Inclusion–Exclusion Principle: Structure, Logic, and Applications Quiz

1) The main purpose of the inclusion–exclusion principle is to:

Simplify algebraic expressions

Count elements in unions of overlapping sets

Compute derivatives

Prove injectivity of functions

It corrects overcounting when sets overlap.

Explanation

It corrects overcounting when sets overlap.

2) If |A|=40, |B|=35, and |A∪B|=60, then |A∩B| equals:

5

10

15

20

40 + 35 − 60 = 15.

Explanation

40 + 35 − 60 = 15.

3) For three sets A, B, C, inclusion–exclusion uses the pattern “sum of singles – sum of pairwise intersections + triple intersection”.

True

False

This is the standard 3-set formula.

Explanation

This is the standard 3-set formula.

4) The first correction term in inclusion–exclusion for three sets corresponds to:

All single sets

All pairwise intersections

The triple intersection

The empty set

After adding singles, subtract pairwise overlaps.

Explanation

After adding singles, subtract pairwise overlaps.

5) For any finite family of sets, inclusion–exclusion signs alternate according to intersection size.

True

False

k-fold intersections have sign (−1)^(k+1).

Explanation

k-fold intersections have sign (−1)^(k+1).

6) In probability terms, 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 the counting formula.

Explanation

Same structure as the counting formula.

7) Match each expression:

|A∪B∪C|

|A∩B∩C|

|Aᶜ|

Union=≥1 set; intersection=all; complement=not in.

Explanation

Union=≥1 set; intersection=all; complement=not in.

Submit

8) Inclusion–exclusion can count elements satisfying none of several properties by subtracting the union from the total.

True

False

|None| = |Total| − |Union|.

Explanation

|None| = |Total| − |Union|.

9) In a class: 30 Algebra, 32 Analysis, 28 Topology, intersections 12, 10, 9, and 5 all three. How many take ≥1?

60

62

64

66

30+32+28 − (12+10+9) + 5 = 64.

Explanation

30+32+28 − (12+10+9) + 5 = 64.

10) For n sets, inclusion–exclusion always uses (2^n − 1) non-empty intersection terms.

True

False

Each non-empty subset of sets gives an intersection term.

Explanation

Each non-empty subset of sets gives an intersection term.

11) For two finite sets A and B, which identity is correct?

|A∪B|=|A|+|B|

|A∪B|=|A|+|B|+|A∩B|

|A∪B|=|A|+|B|−|A∩B|

|A∪B|=|A∩B|

We must subtract the intersection once.

Explanation

We must subtract the intersection once.

12) If |A|=30, |B|=25, |C|=20, all pairwise intersections are 5, and |A∩B∩C|=2, then |A∪B∪C| equals:

57

60

62

65

30+25+20 − (5+5+5) + 2 = 62.

Explanation

30+25+20 − (5+5+5) + 2 = 62.

13) Among the positive integers from 1 to 100, how many are divisible by 2 or 5?

50

55

60

65

50 + 20 − 10 = 60.

Explanation

50 + 20 − 10 = 60.

14) To apply inclusion–exclusion for three sets A, B, C, you need to know:

|A|, |B|, |C|

|A∩B|, |A∩C|, |B∩C|

|A∩B∩C|

|Aᶜ|, |Bᶜ|, |Cᶜ|

Single, pairwise, and triple intersections are used.

Explanation

Single, pairwise, and triple intersections are used.

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15) Conceptually, inclusion–exclusion is needed because:

Unions are always disjoint

Counting single sets avoids overcounting

Overlaps cause multiple counting

Intersections are empty

It corrects for repeated counts.

Explanation

It corrects for repeated counts.

Inclusion–Exclusion Principle: Structure, Logic, and Applications Quiz

1) The main purpose of the inclusion–exclusion principle is to:

2)

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

2) If |A|=40, |B|=35, and |A∪B|=60, then |A∩B| equals:

3) For three sets A, B, C, inclusion–exclusion uses the pattern “sum of singles – sum of pairwise intersections + triple intersection”.

4) The first correction term in inclusion–exclusion for three sets corresponds to:

5) For any finite family of sets, inclusion–exclusion signs alternate according to intersection size.

6) In probability terms, for events A and B, inclusion–exclusion gives:

7) Match each expression:

8) Inclusion–exclusion can count elements satisfying none of several properties by subtracting the union from the total.

9) In a class: 30 Algebra, 32 Analysis, 28 Topology, intersections 12, 10, 9, and 5 all three. How many take ≥1?

10) For n sets, inclusion–exclusion always uses (2^n − 1) non-empty intersection terms.

11) For two finite sets A and B, which identity is correct?

12) If |A|=30, |B|=25, |C|=20, all pairwise intersections are 5, and |A∩B∩C|=2, then |A∪B∪C| equals:

13) Among the positive integers from 1 to 100, how many are divisible by 2 or 5?

14) To apply inclusion–exclusion for three sets A, B, C, you need to know:

15) Conceptually, inclusion–exclusion is needed because: