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Sets, Venn Diagrams
& Overlap

Master the Inclusion-Exclusion Principle, read Venn diagrams with confidence, and solve overlap problems quickly on the GRE.

GRE Quantitative

What Is a Set?

A set is a well-defined collection of distinct objects, called elements or members. On the GRE, sets usually contain people, numbers, or categories.

Notation
A set is written with curly braces: \( A = \{1, 2, 3\} \). The symbol \( \in \) means "is an element of."
Cardinality
The number of elements in set \( A \) is written \( |A| \). If \( A = \{1,2,3\} \), then \( |A| = 3 \).

Key Set Operations

Union \( A \cup B \)
All elements in \( A \) or \( B \) (or both). Think: everything combined.
Intersection \( A \cap B \)
Only elements in both \( A \) and \( B \). This is the overlap region.
Complement \( A^c \)
Everything not in \( A \). Relative to a universal set \( U \).

Anatomy of a Venn Diagram

U (Universal Set) Only A
\( A \cap B \)
 Only B A B Neither

The Inclusion-Exclusion Principle

When you simply add \( |A| + |B| \), every element in the overlap gets counted twice. Subtracting \( |A \cap B| \) corrects for that double-count.

|A| + |B| −
\( A \cap B \)

Adding the "Neither" Region

Most GRE problems give you a total number of people in a group. Some belong to neither category. The four regions always add up to the total:

1
Only A = \( |A| - |A \cap B| \)
2
Both = \( |A \cap B| \)
3
Only B = \( |B| - |A \cap B| \)
4
Neither = \( \text{Total} - |A \cup B| \)

Union vs. Intersection

Union \( A \cup B \) Intersection \( A \cap B \)
Meaning In A or B (or both) In A and B (both)
Diagram Region Entire shaded area Only the overlap lens
Size Relationship \( \geq \max(|A|, |B|) \) \( \leq \min(|A|, |B|) \)
GRE Keyword "at least one," "either" "both," "also"
Formula \( |A|+|B|-|A\cap B| \) Often the unknown to solve for

How to Solve Overlap Problems

1
Identify the sets. What are the two (or three) groups? What is the total?
2
Draw a Venn Diagram. Label each region: Only A, Both, Only B, Neither.
3
Fill in what you know. Start with the overlap (\( A \cap B \)) if given, then work outward.
4
Apply Inclusion-Exclusion. Use \( |A \cup B| = |A| + |B| - |A \cap B| \) to find the unknown.
5
Check: all four regions must sum to the total. If they don't, recheck your work.

Example 1: Finding the Overlap

In a class of 60 students, 35 study French and 30 study Spanish. Every student studies at least one language. How many study both?

Total = 60 30 5 25 Only F Both Only S French (35) Spanish (30)

Example 2: With a "Neither" Group

Of 100 employees, 55 drink coffee, 40 drink tea, and 15 drink neither. How many drink both?

Total = 100 45 10 30 Only C Both Only T 15 Neither Coffee (55) Tea (40)

Example 3: Finding "At Least One"

A survey of 200 people finds 120 own a car, 80 own a bike, and 40 own both. How many own at least one? How many own neither?

1
\( |C \cup B| = 120 + 80 - 40 = \) 160 own at least one.
2
Neither \( = 200 - 160 = \) 40 own neither.
3
Check: \( 80 + 40 + 40 + 40 = 200 \) ✓

Three-Set Inclusion-Exclusion

A B C
\( A \cap B \cap C \)

The logic is the same: add each set, subtract each pairwise overlap, then add back the triple overlap (which was subtracted too many times).

Mutually Exclusive vs. Overlapping

Mutually Exclusive
\( A \cap B = \emptyset \), so \( |A \cup B| = |A| + |B| \). The circles don't overlap. Example: even numbers and odd numbers.
Overlapping
\( A \cap B \neq \emptyset \). You must subtract the overlap. Example: students who play soccer and students who play basketball.
A B Mutually Exclusive A B Overlapping

Subsets and Special Cases

If \( A \subseteq B \), then every element of \( A \) is also in \( B \). This means \( A \cap B = A \) and \( |A \cup B| = |B| \).

A B A ⊆ B : all of A is inside B

Common Mistakes to Avoid

✕ Double-Counting the Overlap
Adding \( |A| + |B| \) without subtracting \( |A \cap B| \). This overcounts everyone in both groups.
✕ Forgetting "Neither"
Assuming everyone belongs to at least one set. Always check: does the problem mention a "neither" group?
✕ Confusing "Only A" with \( |A| \)
\( |A| \) includes overlap members. "Only A" \( = |A| - |A \cap B| \). These are different numbers.

Formula Reference Sheet

Quantity Formula
Union (2 sets) \( |A| + |B| - |A \cap B| \)
Only A \( |A| - |A \cap B| \)
Only B \( |B| - |A \cap B| \)
Neither \( \text{Total} - |A \cup B| \)
Union (3 sets) \( |A|+|B|+|C|-|A\cap B|-|A\cap C|-|B\cap C|+|A\cap B\cap C| \)
Four-region check \( \text{Only A} + \text{Both} + \text{Only B} + \text{Neither} = \text{Total} \)

Question 1 of 5

In a group of 80 people, 50 like chocolate and 35 like vanilla. If everyone likes at least one flavor, how many like both?

Question 2 of 5

Of 150 students, 90 play soccer, 60 play basketball, and 30 play both. How many play neither sport?

Question 3 of 5

If \( |A| = 40 \), \( |B| = 25 \), and \( |A \cap B| = 10 \), what is the number of elements in Only \( A \) (in \( A \) but not \( B \))?

Question 4 of 5

Sets \( P \) and \( Q \) are mutually exclusive with \( |P| = 22 \) and \( |Q| = 18 \). What is \( |P \cup Q| \)?

Question 5 of 5

In a company of 200 employees, 110 speak English, 85 speak Hindi, 50 speak both, and the rest speak neither. How many speak at least one of the two languages?