 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2013/discrete-mathematics-probability-and-continuous-mathematics/1/c/iii/solution

Use the symmetric difference formula from part (b), followed by the set difference identity from part (c)(ii):

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

Taking both complements swaps the two ways to belong to exactly one set, leaving their union unchanged:

\overline{A} \oplus \overline{B} = A \oplus B .

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