 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2015/discrete-mathematics-continuous-mathematics-and-probability/1/a/ii/solution

U

x \in U

x \in \overline{A \cap B} x \in \overline{A \cup B} ⟺ \neg (x \in A \land x \in B) ⟺ (x \in / A \lor x \in / B) ⟺ x \in \overline{A} \cup \overline{B}, ⟺ \neg (x \in A \lor x \in B) ⟺ (x \in / A \land x \in / B) ⟺ x \in \overline{A} \cap \overline{B} .

Equality of membership for every element proves the set identities

\overline{A \cap B} = \overline{A} \cup \overline{B}, \overline{A \cup B} = \overline{A} \cap \overline{B} .

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