Intersection of Sets
The intersection of two sets is the set of all elements that belong to both sets simultaneously, written A ∩ B.
Formula
A \cap B = \{x : x \in A \text{ and } x \in B\}
Definition
The intersection of two sets keeps only the elements that appear in both sets at the same time. Formally, $A \cap B = \{x : x \in A \text{ and } x \in B\}$, with properties $A \cap A = A$, $A \cap \emptyset = \emptyset$, commutativity, associativity, and the distributive law $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$. For a collection $\{A_i\}$, $\bigcap_{i \in I} A_i = \{x : x \in A_i \text{ for all } i \in I\}$; in topology, open sets are closed under finite intersection (but not necessarily arbitrary intersection), and in probability $P(A \cap B) = P(A)P(B)$ characterizes independent events.
Example
For $A = \{1, 2, 3, 4\}$ and $B = \{3, 4, 5, 6\}$: $A \cap B = \{3, 4\}$, only the elements in both sets. If $A$ = multiples of $2$ and $B$ = multiples of $3$ (in positive integers), $A \cap B$ = multiples of $6$ = $\{6, 12, 18, \ldots\}$. An infinite example: $\bigcap_{n=1}^{\infty} (0, 1/n) = \emptyset$, the intersection of open intervals shrinks to nothing, but $\bigcap_{n=1}^{\infty} [0, 1/n] = \{0\}$ by the Nested Intervals theorem.
Key Insight
In a Venn diagram, the intersection is the overlapping region in the middle of the two circles, and intersection corresponds to logical AND; together with union and complement it forms the Boolean algebra structure of set operations. The Nested Intervals theorem (non-empty intersection of a decreasing sequence of closed bounded intervals) is a key tool for constructing real numbers and proving completeness of $\mathbb{R}$.