Îți voi arăta cum să procedezi la primele 2, iar ultimele le faci singur:
Definiții și idei folosite:
1) [tex]\text{ $\wedge$ - inseamna cuvantul "si", iar $\lor$ - sau}.[/tex]
2) [tex]x \in A \cap B \iff (x \in A) \wedge (x \in B)[/tex]
3) [tex]x \in (A \cup B) \iff (x \in A) \lor (x \in B)[/tex]
4) [tex](a \wedge b) \lor c = (a \lor c) \wedge (b \lor c)[/tex]
5) [tex](a \lor b) \wedge c = (a \wedge c) \lor (b \wedge c)[/tex]
6) [tex](A \subseteq B) \wedge (B \subseteq A) \iff A = B[/tex]
7) [tex]\text{Daca } a \in A \implies a \in A \cup B, \text{ pentru oricare multime $B$}.[/tex]
a) [tex](A \cup B) \cap A = A[/tex]
[tex]\text{Fie } x \in (A \cup B) \cap A \implies x \in A \cup B \text{ }\wedge\text{ } x \in A\\((x \in A) \lor (x \in B)) \wedge (x \in A) \implies ((x \in A) \wedge (x \in A)) \lor ((x \in B) \wedge (x \in A)) \implies (x \in A) \lor (x \in A \cap B) \implies x \in A \implies (A \cup B) \cap A \subseteq A.[/tex]
[tex]\text{Fie }x \in A \implies (x \in A \cup B) \wedge (x \in A) => x \in (A \cup B) \cap A \implies (A \cup B) \cap A \subseteq A\\\boxed{A = (A \cup B) \cap A}[/tex]
b) [tex](A \cap B) \cup A = A[/tex]
[tex]\text{Fie }x \in (A \cap B) \cup A \implies (x \in A \cap B) \lor (x \in A) \implies ((x \in A) \wedge (x \in B)) \lor (x \in A) \implies ((x \in A) \lor (x \in A)) \wedge ((x \in B) \lor (x \in A)) \implies (x \in A) \wedge ((x \in B) \lor (x \in A)) \implies x \in A \implies (A \cap B) \cup A \subseteq A.[/tex]
[tex]\text{Fie }x \in A \implies (x \in (A \cap B)) \lor (x \in A) \implies A \subseteq (A \cap B) \cup A\\\boxed{A = (A \cap B) \cup A}[/tex]
