Inclusion Exclusion Quiz: Two Set Inclusion Exclusion Basics

Grade 12th

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| Attempts: 12 | Questions: 20 | Updated: May 19, 2026

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1) Out of 100 students, 64 take Math, 58 take Science, and 30 take both. How many take neither?

A∪B = 64 + 58 - 30 = 92. Students taking neither = 100 - 92 = 8. Option A gives 6, option C gives 10, option D gives 12.

Explanation

A∪B = 64 + 58 - 30 = 92. Students taking neither = 100 - 92 = 8. Option A gives 6, option C gives 10, option D gives 12.

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

A∪B = 30

Only-A count is 9

Only-B count is 12

A∩B = 6

A∪B = 18 + 21 - 9 = 30, confirming A. Only-A = 18 - 9 = 9, confirming B. Only-B = 21 - 9 = 12, confirming C. Option D states A∩B = 6 but the given value is 9, making it false.

Explanation

A∪B = 18 + 21 - 9 = 30, confirming A. Only-A = 18 - 9 = 9, confirming B. Only-B = 21 - 9 = 12, confirming C. Option D states A∩B = 6 but the given value is 9, making it false.

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

Rearranging: B = A∪B - A + A∩B = 70 - 45 + 20 = 45. Option A gives 35, option B gives 40, option D gives 50.

Explanation

Rearranging: B = A∪B - A + A∩B = 70 - 45 + 20 = 45. Option A gives 35, option B gives 40, option D gives 50.

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4) At a school of 200 students, 120 play Soccer, 90 play Basketball, and 50 play both. What is A∪B?

140

150

160

170

A∪B = 120 + 90 - 50 = 160. The 50 who play both are subtracted once to avoid double-counting. Option A gives 140, option B gives 150, option D gives 170.

Explanation

A∪B = 120 + 90 - 50 = 160. The 50 who play both are subtracted once to avoid double-counting. Option A gives 140, option B gives 150, option D gives 170.

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5) If the sizes of A and B are equal and A∪B equals the size of A, then B is a subset of A.

True

False

The answer is True. A∪B = A means adding B introduces no new elements so every element of B is already in A. Combined with A and B having equal size, the two sets must be equal, meaning B is fully contained within A.

Explanation

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6) If 15 are only in A, 12 are only in B, and 9 are in both, what is A∪B?

A∪B = only-A + only-B + both = 15 + 12 + 9 = 36. The union consists of three non-overlapping regions. Option A gives 33, option B gives 34, option C gives 35.

Explanation

A∪B = only-A + only-B + both = 15 + 12 + 9 = 36. The union consists of three non-overlapping regions. Option A gives 33, option B gives 34, option C gives 35.

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

A∪B is at most A + B

A∪B is at least the size of the larger set

A∩B is at most the size of the smaller set

A∩B is at least A + B

The union cannot exceed the sum since the intersection prevents double-counting, confirming A. The union must contain at least the larger set, confirming B. The intersection cannot exceed the smaller set, confirming C. Option D is false — the intersection is at most the smaller set, never at least the sum of both.

Explanation

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

With A∩B = 0 the sets are disjoint so A∪B = A + B = 12 + 17 = 29. Option A gives 27, option B gives 28, option D gives 30.

Explanation

With A∩B = 0 the sets are disjoint so A∪B = A + B = 12 + 17 = 29. Option A gives 27, option B gives 28, option D gives 30.

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9) If A=19, B=23, and A∪B=35, what is A∩B?

A∩B = A + B - A∪B = 19 + 23 - 35 = 7. Option A gives 5, option B gives 6, option D gives 8.

Explanation

A∩B = A + B - A∪B = 19 + 23 - 35 = 7. Option A gives 5, option B gives 6, option D gives 8.

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10) For any sets A and B, the size of A∩B is always less than or equal to the size of the smaller set.

True

False

The answer is True. The intersection contains only elements belonging to both sets so it cannot exceed either set individually. Every element of the intersection must belong to both sets, bounding it by the smaller of the two.

Explanation

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11) Given A=30, B=25, and A∩B=12, what is A∪B?

A∪B = A + B - A∩B = 30 + 25 - 12 = 43. Option A gives 41, option C gives 45, option D gives 47, none of which correctly subtract the intersection. Without subtracting the overlap, elements in both sets would be counted twice.

Explanation

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

Exactly 12 elements are in A∩B

Exactly 18 elements are only in A

Exactly 10 elements are only in B

Exactly 40 elements are in A∪B

A∩B = 30 + 22 - 40 = 12, confirming A. Only-A = 30 - 12 = 18, confirming B. Only-B = 22 - 12 = 10, confirming C. The union contains exactly 40 elements by the given premise, confirming D.

Explanation

A∩B = 30 + 22 - 40 = 12, confirming A. Only-A = 30 - 12 = 18, confirming B. Only-B = 22 - 12 = 10, confirming C. The union contains exactly 40 elements by the given premise, confirming D.

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

The minimum union occurs when the smaller set is entirely contained within the larger, giving A∪B = max(A,B) = 25. Option A gives 20 which is impossible since B = 25 exceeds it. Options C and D both exceed the minimum.

Explanation

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14) Given A=40, B=35, and A∩B=22, how many elements are in B only?

Elements only in B = B - A∩B = 35 - 22 = 13. These are elements in B not shared with A. Option A gives 11, option B gives 12, option D gives 14.

Explanation

Elements only in B = B - A∩B = 35 - 22 = 13. These are elements in B not shared with A. Option A gives 11, option B gives 12, option D gives 14.

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15) If the size of A∪B equals the size of A plus the size of B, then A and B are disjoint.

True

False

The answer is True. From A∪B = A + B - A∩B, if A∪B equals A + B then A∩B must equal 0. A zero intersection means the sets share no elements, which is the definition of disjoint.

Explanation

The answer is True. From A∪B = A + B - A∩B, if A∪B equals A + B then A∩B must equal 0. A zero intersection means the sets share no elements, which is the definition of disjoint.

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

A∪B = 18 + 14 - 7 = 25. The 7 who take both are subtracted once to avoid counting them twice. Option A gives 23, option B gives 24, option D gives 26.

Explanation

A∪B = 18 + 14 - 7 = 25. The 7 who take both are subtracted once to avoid counting them twice. Option A gives 23, option B gives 24, option D gives 26.

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17) Select all correct identities for two sets A and B.

A∪B = A + B - A∩B

A∩B = A + B - A∪B

If A and B are disjoint then A∪B = A + B

A∩B = A∪B - A - B

Option A is the inclusion-exclusion formula, confirming A. Option B is the same formula rearranged to solve for the intersection, confirming B. Option C follows when A∩B = 0 for disjoint sets, confirming C. Option D has the wrong sign — the correct rearrangement gives A∩B = A + B - A∪B, not A∪B - A - B, which produces a negative result in most cases.

Explanation

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

Rearranging: A∩B = A + B - A∪B = 27 + 26 - 45 = 8. Option A gives 6, option B gives 7, option D gives 9.

Explanation

Rearranging: A∩B = A + B - A∪B = 27 + 26 - 45 = 8. Option A gives 6, option B gives 7, option D gives 9.

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19) In a class of 50, 28 study Art and 23 study Music with 11 studying both. What is the size of A∪B?

A∪B = 28 + 23 - 11 = 40. The 11 students studying both are counted once in each group so they must be subtracted once. Option A gives 36, option B gives 38, option D gives 42.

Explanation

A∪B = 28 + 23 - 11 = 40. The 11 students studying both are counted once in each group so they must be subtracted once. Option A gives 36, option B gives 38, option D gives 42.

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20) If A and B are disjoint, then the size of A∪B equals the size of A plus the size of B.

True

False

The answer is True. Disjoint sets share no elements so A∩B = 0. The formula gives A∪B = A + B - 0 = A + B. No elements are double-counted because the two sets have nothing in common.

Explanation

The answer is True. Disjoint sets share no elements so A∩B = 0. The formula gives A∪B = A + B - 0 = A + B. No elements are double-counted because the two sets have nothing in common.

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