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