Prove that \((A \cap B)\backslash (A \cap C) = A \cap (B\backslash C)\) where A, B and C are three subsets of the universal set U.

\((A \cap B)\backslash (A \cap C) = (A \cap B) \cap (A \cap C)'\)     M1

\( = (A \cap B) \cap (A' \cup C')\)     A1

\( = (A \cap B \cap A') \cup (A \cap B \cap C')\)     A1

\( = (A \cap A' \cap B) \cup (A \cap B \cap C')\)     A1

\( = (\emptyset  \cap B) \cup (A \cap B \cap C')\)     (A1)

\( = \emptyset  \cup (A \cap B \cap C')\)

\( = \left( {A \cap (B \cap C')} \right)\)     A1

\( = A \cap (B\backslash C)\)     AG

Note: Do not accept proofs by Venn diagram.

[6 marks]

Venn diagram ‘proof’ are not acceptable. Those who used de Morgan’s laws usually were successful in this question.