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.