Venn Diagram
A Venn diagram is a visual representation of sets using overlapping circles inside a rectangle to show unions, intersections, and complements.
Definition
A Venn diagram uses overlapping circles to show how sets relate: elements shared by both sets go in the overlapping middle region, and elements in only one set go in the non-overlapping parts. A two-set Venn diagram partitions the universe $U$ into four regions: only in $A$, only in $B$, in both ($A \cap B$), and in neither (complement of $A \cup B$); three-set diagrams produce eight regions, and they are used to verify or discover set identities visually. Diagrams for $n$ sets require regions for all $2^n$ possible intersection combinations, and constructing valid Venn diagrams for $n \ge 4$ requires non-circular shapes; symmetric Venn diagrams for prime $n$ have elegant rotational symmetry.
Example
Draw two overlapping circles labeled "can swim" and "can fly": ducks go in the middle, fish go in the swim-only region, eagles go in the fly-only region. To verify De Morgan's law $(A \cup B)' = A' \cap B'$: shade $A \cup B$ then take its complement, and separately shade $A'$ and $B'$ then intersect, both producing the same region outside both circles. A symmetric Venn diagram for $5$ sets uses five congruent ellipses, and for $n = 11$, a symmetric diagram with $11$-fold rotational symmetry was discovered only in 2004.
Key Insight
Venn diagrams make abstract set relationships visible and concrete, the most widely used tool for visualizing logic and set operations; they prove identities by region counting, since if two expressions always shade the same regions, they are equal for all sets $A, B$. The combinatorial and geometric complexity of multi-set Venn diagrams connects to graph theory (Hamiltonian cycles on the Boolean lattice) and combinatorics.