Sets and Venn Diagrams
Translate the set expression or fill the smallest overlap first—then count each region exactly once.
Sets and Venn Diagrams
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UPCAT set problems test translation and overlap control
Union A∪B includes A, B, and their overlap. Intersection A∩B keeps only what the sets share. Complement A′ means outside A but still inside the universal set.
|A∪B| = |A| + |B| − |A∩B||A∪B∪C| = |A|+|B|+|C| − |A∩B|−|A∩C|−|B∩C| + |A∩B∩C|For three sets, subtract the pairwise overlaps, then add the center back once. It was added three times with the set totals and removed three times with the pairwise subtractions.
Translate before calculating
For (A∪B)∩C′, first keep A or B, then remove every part inside C.
Start at the center
In three-set problems, fill the all-three region before converting pairwise totals into “exactly two” regions.
Center → pair-only regions → single-only regions → neither
Example: In a club survey, 15 students belong to both A and B, and 6 of those students also belong to C. The A-and-B-only region is 15−6 = 9. Repeat this subtraction for the other pairwise totals before finding the single-only regions.
Why it works
Published intersection totals normally include the center. Separating the center first prevents it from being counted repeatedly.
Five forms you should recognize
Problem: Forty-five students joined Math Club, 38 joined Science Club, and 15 joined both. How many joined at least one club?
|M∪S| = 45 + 38 − 15 = 68Why: Adding 45 and 38 counts the 15 students in both clubs twice, so subtract the overlap once.
Problem: Of 68 students who joined Math or Science, 45 joined Math and 38 joined Science. How many joined both?
|M∩S| = 45 + 38 − 68 = 15Why: The amount by which the two set totals exceed the union is the duplicated overlap.
Problem: Thirty-eight students use app A, 32 use B, and 27 use C. The pairwise intersections are 15, 12, and 11, while 6 use all three. How many use at least one app?
38 + 32 + 27 − 15 − 12 − 11 + 6 = 65Why: Subtract the three pairwise overlaps, then restore the all-three group once.
Problem: The pairwise totals are 18, 15, and 16, and 7 students belong to all three sets. How many belong to exactly two?
(18−7) + (15−7) + (16−7) = 28Why: Each pairwise total includes the same seven students in the center.
Problem: In a group of 100, set A has 64 members and set B has 53. What is the smallest possible overlap?
64 + 53 − 100 = 17Why: The 117 memberships cannot fit into 100 distinct people without at least 17 belonging to both sets.
Check before you commit
- Treating “or” as exactly one
- Forgetting that pairwise totals include the triple intersection
- Subtracting the all-three region instead of adding it back in the union formula
- Using a set total as an “only” region
- Ignoring the universal-set limit
- Accepting impossible survey data
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Sets and Venn Diagrams FAQ
What does “or” mean in sets?
Inclusive or: A or B or both, unless the problem explicitly says exactly one.
Why is the triple intersection added back?
Adding the three set totals counts it three times, while subtracting the three pairwise intersections removes it three times; adding it once leaves one correct count.
How do I find a minimum possible overlap?
Add the two set totals. Any amount beyond the universal-set size must be shared.
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