 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/b/solution

Use De Morgan's laws to remove the complement of a set, then distribute the two unions:

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

The first and last intersections in the middle line are the empty set. The remaining union is precisely the collection of elements belonging to exactly one of the original sets. Therefore both expressions define the same symmetric difference:

(A \cup B) ∖ (A \cap B) = (A ∖ B) \cup (B ∖ A) = A \oplus B .

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