Intersection of Sets

The intersection of two sets $A$ and $B$ is the set of all elements common to both $A$ and $B$.

• Notation: $A\cap B$
• Definition (Set-Builder Notation):

$$A\cap B=\{x\mid x\in A\mathrm{and}x\in B\}$$

• Example:
  • $A=\{1,2,3\},B=\{3,4,5\}$
  • $A\cap B=\{3\}$

Properties:

• Commutative: $A\cap B=B\cap A$
• Associative: $(A\cap B)\cap C=A\cap (B\cap C)$
• Idempotent: $A\cap A=A$
• Identity: $A\cap U=A$, where $U$ is the Universal Set.