added, pairs subtracted, triple added.

Explanation

Singles added, pairs subtracted, triple added.

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7) Inclusion–Exclusion can be used to compute:

Number of integers divisible by 2 or 3

Number of students taking two or more classes

The derivative of a function

The probability of at least one event occurring

All involve counting unions or probabilities of unions.

Explanation

All involve counting unions or probabilities of unions.

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8) Select all correct statements:

Inclusion–Exclusion alternates between addition and subtraction

Inclusion–Exclusion formulas become shorter for more sets

Inclusion–Exclusion can be applied in both counting and probability

Inclusion–Exclusion requires knowing intersection sizes

It alternates signs, works in probability, and needs intersection information.

Explanation

It alternates signs, works in probability, and needs intersection information.

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9) True or False: For two sets, the intersection is subtracted once.

True

False

The overlap must be removed once to correct double-counting.

Explanation

The overlap must be removed once to correct double-counting.

10) True or False: Inclusion–Exclusion does not require knowledge of intersections.

True

False

Intersection sizes are essential.

Explanation

Intersection sizes are essential.

11) True or False: When sets are disjoint, Inclusion–Exclusion reduces to simple addition.

True

False

If intersections are empty, nothing needs subtracting.

Explanation

If intersections are empty, nothing needs subtracting.

12) True or False: The more overlap sets have, the more important Inclusion–Exclusion becomes.

True

False

Greater overlap increases overcounting.

Explanation

Greater overlap increases overcounting.

13) Match the Following

Counting each set individually

Removing double-counted overlaps

Adding back the triple overlap

Singles added, pairs subtracted, triples added.

Explanation

Singles added, pairs subtracted, triples added.

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14) Match the Following

Count each group on its own

Remove the overlaps between each pair of groups

Add back the overlap shared by all groups

Standard Inclusion–Exclusion ordering.

Explanation

Standard Inclusion–Exclusion ordering.

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15) A school has 120 students, 70 take English, 60 take History, 30 take both. How many take at least one?

90

100

120

70

70 + 60 − 30 = 100.

Explanation

70 + 60 − 30 = 100.

Inclusion–Exclusion Principle Essentials Quiz

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