GMAT Focus Edition — Data Insights: Table Analysis · Graphics Interpretation · Multi-Source Reasoning · Two-Part Analysis
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Data Insights Lesson 14 of 20

Lesson 14:
Venn Diagrams & Set Logic

Master union, intersection, and complement in Venn diagram problems. Apply the inclusion-exclusion principle for overlapping categories.

⏱ 50 mins
🎯 DI 75 to 88
📚 Prereq: Lessons 1–5
Note: Lesson 14 of 20 — Venn Diagrams & Set Logic.
1

Core Concepts: Venn Diagrams & Set Logic

Master union, intersection, and complement in Venn diagram problems. Apply the inclusion-exclusion principle for overlapping categories.

Key Framework
|A∪B| = |A| + |B| − |A∩B|
Inclusion-exclusion: subtract the overlap to avoid double-counting.
Complement: A' = Total − A
Elements not in A. Useful for "not both" and "neither" questions.
Three sets: add all three, subtract all pairs, add the triple overlap
Extension of inclusion-exclusion to three circles.
Draw the Venn before computing
Label all regions and fill in known values from outside in.
2

Application Strategy

Step-by-Step Approach
①
Identify the question type and the specific metric being asked about.
②
Locate the relevant data in the chart, table, or tab.
③
Apply the key formula or logic rule from the framework above.
④
Verify that your answer satisfies the question as stated.
3

Visual Reference Diagram

Visual Framework for Lesson 14
📊
|A∪B| = |A| + |B| − |A∩B|
📈
Complement: A' = Total − A
🔍
Three sets: add all three, subtract all pairs, add the triple overlap
📋
Draw the Venn before computing
4

Quick Reference Rules

▸
|A∪B| = |A| + |B| − |A∩B|: Inclusion-exclusion: subtract the overlap to avoid double-counting.
▸
Complement: A' = Total − A: Elements not in A. Useful for "not both" and "neither" questions.
▸
Three sets: add all three, subtract all pairs, add the triple overlap: Extension of inclusion-exclusion to three circles.
▸
Draw the Venn before computing: Label all regions and fill in known values from outside in.
5

10 Traps for Lesson 14

⚠ Confusing the key formula for this topic

Review the core rule: |A∪B| = |A| + |B| − |A∩B|

⚠ Reading the wrong data series

Always verify axis labels and legends before extracting values.

⚠ Ignoring units or scale

Check units on every axis before computing ratios or changes.

⚠ Treating estimates as exact

GI data requires interpolation — use approximate values and pick the closest answer.

⚠ Forgetting the denominator

Complement: A' = Total − A — always identify what you're dividing by.

⚠ Applying simple average when weighted is needed

If group sizes differ, compute weighted average, not simple mean.

⚠ "All" statements need every row verified

One counterexample makes a universal statement False.

⚠ Scope: data sample ≠ full population

Conclusions are limited to the data range provided, not broader populations.

⚠ Confusing absolute and relative measures

Absolute change ($) and relative change (%) answer different questions.

⚠ Not using the complement when helpful

P(at least one) = 1 − P(none). In set logic, use complement for "not" conditions.

✦

10 Practice Questions

Q1 of 10
GI~650

In a group of 60 people, 35 speak French, 28 speak German, and 12 speak both. How many speak neither?

Explanation: 9. French ∪ German = 35+28−12 = 51. Neither = 60−51 = 9.
Q2 of 10
GI~600

A Venn diagram has Set A = 40 elements, Set B = 30, Set A∩B = 15. |A∪B| = ?

Explanation: 55. |A∪B| = 40+30−15 = 55.
Q3 of 10
GI~600

In a class of 100 students: 60 take Math, 50 take Science, 30 take both. How many take Math only (not Science)?

Explanation: 30. Math only = Math total − Both = 60−30 = 30.
Q4 of 10
GI~750

Three sets A, B, C. |A|=20, |B|=25, |C|=15, |A∩B|=8, |A∩C|=5, |B∩C|=6, |A∩B∩C|=3. |A∪B∪C| = ?

Explanation: 44. |A∪B∪C| = 20+25+15−8−5−6+3 = 44.
Q5 of 10
GI~700

A pie chart shows 3 overlapping customer segments. The total without any overlap would be 500. With overlaps removed (using inclusion-exclusion), the actual unique customer count is 380. The total counted in overlaps is:

Explanation: 120. Overcounted = raw total − unique = 500−380 = 120 customers were counted in overlaps.
Q6 of 10
GI~650

In a survey, 70% like Product X, 60% like Product Y, and 40% like both. What % like neither?

Explanation: 10%. X∪Y = 70+60−40 = 90%. Neither = 100−90 = 10%.
Q7 of 10
GI~600

Set A has 15 elements. Set B is a subset of A with 8 elements. |A − B| = ?

Explanation: 7. A − B = elements in A but not in B = 15−8 = 7.
Q8 of 10
GI~700

A Venn diagram shows: Only A = 10, Only B = 15, Only C = 12, A∩B only = 5, A∩C only = 3, B∩C only = 4, All three = 2. Total = ?

Explanation: 51. Total = 10+15+12+5+3+4+2 = 51. (Sum all distinct regions of the Venn diagram)
Q9 of 10
GI~600

Of 200 employees, 120 use Slack, 90 use Teams, 50 use both. How many use Slack OR Teams (or both)?

Explanation: 160. |Slack ∪ Teams| = 120+90−50 = 160.
Q10 of 10
GI~600

A student draws a Venn diagram with two circles, A and B, that do not overlap at all. What does this mean?

Explanation: A and B are mutually exclusive. Non-overlapping circles in a Venn diagram mean the two sets share no elements: A∩B = ∅ (empty set).
Lesson Summary
|A∪B| = |A| + |B| − |A∩B|

Inclusion-exclusion: subtract the overlap to avoid double-counting.

Complement: A' = Total − A

Elements not in A. Useful for "not both" and "neither" questions.

Three sets: add all three, subtract all pairs, add the triple overlap

Extension of inclusion-exclusion to three circles.

Draw the Venn before computing

Label all regions and fill in known values from outside in.

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