# Set Identities

• Sets: $$A$$, $$B$$, $$C$$
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: $$A\backslash B$$
1. $$A \subset I$$
2. $$A \subset A$$
3. $$A = B,$$ if $$A \subset B$$ and $$B \subset A$$
4. Empty set $$\emptyset \subset A$$
5. Union of sets $$C = A \cup B =$$ $$\left\{ {x \mid x \in A\;\text{or}\;x \in B} \right\}$$
6. Commutativity of union $$A \cup B = B \cup A$$
7. Associativity of union $$A \cup \left( {B \cup C} \right) =$$ $$\left( {A \cup B} \right) \cup C$$
8. Intersection of sets $$C = A \cap B =$$ $$\left\{ {x \mid x \in A\;\text{and}\;x \in B} \right\}$$
9. Commutativity of intersection $$A \cap B = B \cap A$$
10. Associativity of intersection $$A \cap \left( {B \cap C} \right) = \left( {A \cap B} \right) \cap C$$
11. Distributivity
$$A \cup \left( {B \cap C} \right) =$$ $$\left( {A \cup B} \right) \cap \left( {A \cup C} \right)$$
$$A \cap \left( {B \cup C} \right) =$$ $$\left( {A \cap B} \right) \cup \left( {A \cap C} \right)$$
12. Idempotency
$$A \cap A = A$$
$$A \cup A = A$$
13. Domination (Intersection of any set with the empty set) $$A \cap \emptyset = \emptyset$$
14. Union of any set with the universal set $$A \cup I = I$$
15. Union of any set with the empty set $$A \cup \emptyset = A$$
16. Intersection of any set with the universal set $$A \cap I = A$$
17. Complement $$\overline A = \left\{ {x \in I \mid x \notin A} \right\}$$
18. Properties of the Complement
$$A \cup \overline A = I$$
$$A \cap \overline A = \emptyset$$
19. De Morgan’s laws
$$\overline {\left( {A \cup B} \right)} = \overline A \cap \overline B$$
$$\overline {\left( {A \cap B} \right)} = \overline A \cup \overline B$$
20. Difference of sets $$C = B\backslash A =$$ $$\left\{ {x \mid x \in B\;\text{and}\;x \notin A} \right\}$$
21. $$B\backslash A = B\backslash \left( {A \cap B} \right)$$
22. $$B\backslash A = B \cap \overline A$$
23. Difference of a set from itself $$A\backslash A = \emptyset$$
24. $$A\backslash B = A,\;$$ $$\text{if}\;\;A \cap B = \emptyset$$
25. $$\left( {A\backslash B} \right) \cap C =$$ $$\left( {A \cap C} \right)\backslash \left( {B \cap C} \right)$$
26. $$\overline A = I\backslash A$$
27. Cartesian product $$C = A \times B =$$ $$\left\{ {\left( {x,y} \right) \mid x \in A\;\text{and}\;y \in B} \right\}$$