Inclusion-Exclusion Principle Reference Cheat Sheet

Key Facts

• For two sets, the inclusion-exclusion formula is $|A \cup B| = |A| + |B| - |A \cap B|$.
• For three sets, the formula is $|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C|$.
• The complement rule is $|A^c| = |U| - |A|$, where $U$ is the universal set.
• For two events in probability, $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.
• If two sets are disjoint, then $A \cap B = \varnothing$ and $|A \cup B| = |A| + |B|$.
• The number in neither of two sets is $|U| - |A \cup B|$.
• The general inclusion-exclusion pattern alternates signs: add single sets, subtract pairwise intersections, add triple intersections, and continue.
• For finite sets $A_1, A_2, \ldots, A_n$, $\left|\bigcup_{i=1}^{n} A_i\right| = \sum |A_i| - \sum |A_i \cap A_j| + \sum |A_i \cap A_j \cap A_k| - \cdots$.

Vocabulary

Set

A set is a collection of distinct objects, numbers, or outcomes.

Union

The union $A \cup B$ contains every element that is in $A$, in $B$, or in both.

Intersection

The intersection $A \cap B$ contains only the elements that are in both $A$ and $B$.

Complement

The complement $A^c$ contains all elements in the universal set $U$ that are not in $A$.

Disjoint Sets

Disjoint sets have no elements in common, so $A \cap B = \varnothing$.

Universal Set

The universal set $U$ is the full collection of elements being considered in a problem.