Advanced Two-Set Inclusion–Exclusion and Set Identities Quiz

1) Rearranged inclusion–exclusion:

|A|=|A∪B|+|B|-|A∩B|

|A|=|A∪B|+|A∩B|-|B|

|A|=|A∩B|-|A∪B|-|B|

|A|=|B|-|A∪B|+|A∩B|

Rearranging |A∪B|=|A|+|B|-|A∩B| gives |A|=|A∪B|+|A∩B|-|B|.

Explanation

Rearranging |A∪B|=|A|+|B|-|A∩B| gives |A|=|A∪B|+|A∩B|-|B|.

Advanced Two-set Inclusionexclusion and Set Identities Quiz - Quiz

Ready to look at inclusion–exclusion from a more advanced, structural point of view? This quiz challenges you to manipulate the two-set formula algebraically, derive rearranged forms, and interpret what equations like ∣A ∪ B∣ = ∣U∣ or ∣A ∪ B∣ = ∣A∣ really say about the relationship between sets. You’ll... see moreuse inclusion–exclusion with complements, symmetric difference, and difference sets, and you’ll spot common logical and algebraic mistakes that occur when the formula is used incorrectly. Along the way, you’ll analyze realistic data scenarios to check whether a professor’s or survey’s claims are even possible. Step by step, you’ll deepen your understanding of how inclusion–exclusion encodes rich set relationships and guards against counting errors at an advanced level.
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2) If |A∪B|=52, |B|=35, and |A∩B|=17, then |A| equals:

|A| = 52 + 17 - 35 = 34.

Explanation

|A| = 52 + 17 - 35 = 34.

3) |A∪B| ≥ (|A| + |B|) / 2.

True

False

Union is always ≥ the average.

Explanation

Union is always ≥ the average.

4) Universe 120: |A|=70, |B|=60, 20 in neither. |A∩B|?

Union=100, intersection=70+60-100=30.

Explanation

Union=100, intersection=70+60-100=30.

5) If |A| + |B| - |A∩B| = |U|, this means:

A and B disjoint

Everyone in exactly one

Everyone in at least one

No one in intersection

Union covers universe.

Explanation

Union covers universe.

6) If P(A∪B)=0.9, P(A)=0.7, P(B)=0.5, intersection is:

0.1

0.2

0.3

0.4

0.7+0.5-0.9=0.3.

Explanation

0.7+0.5-0.9=0.3.

7) If |A∪B| = |A| + |B|, then:

A ⊆ B

B ⊆ A

A and B disjoint

A = B

Intersection=0.

Explanation

Intersection=0.

8) How many integers 1–100 divisible by 4 or 9?

25 + 11 - 5 = 31.

Explanation

25 + 11 - 5 = 31.

9) 80 know Python, 70 SQL, 50 both. At least one?

100

110

120

130

80+70-50=100.

Explanation

80+70-50=100.

10) Common inclusion–exclusion mistakes:

Forgetting to subtract |A∩B|

Using +|A∩B|

Assuming |A∩B|=|A||B|

Assuming |A∪B|=max(|A|,|B|)

All violate correct formula.

Explanation

All violate correct formula.

Submit

11) Match expressions:

∣A∪B∣

∣A∩B∣

∣A△B∣

Correct meanings.

Explanation

Correct meanings.

Submit

12) Universe 60: |A|=40, |B|=35, |A∩B|=20. Neither?

Union=55, neither=5.

Explanation

Union=55, neither=5.

13) 40 people: 30 A, 25 B, 5 both. Possible?

True

False

Sum-overlap=50 > 40 impossible.

Explanation

Sum-overlap=50 > 40 impossible.

14) Exactly one of A, B occurs:

P(A)+P(B)

P(A∪B)

P(A)+P(B)-2P(A∩B)

P(A∩B)

Symmetric difference formula.

Explanation

Symmetric difference formula.

15) Class 80: 55 Algebra, 50 Calculus, 15 fail both. Both passed?

Union=65, intersection=40.

Explanation

Union=65, intersection=40.

Advanced Two-Set Inclusion–Exclusion and Set Identities Quiz

1) Rearranged inclusion–exclusion:

2)

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

2) If |A∪B|=52, |B|=35, and |A∩B|=17, then |A| equals:

3) |A∪B| ≥ (|A| + |B|) / 2.

4) Universe 120: |A|=70, |B|=60, 20 in neither. |A∩B|?

5) If |A| + |B| - |A∩B| = |U|, this means:

6) If P(A∪B)=0.9, P(A)=0.7, P(B)=0.5, intersection is:

7) If |A∪B| = |A| + |B|, then:

8) How many integers 1–100 divisible by 4 or 9?

9) 80 know Python, 70 SQL, 50 both. At least one?

10) Common inclusion–exclusion mistakes:

11) Match expressions:

12) Universe 60: |A|=40, |B|=35, |A∩B|=20. Neither?

13) 40 people: 30 A, 25 B, 5 both. Possible?

14) Exactly one of A, B occurs:

15) Class 80: 55 Algebra, 50 Calculus, 15 fail both. Both passed?