Intersection
Definition
The intersection of two sets \(A\) and \(B\), written \(A \cap B\), is the set of all elements that are in both \(A\) and \(B\).
\[A \cap B = \{x : x \in A \land x \in B\}\]
Properties
Commutativity
\[A \cap B = B \cap A\]
Associativity
\[(A \cap B) \cap C = A \cap (B \cap C)\]
Identity
\[A \cap \emptyset = \emptyset\]
Idempotence
\[A \cap A = A\]
Distributive Laws
\[A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$$ $$A \cup (B \cap C) = (A \cup B) \cap (A \cup C)\]
Examples
Example 1
\[\{1, 2, 3\} \cap \{2, 3, 4\} = \{2, 3\}\]
Example 2
\[\{1, 2\} \cap \{3, 4\} = \emptyset\]
Disjoint Sets
Two sets are disjoint if their intersection is empty: $\(A \cap B = \emptyset\)$