# Set Theory Results

Consider that $A$,$B$ and $C$ are the sets, then

1. $A \subseteq A \cup B$ and $B \subseteq A \cup B$

2. $A \cup \emptyset = A$

3. $A \cap \emptyset = \emptyset$

4. $A \subseteq C$ and $B \subseteq C$ then $A \cup B \subseteq C$

5. $A \subset B$ if and only if $A \cup B = B$

6. $A \subset B$ if and only if $A \cap B = A$

7. $A \cap B \subseteq A$ and $A \cap B \subseteq B$

8. If $A \subseteq B$ and $A \subseteq C$ then $A \subseteq B \cap C$

9. $A \cap U = A$

10.$A \cup U = U$

11.${({A^C})^C} = A$

12.${\emptyset ^C} = U$ and ${U^C} = \emptyset$

13.$A \cup {A^C} = U$

14. $A \cap {A^C} = \emptyset$

15. If $A \subseteq B$ then ${B^C} \subseteq {A^C}$

16. If  $x \in A$ and $x \in B$ then $x \in A \cap B$

17. If $x \in A$ or $x \in B$ then $x \in A \cup B$

18. $A – B = A \cap {B^C}$

19. $A – (A – B) = A \cap B$

20. $A – (A – B) = A – (A \cap {B^C})$