 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-resit/1/b/solution

A \oplus B = (A ∖ B) \cup (B ∖ A)

. Put

D = A \oplus B

and

E = A \cap B

. Distribution of set intersection over set union gives

D \cap E = (A \cap B^{c} \cap A \cap B) \cup (B \cap A^{c} \cap A \cap B) = \emptyset .

For disjoint sets, set difference removes nothing, so

D \oplus E = (D ∖ E) \cup (E ∖ D) = D \cup E

. Consequently

(A \oplus B) \oplus (A \cap B) = (A \cap B^{c}) \cup (B \cap A^{c}) \cup (A \cap B) = A \cup (B \cap A^{c}) = (A \cup B) \cap (A \cup A^{c}) = A \cup B .

Here the second equality combines

A \cap B^{c}

with

A \cap B

; the last uses

A \cup A^{c} = U

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