Sets: \(A\), \(B\), \(C\)

Universal set: \(I\)

Complement: \(\overline A \)

Proper subset: \(A \subset B\)

Universal set: \(I\)

Complement: \(\overline A \)

Proper subset: \(A \subset B\)

Empty set: \(\emptyset \)

Union of sets: \(A \cup B\)

Intersection of sets: \(A \cap B\)

Difference of sets: \(A\backslash B\)

Union of sets: \(A \cup B\)

Intersection of sets: \(A \cap B\)

Difference of sets: \(A\backslash B\)

- \(A \subset I\)
- \(A \subset A\)
- \(A = B,\) if \(A \subset B\) and \(B \subset A\)
- Empty set \(\emptyset \subset A\)
- Union of sets \(C = A \cup B =\) \( \left\{ {x \mid x \in A\;\text{or}\;x \in B} \right\}\)
- Commutativity of union \(A \cup B = B \cup A\)
- Associativity of union \(A \cup \left( {B \cup C} \right) =\) \( \left( {A \cup B} \right) \cup C\)
- Intersection of sets \(C = A \cap B =\) \( \left\{ {x \mid x \in A\;\text{and}\;x \in B} \right\}\)
- Commutativity of intersection \(A \cap B = B \cap A\)
- Associativity of intersection \(A \cap \left( {B \cap C} \right) = \left( {A \cap B} \right) \cap C\)
- Distributivity

\(A \cup \left( {B \cap C} \right) =\) \( \left( {A \cup B} \right) \cap \left( {A \cup C} \right)\)

\(A \cap \left( {B \cup C} \right) =\) \( \left( {A \cap B} \right) \cup \left( {A \cap C} \right)\) - Idempotency

\(A \cap A = A\)

\(A \cup A = A\) - Domination (Intersection of any set with the empty set) \(A \cap \emptyset = \emptyset \)
- Union of any set with the universal set \(A \cup I = I\)
- Union of any set with the empty set \(A \cup \emptyset = A\)
- Intersection of any set with the universal set \(A \cap I = A\)
- Complement \(\overline A = \left\{ {x \in I \mid x \notin A} \right\}\)
- Properties of the Complement

\(A \cup \overline A = I\)

\(A \cap \overline A = \emptyset \) - De Morgan’s laws

\(\overline {\left( {A \cup B} \right)} = \overline A \cap \overline B \)

\(\overline {\left( {A \cap B} \right)} = \overline A \cup \overline B \) - Difference of sets \(C = B\backslash A =\) \( \left\{ {x \mid x \in B\;\text{and}\;x \notin A} \right\}\)
- \(B\backslash A = B\backslash \left( {A \cap B} \right)\)
- \(B\backslash A = B \cap \overline A \)
- Difference of a set from itself \(A\backslash A = \emptyset \)
- \(A\backslash B = A,\;\) \(\text{if}\;\;A \cap B = \emptyset \)
- \(\left( {A\backslash B} \right) \cap C =\) \( \left( {A \cap C} \right)\backslash \left( {B \cap C} \right)\)
- \(\overline A = I\backslash A\)
- Cartesian product \(C = A \times B =\) \( \left\{ {\left( {x,y} \right) \mid x \in A\;\text{and}\;y \in B} \right\}\)