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

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

Explanation

It corrects overcounting when sets overlap.

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About This Quiz
Inclusionexclusion Principle: Structure, Logic, And Applications Quiz - Quiz

Think you know how to manage overlapping sets? This quiz takes you deeper into the structure behind inclusion–exclusion, where unions, pairwise intersections, and triple overlaps must be handled in a precise order. You’ll use the principle to compute probabilities, analyze real-world counting situations, and interpret formulas involving residues, divisibility, and... see morestudent-course enrollments. By stepping through alternation of signs, logical reasoning, and multi-set interactions, you’ll see how inclusion–exclusion forms the backbone of many combinatorial arguments. By the end, you'll have a stronger grasp of how this method corrects overcounting across increasingly complex situations. see less

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

Explanation

We must subtract the intersection once.

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

Explanation

40 + 35 − 60 = 15.

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

Explanation

This is the standard 3-set formula.

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

Explanation

After adding singles, subtract pairwise overlaps.

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

Explanation

30+25+20 − (5+5+5) + 2 = 62.

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

Explanation

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

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

Explanation

Same structure as the counting formula.

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

Explanation

50 + 20 − 10 = 60.

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

Explanation

Single, pairwise, and triple intersections are used.

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11) Match each expression:

Explanation

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

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

Explanation

|None| = |Total| − |Union|.

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

Explanation

30+32+28 − (12+10+9) + 5 = 64.

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14) Conceptually, inclusion–exclusion is needed because:

Explanation

It corrects for repeated counts.

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

Explanation

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

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The main purpose of the inclusion–exclusion principle is to:
For two finite sets A and B, which identity is correct?
If |A|=40, |B|=35, and |A∪B|=60, then |A∩B| equals:
For three sets A, B, C, inclusion–exclusion uses the pattern “sum...
The first correction term in inclusion–exclusion for three sets...
If |A|=30, |B|=25, |C|=20, all pairwise intersections are 5, and...
For any finite family of sets, inclusion–exclusion signs alternate...
In probability terms, for events A and B, inclusion–exclusion gives:
Among the positive integers from 1 to 100, how many are divisible by 2...
To apply inclusion–exclusion for three sets A, B, C, you need to...
Match each expression:
Inclusion–exclusion can count elements satisfying none of several...
In a class: 30 Algebra, 32 Analysis, 28 Topology, intersections 12,...
Conceptually, inclusion–exclusion is needed because:
For n sets, inclusion–exclusion always uses (2^n − 1) non-empty...
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