Advanced Inclusion–Exclusion and Overlap-Correction Theory Quiz

1) The inclusion–exclusion principle is primarily used to:

Compute intersections of disjoint sets

Count the size of complicated complements

Correct overcounting when sets overlap

Factor polynomials

It adjusts for elements counted more than once in overlaps.

Explanation

It adjusts for elements counted more than once in overlaps.

2) For two finite sets A and B, which rearranged identity follows from inclusion–exclusion?

|A∩B|=|A|+|B|

|A∩B|=|A|+|B|−|A∪B|

|A∩B|=|A∪B|−|A^c|

|A∩B|=|A∪B|

Direct rearrangement from |A∪B| formula.

Explanation

Direct rearrangement from |A∪B| formula.

3) If |A|=25, |B|=30, and |A∩B|=8, then |A∪B| equals:

47

55

63

3

25 + 30 − 8 = 47.

Explanation

25 + 30 − 8 = 47.

4) If |A∪B|=40, |A|=23, and |B|=28, then |A∩B| equals:

9

11

13

17

23 + 28 − 40 = 11.

Explanation

23 + 28 − 40 = 11.

5) In inclusion–exclusion, intersections of an odd number of sets appear with a plus sign.

True

False

Sign is (−1)^(k+1).

Explanation

Sign is (−1)^(k+1).

6) For three sets A, B, C, the term |A∩B∩C| appears with:

A minus sign

A plus sign

No sign

It does not appear

Triple intersections are added.

Explanation

Triple intersections are added.

7) How many integers from 1 to 50 are divisible by 3 or 5?

23

25

27

29

16 + 10 − 3 = 23.

Explanation

16 + 10 − 3 = 23.

8) How many integers from 1 to 60 are divisible by 2 or 3?

30

40

50

45

30 + 20 − 10 = 40.

Explanation

30 + 20 − 10 = 40.

9) For events A and B, inclusion–exclusion gives:

P(A∪B)=P(A)P(B)

P(A∪B)=P(A)+P(B)

P(A∪B)=P(A)+P(B)−P(A∩B)

P(A∪B)=P(A∩B)

Same structure as sets.

Explanation

Same structure as sets.

10) To find the probability that none occur, subtract the union from 1.

True

False

P(none)=1−P(union).

Explanation

P(none)=1−P(union).

11) Inclusion–exclusion for three events A, B, C gives:

Sum of singles only

Sum of singles − pairwise intersections + triple intersection

Product of probabilities

Difference of probabilities

Exact 3-event formula.

Explanation

Exact 3-event formula.

12) Match the step with its role in inclusion–exclusion for three sets.

Add ∣A∣+∣B∣+∣C∣

Subtract ∣A∩B∣, ∣A∩C∣, ∣B∩C∣

Add ∣A∩B∩C∣

Singles add, pairs subtract, triple adds.

Explanation

Singles add, pairs subtract, triple adds.

Submit

13) A standard way to define sets for inclusion–exclusion is to:

Partition arbitrarily

Let each set represent numbers with one property

Use equal-size sets

Use only disjoint sets

Each set corresponds to one property.

Explanation

Each set corresponds to one property.

14) Inclusion–exclusion can count elements satisfying at least r properties.

True

False

Requires grouping intersections appropriately.

Explanation

Requires grouping intersections appropriately.

15) 90 students: 36 French, 32 German, 30 Spanish; intersections 14, 12, 10; triple 8. How many study ≥1?

64

68

70

72

36+32+30 − (14+12+10) + 8 = 70.

Explanation

36+32+30 − (14+12+10) + 8 = 70.

Advanced Inclusion–Exclusion and Overlap-Correction Theory Quiz

1) The inclusion–exclusion principle is primarily used to:

2)

What first name or nickname would you like us to use?

2) For two finite sets A and B, which rearranged identity follows from inclusion–exclusion?

3) If |A|=25, |B|=30, and |A∩B|=8, then |A∪B| equals:

4) If |A∪B|=40, |A|=23, and |B|=28, then |A∩B| equals:

5) In inclusion–exclusion, intersections of an odd number of sets appear with a plus sign.

6) For three sets A, B, C, the term |A∩B∩C| appears with:

7) How many integers from 1 to 50 are divisible by 3 or 5?

8) How many integers from 1 to 60 are divisible by 2 or 3?

9) For events A and B, inclusion–exclusion gives:

10) To find the probability that none occur, subtract the union from 1.

11) Inclusion–exclusion for three events A, B, C gives:

12) Match the step with its role in inclusion–exclusion for three sets.

13) A standard way to define sets for inclusion–exclusion is to:

14) Inclusion–exclusion can count elements satisfying at least r properties.

15) 90 students: 36 French, 32 German, 30 Spanish; intersections 14, 12, 10; triple 8. How many study ≥1?