Inclusion–Exclusion for Two Sets (Applied & Algebraic Focus)

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| Questions: 20 | Updated: Dec 17, 2025
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1) In a survey, |M| = 60 like Music, |A| = 45 like Art, |M ∩ A| = 25. Which statements are true?

Explanation

|M ∪ A| = 60 + 45 − 25 = 80 

(B). Exactly-one = (60 − 25) + (45 − 25) = 35 + 20 = 55 

(C). A) adds without subtracting the overlap; D) is false since |M ∩ A| = 25 > 0.

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About This Quiz
Inclusionexclusion For Two Sets (Applied & Algebraic Focus) - Quiz

Can you use inclusion–exclusion beyond basic set diagrams? In this quiz, you’ll apply the idea to algebraic expressions and more complex counting setups. You’ll begin with straightforward numerical cases and move on to multi-step problems where expressions, logic, and structure matter. As you work, you’ll see how algebra and set... see moretheory connect, and how the principle helps keep your reasoning organized.
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2) |A| = 40, |B| = 35, |A ∩ B| = 15. Select the true statements.

Explanation

Union = 40 + 35 − 15 = 60 (A). 

Exactly-one = (40 − 15) + (35 − 15) = 25 + 20 = 45 (B). 

Rearranging the formula gives (C). Intersection ≠ 0, so not disjoint.

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3) |A ∪ B| = 50, |A| = 30, |B| = 35. Which are correct?

Explanation

Intersection = 30 + 35 − 50 = 15 

(A). Exactly-one = (30 − 15) + (35 − 15) = 15 + 20 = 35 

(B). C and D contradict the data.

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4) 70 students take Chemistry (C), 50 take Biology (B), and 20 take both. Which statements are correct?

Explanation

Union =70+50−20=100 (A). 

Exactly-one =(70−20)+(50−20)=50+30=80 (B). 

Intersection =20 (C). 

Not disjoint because the intersection is nonzero, so D is false.

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5) Assume xxx is a non-negative integer (counts), i.e., x ≥ 0. |A| = x + 3, |B| = 2x, |A ∩ B| = x. Find the true relations.

Explanation

Union = (x+3) + 2x − x = 2x (A). 

Exactly-one = (x + 3 - x) + (2x - x) = 3 + x (B). 

Intersection = x (C). 

Not disjoint (intersection ≠0).

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6) Probabilities: P(A) = 0.4, P(B) = 0.5, P(A ∩ B) = 0.1. Select the true statements.

Explanation

Union = 0.4 + 0.5 − 0.1 = 0.8 (A). Exactly-one = 0.4 + 0.5 − 2(0.1) = 0.7 (B, C). Disjoint would reuire intersection 0, which is false.

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7) |A| = 28, |B| = 25, |A ∩ B| = 9. Which statements hold?

Explanation

Union = 28 + 25 − 9 = 44 (A). Exactly-one = (28 − 9) + (25 − 9) = 19 + 16 = 35 (B). Intersection non-zero, so not disjoint.

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8) If P(A ∪ B) = 0.7, P(A) = 0.5, P(B) = 0.4, then A and B are disjoint.

Explanation

Using P(A ∪ B) = P(A) + P(B) − P(A ∩ B): 0.7 = 0.5 + 0.4 − P(A ∩ B) ⇒ P(A ∩ B) = 0.2 > 0. Hence not disjoint.

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9) When |A ∪ B| > |A| + |B|, the counting is incorrect.

Explanation

Formula gives |A ∪ B| = |A| + |B| − |A ∩ B| ≤ |A| + |B|, since intersection ≥ 0.

If the union is larger, an error occurred — you likely added incorrectly.

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10) If |A| = |B|, then |A ∩ B| must eual |A|.

Explanation

Feedback: Two equal-sized sets can have any overlap from 0 to |A|. Equality of sizes does not imply complete overlap.

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11) The Inclusion–Exclusion formula is symmetric in A and B.

Explanation

Switching A and B does not change |A | + |B| − |A ∩ B|; the union is commutative.

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12) If |A ∩ B| = 0, then |A ∪ B| = |A| + |B|.

Explanation

No overlap means nothing to subtract, so the union is the simple sum.

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13) The Inclusion–Exclusion principle applies to sets, probabilities, and logic events.

Explanation

All three obey the same union–intersection structure. Example: P(A ∨ B) = P(A) + P(B) − P(A ∧ B).

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14) If |A ∪ B| = 15 and |A| = 10, |B| = 8, then |A ∩ B| = 3.

Explanation

Feedback: |A ∩ B| = 10 + 8 − 15 = 3, so three elements belong to both sets.

Submit
15) |A ∪ B| = |A| + |B| − _____.

Explanation

Subtract the overlap once to correct double-counting.

Submit
16) If |A| = 25, |B| = 20, |A ∩ B| = 5, then |A ∪ B| = _____.

Explanation

25 + 20 − 5 = 40. Without subtracting the overlap, you’d overcount by 5

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17) In a class of 100, 70 take Chemistry, 50 take Biology. Number taking both = _____.

Explanation

70 + 50 − 100 = 20. That means 20 students appear in both groups.

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18) When sets are disjoint, |A ∩ B| = _____.

Explanation

Disjoint means no common members; intersection is the empty set with size 0.

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19) If P(A) = 0.3, P(B) = 0.6, P(A ∩ B) = 0.2, then P(A ∪ B) = _____.

Explanation

0.3 + 0.6 − 0.2 = 0.7. So the probability of A or B occurring is 70%.

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20) |A ∩ B| represents the number of elements _____ to both sets.

Explanation

The intersection is the collection of common members shared by A and B.

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In a survey, |M| = 60 like Music, |A| = 45 like Art, |M ∩ A| = 25....
|A| = 40, |B| = 35, |A ∩ B| = 15. Select the true statements.
|A ∪ B| = 50, |A| = 30, |B| = 35. Which are correct?
70 students take Chemistry (C), 50 take Biology (B), and 20 take both....
Assume xxx is a non-negative integer (counts), i.e., x ≥ 0. |A| = x...
Probabilities: P(A) = 0.4, P(B) = 0.5, P(A ∩ B) = 0.1. Select the...
|A| = 28, |B| = 25, |A ∩ B| = 9. Which statements hold?
If P(A ∪ B) = 0.7, P(A) = 0.5, P(B) = 0.4, then A and B are...
When |A ∪ B| > |A| + |B|, the counting is incorrect.
If |A| = |B|, then |A ∩ B| must eual |A|.
The Inclusion–Exclusion formula is symmetric in A and B.
If |A ∩ B| = 0, then |A ∪ B| = |A| + |B|.
The Inclusion–Exclusion principle applies to sets,...
If |A ∪ B| = 15 and |A| = 10, |B| = 8, then |A ∩ B| = 3.
|A ∪ B| = |A| + |B| − _____.
If |A| = 25, |B| = 20, |A ∩ B| = 5, then |A ∪ B| = _____.
In a class of 100, 70 take Chemistry, 50 take Biology. Number taking...
When sets are disjoint, |A ∩ B| = _____.
If P(A) = 0.3, P(B) = 0.6, P(A ∩ B) = 0.2, then P(A ∪ B) =...
|A ∩ B| represents the number of elements _____ to both sets.
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