 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/a/i/solution

Write

C^{c} = U ∖ C

(A \cap B) ∖ (A \cap C) = (A \cap B) \cap (A \cap C)^{c} = (A \cap B) \cap (A^{c} \cup C^{c}) = (A \cap B \cap A^{c}) \cup (A \cap B \cap C^{c}) = \emptyset \cup (A \cap (B \cap C^{c})) = A \cap (B ∖ C) .

The second step is De Morgan's laws; the third distributes set intersection over set union. The fourth uses

A \cap A^{c} = \emptyset

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