Intersection of Sets
Calculus & Advanced MathThe 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.
Example
$A = \{1, 2, 3, 4\}$ and $B = \{3, 4, 5, 6\}$. $A \cap B = \{3, 4\}$. Only $3$ and $4$ are in both sets.
Key Insight
In a Venn diagram, the intersection is the overlapping region in the middle of the two circles.
Definition
$A \cap B = \{x : x \in A \text{ and } x \in B\}$. Properties: $A \cap A = A$, $A \cap \emptyset = \emptyset$, commutative, associative. Distributive laws: $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$.
Example
$A$ = multiples of $2$, $B$ = multiples of $3$ (in positive integers). $A \cap B$ = multiples of $6$ = $\{6, 12, 18, \ldots\}$.
Key Insight
Intersection corresponds to logical AND. Together with union and complement, it forms the Boolean algebra structure of set operations.
Definition
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). In probability, $P(A \cap B) = P(A)P(B)$ characterizes independent events.
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
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 R.