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Theorem (Union And Intersection Properties)

Let $A,B,C$ be sets. Then the following facts are true.

  1. $A\subseteq A$ (Every set is a subset of itself.)
  2. $A\subseteq A\cup B$ (A set is a subset of the union of itself and another set.)
  3. $A\cap B\subseteq A$ (The intersection of two sets is a subset of the first set.)
  4. $A\cup B = B\cup A$ (Set unions are commutative.)
  5. $A\cap B = B\cap A$ (Set intersections are commutative.)
  6. $A\cup (B\cup C)=(A\cup B)\cup C$ (Set unions are associative.)
  7. $A\cap (B\cap C)=(A\cap B)\cap C$ (Set intersections are associative.)
  8. $A\cup (B\cap C)=(A\cup B)\cap(A\cup C)$
  9. $A\cap (B\cup C)=(A\cap B)\cup(A\cap C)$

The last two we call the distributive laws for unions and intersections.