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