 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2017/discrete-mathematics-continuous-mathematics-and-probability/1/b/ii/solution

x \in A \oplus B

, the definition of symmetric difference puts it in exactly one of

A, B

. It is therefore in their set union and outside their set intersection, giving

x \in (A \cup B) ∖ (A \cap B)

Conversely, an element of

(A \cup B) ∖ (A \cap B)

lies in at least one of the two sets but not both. It consequently belongs to

A ∖ B

or to

B ∖ A

, and hence to

A \oplus B

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