Prove that set difference is not associative.

we are trying to prove \((A\backslash B)\backslash C \ne A\backslash (B\backslash C)\)     M1(A1)

\({\text{LHS}} = (A \cap B')\backslash C\)     (A1)

\( = (A \cap B') \cap C'\)     A1

\({\text{RHS}} = A\backslash (B \cap C')\)

\( = A \cap (B \cap C')'\)     (A1)

\( = A \cap (B' \cup C)\)     A1

as LHS does not contain any element of C and RHS does, \({\text{LHS}} \ne {\text{RHS}}\)     R1

hence set difference is not associative     AG

Note: Accept answers which use a proof containing a counter example.

Total [7 marks]

This question was found difficult by a large number of candidates, but a number of correct solutions were seen. A number of candidates who understood what was required failed to gain the final reasoning mark. Many candidates seemed to be ill-prepared to deal with this style of question.