Counting with Three Overlapping Sets Quiz

1) In the inclusion–exclusion formula for three sets A,B,C, which term has a positive sign?

∣A∩B∣

∣A∩C∣

∣B∩C∣

∣A∩B∩C∣

Only the triple intersection is added.

Explanation

Only the triple intersection is added.

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∣?

48

50

52

54

25+22+20 − (7+6+5) + 3 = 52.

Explanation

25+22+20 − (7+6+5) + 3 = 52.

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

True

False

Signs alternate by intersection size.

Explanation

Signs alternate by intersection size.

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

0

1

2

3

Solving gives x = 1.

Explanation

Solving gives x = 1.

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

In exactly one

In at least one

In all three

In none

Union means at least one.

Explanation

Union means at least one.

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

True

False

Triple is contained in each pairwise.

Explanation

Triple is contained in each pairwise.

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

90

95

100

105

Computed as 95.

Explanation

Computed as 95.

8) How many speak none?

15

20

25

30

120−95=15.

Explanation

120−95=15.

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

Singles

Pairwise

Triple

Complement union

All appear in formula.

Explanation

All appear in formula.

Submit

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

60

62

64

66

Result is 64.

Explanation

Result is 64.

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

True

False

Triple must be empty.

Explanation

Triple must be empty.

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

All intersections empty

Only pairwise empty

Only triple empty

One empty

Means no overlaps.

Explanation

Means no overlaps.

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

Sum pairs

Sum pairs−3 triple

Sum singles

Triple

Subtract triple thrice.

Explanation

Subtract triple thrice.

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

Sum

Sum−pairs

Full IE

Triple only

Matches formula.

Explanation

Matches formula.

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

True

False

Need intersections.

Explanation

Need intersections.

Counting with Three Overlapping Sets Quiz

1) In the inclusion–exclusion formula for three sets A,B,C, which term has a positive sign?

2)

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

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∣?

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

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

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

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

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

8) How many speak none?

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

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

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

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

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

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

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