Inclusion Exclusion Quiz: Two Set Inclusion Exclusion Basics

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| Questions: 20 | Updated: Dec 17, 2025
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1) Given |A|=30, |B|=25, and |A ∩ B|=12, what is |A ∪ B|?

Explanation

Use |A ∪ B| = |A| + |B| − |A ∩ B| = 30 + 25 − 12 = 43.

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About This Quiz
Inclusion Exclusion Quiz: Two Set Inclusion Exclusion Basics - Quiz

How can you handle overlapping sets without double-counting? In this quiz, you’ll explore the two-set inclusion–exclusion principle and learn how it corrects overcounts by subtracting shared elements. You’ll practice interpreting Venn diagrams, organizing information, and applying the formula to counting problems involving categories, memberships, or survey data. Step by step,... see moreyou’ll strengthen your ability to reason through overlaps clearly and understand why inclusion–exclusion provides accurate totals when sets intersect.
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2) If A and B are disjoint, then |A ∪ B| = |A| + |B|.

Explanation

Disjoint sets have |A ∩ B|=0, so |A ∪ B| = |A| + |B| − 0 = |A| + |B|.

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3) In a class of 50, |A|=28 study Art and |B|=23 study Music, with |A ∩ B|=11. Then |A ∪ B| = ____.

Explanation

Compute 28 + 23 − 11 = 40. So 40 students take Art or Music (or both).

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4) Suppose |A ∪ B|=45, |A|=27, and |B|=26. What is |A ∩ B|?

Explanation

Rearrange: |A ∩ B| = |A| + |B| − |A ∪ B| = 27 + 26 − 45 = 8.

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5) Select all correct identities/inequalities for two sets A and B.

Explanation

The two-set inclusion–exclusion identity holds, and it rearranges to |A ∩ B| = |A| + |B| − |A ∪ B|. Disjoint sets give |A ∩ B|=0, so the union is the sum. Also, |A ∪ B| ≤ |A| + |B| since overlap prevents double counting.

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6) In a survey, 18 take Art and 14 take Music; 7 take both. How many take at least one?

Explanation

|A ∪ B| = 18 + 14 − 7 = 25.

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7) If |A ∪ B| = |A| + |B|, then A and B are disjoint.

Explanation

Equality occurs exactly when |A ∩ B| = 0, i.e., disjoint sets.

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8) Given |A|=40, |B|=35, and |A ∩ B|=22, the number of elements in B only is ____.

Explanation

B only = |B| − |A ∩ B| = 35 − 22 = 13.

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9) Given |A|=20 and |B|=25, what is the minimum possible value of |A ∪ B|?

Explanation

The union is at least the larger of the two sizes: min |A ∪ B| = max(|A|,|B|) = 25 (when one set is contained in the other).

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10) Suppose |A|=30, |B|=22, and |A ∪ B|=40. Select all statements that must be true.

Explanation

Compute |A ∩ B| = 30 + 22 − 40 = 12. Then only‑A = 30 − 12 = 18, only‑B = 22 − 12 = 10, and by premise the union has 40.

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11) Out of 100 students, 64 take Math (A), 58 take Science (B), and 30 take both. How many take neither?

Explanation

|A ∪ B| = 64 + 58 − 30 = 92. Neither = 100 − 92 = 8.

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12) For any sets A and B, |A ∩ B| ≤ min(|A|,|B|).

Explanation

The intersection cannot exceed either set’s size, so it is bounded by the smaller size.

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13) If |A|=19, |B|=23, and |A ∪ B|=35, then |A ∩ B| = ____.

Explanation

Compute 19 + 23 − 35 = 7.

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14) If |A ∩ B|=0 with |A|=12 and |B|=17, what is |A ∪ B|?

Explanation

Disjoint sets add: |A ∪ B| = 12 + 17 = 29.

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15) Select all statements that are always true for any sets A and B.

Explanation

Union cannot exceed the sum, must be at least the larger set; intersection cannot exceed the smaller. The other two statements are false in general.

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16) If 15 are only in A, 12 are only in B, and 9 are in both, then |A ∪ B| equals:

Explanation

Union = only‑A + only‑B + both = 15 + 12 + 9 = 36.

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17) If |A|=|B| and |A ∪ B|=|A|, then B ⊆ A.

Explanation

|A ∪ B|=|A| implies adding B does not increase size, so B is contained in A.

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18) At a school of 200 students, 120 play Soccer (A), 90 play Basketball (B), and 50 play both. Then |A ∪ B| = ____.

Explanation

Compute 120 + 90 − 50 = 160 students play at least one of the two.

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19) Given |A ∪ B|=70, |A|=45, and |A ∩ B|=20, what is |B|?

Explanation

Solve |A ∪ B| = |A| + |B| − |A ∩ B| ⇒ |B| = 70 − 45 + 20 = 45.

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20) Given |A|=18, |B|=21, and |A ∩ B|=9, choose all correct quantities.

Explanation

Compute union: 18 + 21 − 9 = 30. Only‑A = 18 − 9 = 9. Only‑B = 21 − 9 = 12. The last two statements are false.

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Given |A|=30, |B|=25, and |A ∩ B|=12, what is |A ∪ B|?
If A and B are disjoint, then |A ∪ B| = |A| + |B|.
In a class of 50, |A|=28 study Art and |B|=23 study Music, with |A ∩...
Suppose |A ∪ B|=45, |A|=27, and |B|=26. What is |A ∩ B|?
Select all correct identities/inequalities for two sets A and B.
In a survey, 18 take Art and 14 take Music; 7 take both. How many take...
If |A ∪ B| = |A| + |B|, then A and B are disjoint.
Given |A|=40, |B|=35, and |A ∩ B|=22, the number of elements in B...
Given |A|=20 and |B|=25, what is the minimum possible value of |A ∪...
Suppose |A|=30, |B|=22, and |A ∪ B|=40. Select all statements that...
Out of 100 students, 64 take Math (A), 58 take Science (B), and 30...
For any sets A and B, |A ∩ B| ≤ min(|A|,|B|).
If |A|=19, |B|=23, and |A ∪ B|=35, then |A ∩ B| = ____.
If |A ∩ B|=0 with |A|=12 and |B|=17, what is |A ∪ B|?
Select all statements that are always true for any sets A and B.
If 15 are only in A, 12 are only in B, and 9 are in both, then |A ∪...
If |A|=|B| and |A ∪ B|=|A|, then B ⊆ A.
At a school of 200 students, 120 play Soccer (A), 90 play Basketball...
Given |A ∪ B|=70, |A|=45, and |A ∩ B|=20, what is |B|?
Given |A|=18, |B|=21, and |A ∩ B|=9, choose all correct quantities.
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