QUANTITATIVE REASONING · GMAT-Q-SET

Overlapping Sets and Venn Reasoning

GMAT lesson development preview · DRAFT · Original learning and assessment material

Concepts and problem-solving strategy

Core mathematical relationship

∣ A \cup B ∣ = ∣ A ∣ + ∣ B ∣ - ∣ A \cap B ∣

Translate the condition

Separate the groups into A only, B only, both, and neither. The overlap is counted twice in the sum of group totals, so subtract it once.

Test the result

Check that the four regions add to the survey total. For a complement question, identify exactly which region or regions are excluded.

Think Like a GMAT Test-Taker

Name each set and mark the intersection before counting a union or complement.

Faster Methods

Use a four-region table to separate A only, B only, both, and neither.

For a complement, count the excluded intersection directly when De Morgan’s law makes it simpler.

Worked example 1

In a survey, 30 people chose option A and 25 chose option B; 10 chose both. How many chose at least one?

∣ A \cup B ∣ = 30 + 25 - 10 = 45

Add the two set counts and subtract the 10 people counted in both sets: 30 + 25 - 10 = 45.

Worked example 2

Among 80 students, 45 study French, 40 study Spanish, and 15 study neither. How many study both?

Union = 80 - 15 = 65, Both = 45 + 40 - 65 = 20

Start with the whole population and remove neither to obtain the union. Adding the language counts includes bilingual students twice, so subtract the union to find the overlap. The regions are twenty-five French only, twenty both, twenty Spanish only, and fifteen neither.

Worked example 3

Among 90 staff, 60 use service A and 50 use service B. Find the smallest possible overlap.

∣ A \cap B ∣ \geq 60 + 50 - 90 = 20

The union cannot contain more than all ninety staff. To attain the minimum, use forty A-only staff, thirty B-only staff and twenty using both. This construction proves the lower bound is achievable.

Common Traps

Subtract the overlap exactly once when finding a union.
Distinguish “neither set” from “not both sets”; these are different events.