Set Identities

Here are some fundamental set identities, presented with mathematical notation:

1. Union and Intersection Identities: #

  • Idempotent Laws: $$ A \cup A = A $$ $$ A \cap A = A $$

  • Identity Laws: $$ A \cup \emptyset = A $$ $$ A \cap U = A $$ (where ( U ) is the universal set) $$ A \cup U = U $$ $$ A \cap \emptyset = \emptyset $$

  • Domination Laws: $$ A \cup U = U $$ $$ A \cap \emptyset = \emptyset $$

2. Complement Laws: #

  • Complement of the Universal Set: $$ U^c = \emptyset $$
  • Complement of the Empty Set: $$ \emptyset^c = U $$
  • Double Complement: $$ (A^c)^c = A $$
  • Union and Intersection with Complement: $$ A \cup A^c = U $$ $$ A \cap A^c = \emptyset $$

3. Commutative Laws: #

  • Union: $$ A \cup B = B \cup A $$

  • Intersection: $$ A \cap B = B \cap A $$

4. Associative Laws: #

  • Union: $$ (A \cup B) \cup C = A \cup (B \cup C) $$

  • Intersection: $$ (A \cap B) \cap C = A \cap (B \cap C) $$

5. Distributive Laws: #

  • Union over Intersection:$$ A \cup (B \cap C) = (A \cup B) \cap (A \cup C) $$
  • Intersection over Union: $$ A \cap (B \cup C) = (A \cap B) \cup (A \cap C) $$

6. De Morgan’s Laws: #

  • Union: $$ (A \cup B)^c = A^c \cap B^c $$
  • Intersection:$$ (A \cap B)^c = A^c \cup B^c $$

7. Absorption Laws: #

  • Union: $$ A \cup (A \cap B) = A $$
  • Intersection:$$ A \cap (A \cup B) = A $$