 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2016/discrete-mathematics-continuous-mathematics-and-probability/1/b/ii/solution

Because

A \subseteq U

A ∖ U

is empty and

U ∖ A = \overline{A}

. Therefore the symmetric difference satisfies

A \oplus U = (A ∖ U) \cup (U ∖ A) = \emptyset \cup \overline{A} = \overline{A} .

A

and

\overline{A}

have union

U

, so

A \oplus \overline{A} = (A ∖ \overline{A}) \cup (\overline{A} ∖ A) = A \cup \overline{A} = U .

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