Inclusion–Exclusion with Three Sets Quiz

1) For three sets A,B,C, the inclusion–exclusion formula for the union is:

|A|+|B|+|C|

|A|+|B|+|C|-|A∩B|-|A∩C|-|B∩C|

|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.

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

|A|+|B|+|C|

|A∩B|+|A∩C|+|B∩C|

|A∩B∩C|

|A∪B∪C|

These are exactly the pairwise overlaps.

Explanation

These are exactly the pairwise overlaps.

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

True

False

Signs alternate with intersection size.

Explanation

Signs alternate with intersection size.

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:

38

41

43

45

20+18+15 − (5+4+3) + 2 = 43.

Explanation

20+18+15 − (5+4+3) + 2 = 43.

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

2

3

4

5

Solve: 40 = 48 − 12 + x → x = 4.

Explanation

Solve: 40 = 48 − 12 + x → x = 4.

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

80

85

90

95

40+35+30 − (10+8+6) + 4 = 85.

Explanation

40+35+30 − (10+8+6) + 4 = 85.

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

10

15

20

25

100 − 85 = 15.

Explanation

100 − 85 = 15.

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

True

False

All needed in formula.

Explanation

All needed in formula.

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

|A|,|B|,|C|

|A∩B|,|A∩C|,|B∩C|

|A∩B∩C|

|A^c|,|B^c|,|C^c|

These appear directly in the formula.

Explanation

These appear directly in the formula.

Submit

10) Match: singles / subtract pairs / add triple

∣A∣+∣B∣+∣C∣

−(∣A∩B∣+∣A∩C∣+∣B∩C∣)

+∣A∩B∩C∣

Correct roles.

Explanation

Correct roles.

Submit

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

36

40

44

48

30+20+12 − (10+6+4) + 2 = 44.

Explanation

30+20+12 − (10+6+4) + 2 = 44.

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

True

False

All intersections zero.

Explanation

All intersections zero.

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

38

40

42

44

22+18+16 − (6+5+4) + 3 = 44.

Explanation

22+18+16 − (6+5+4) + 3 = 44.

14) Exactly one of A,B,C:

|A|+|B|+|C|

|A|+|B|+|C| − 2(sum pairs) + 3|triple|

|A∪B∪C|

|A∩B∩C|

Remove pair overlaps twice, add triple three times.

Explanation

Remove pair overlaps twice, add triple three times.

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

True

False

Disjoint case gives max; overlaps reduce union.

Explanation

Disjoint case gives max; overlaps reduce union.

Inclusion–Exclusion with Three Sets Quiz

1) For three sets A,B,C, the inclusion–exclusion formula for the union is:

2)

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

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

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

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:

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

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

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

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

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

10) Match: singles / subtract pairs / add triple

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

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

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

14) Exactly one of A,B,C:

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