Set theory studies collections of objects. In aptitude questions, sets are commonly used to solve survey and Venn-diagram problems involving people who belong to one, two, or three groups.

Basic terms

• A set is a well-defined collection of distinct objects, called elements.
• The universal set, U, contains all objects under discussion.
• The empty set, denoted by empty braces, contains no elements.
• A is a subset of B when every element of A belongs to B.
• The number of elements in A is written as n(A).

Set operations

• Union, A union B: elements in A or B or both.
• Intersection, A intersection B: elements common to both A and B.
• Complement, A': elements in U that are not in A.
• Difference, A - B: elements in A but not in B.
• Disjoint sets have no common elements, so their intersection is empty.

Two-set Venn diagrams

A Venn diagram uses overlapping circles to represent sets. Place the common elements in the overlap first, then fill in the values that belong only to each individual set.

• n(A union B) = n(A) + n(B) - n(A intersection B)
• Only A = n(A) - n(A intersection B)
• Neither = n(U) - n(A union B)

Three-set Venn diagrams

For three sets A, B, and C, fill the central region belonging to all three sets first. Then fill each two-set-only overlap before the regions belonging to only one set.

n(A union B union C) = n(A) + n(B) + n(C) - n(A intersection B) - n(B intersection C) - n(C intersection A) + n(A intersection B intersection C).

Useful laws

• Commutative laws: A union B = B union A and A intersection B = B intersection A.
• Associative laws: grouping does not change union or intersection.
• De Morgan's laws: (A union B)' = A' intersection B' and (A intersection B)' = A' union B'.
• n(A') = n(U) - n(A).

How to solve questions

1. Define the universal set and identify each group in the question.
2. Draw circles for the sets and write all common values in overlaps first.
3. Use the union formula to avoid double counting shared elements.
4. For three sets, begin with the intersection of all three sets.
5. Find the number outside all circles by subtracting the union from the total.