Set Laws/Identities
The table below contains several set laws or identities, which are pairs of sets that are equal to each other (we will show that these laws are true in Topic 3:)
| Name | Intersection ($\boldsymbol{\cap}$) version | Union ($\boldsymbol{\cup}$) version |
|---|---|---|
| Identity Law | $A \cap U = U \cap A = A$ | $A \cup \emptyset = \emptyset \cup A = A$ |
| Null Law | $A \cap \emptyset = \emptyset$ | $A \cup U = U$ |
| Idempotent Law | $A \cap A = A$ | $A \cup A = A$ |
| Complement Law | $A \cap A^c = \emptyset$ | $A \cup A^c = U$ |
| Commutative Law | $A \cap B = B \cap A$ | $A \cup B = B \cup A$ |
| Associative Law | $(A \cap B) \cap C = A \cap (B \cap C)$ | $(A \cup B) \cup C = A \cup (B \cup C)$ |
| Distributive Law | $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$ | $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$ |
| Absorption Law | $A \cap (A \cup B) = A$ | $A \cup (A \cap B) = A$ |
| DeMorgan's Law | $(A \cap B)^c = A^c \cup B^c$ | $(A \cup B)^c = A^c \cap B^c$ |
| Double Complement (Involution) Law | ${(A^c)}^c = {A^c}^c = A$ | |
These notes by Miriam Briskman are licensed under CC BY-NC 4.0 and based on sources.