Definitions:

Universal set : Ι
Empty set: Φ

Union of sets $A \cup B = \left\{x : x \in A ~~ or ~~ x \in B \right\}$ Intersection of sets $A \cap B = \left\{x : x \in A ~~ and ~~ x \in B \right\}$ Complement $A' = \left\{ x \in I : x \not \in A \right\}$ Difference of sets $A \setminus B = \left\{x : x \in A ~~ and ~~ x \not \in B \right\}$ Cartesian product $A \times B = \left\{ (x,y) : x \in A ~~ and ~~ y \in B \right\}$

Set identities involving union

Commutativity $A \cup B = B \cup A$ Associativity $A \cup \left(B \cup C \right) = \left( A \cup B \right) \cup C$ Idempotency $A \cup A = A$

Set identities involving intersection

Commutativity $A \cap B = B \cap A$ Associativity $A \cap \left(B \cap C \right) = \left( A \cap B \right) \cap C$ Idempotency $A \cap A = A$

Set identities involving union and intersection

Distributivity $A \cup \left(B \cap C \right) = \left(A \cup B \right) \cap \left(A \cup C \right)$ $A \cap \left(B \cup C \right) = \left(A \cap B \right) \cup \left(A \cap C \right)$ Domination $A \cap \varnothing = \varnothing$ $A \cup I = I$ Identity $A \cup \varnothing = \varnothing$ $A \cap I = A$

Set identities involving union, intersection and complement

Complement of intersection and union $A \cup A' = I$ $A \cap A' = \varnothing$ De Morgan's laws $\left( A \cup B \right)' = A' \cap B~'$ $\left(A \cap B \right)' = A' \cup B~'$ Set identities involving difference $B \setminus A = B \setminus \left( A \cup B \right)$ $B \setminus A = B \cap A'$ $A \setminus A = \varnothing$ $\left(A \setminus B \right) \cap C = \left(A \cap C \right) \setminus \left(B \cap C \right)$ $A' = I \setminus A$