Determine, using Venn diagrams, whether the following statements are true.

(i)     \(A' \cup B' = (A \cup B)'\)

(ii)     \((A\backslash B) \cup (B\backslash A) = (A \cup B)\backslash (A \cap B)\)

Prove, without using a Venn diagram, that \(A\backslash B\) and \(B\backslash A\) are disjoint sets.

(a)     (i)

    A1     A1

since the shaded regions are different, \(A' \cup B' \ne (A \cup B)'\)     R1

\( \Rightarrow \) not true

(ii)

     A1

     A1

since the shaded regions are the same \((A\backslash B) \cup (B\backslash A) = (A \cup B)\backslash (A \cap B)\)     R1

\( \Rightarrow \) true 

[6 marks]

\(A\backslash B = A \cup B'\) and \(B\backslash A = B \cap A'\)     (A1)

consider \(A \cap B' \cap B \cap A'\)     M1

now \(A \cap B' \cap B \cap A' = \emptyset \)     A1

since this is the empty set, they are disjoint     R1

Note: Accept alternative valid proofs.

[4 marks]

Part (a) was accessible to most candidates, but a number drew incorrect Venn diagrams. In some cases the clarity of the diagram made it difficult to follow what the candidate intended. Candidates found (b) harder, although the majority made a reasonable start to the proof. Once again a number of candidates were let down by poor explanation.

Part (a) was accessible to most candidates, but a number drew incorrect Venn diagrams. In some cases the clarity of the diagram made it difficult to follow what the candidate intended. Candidates found (b) harder, although the majority made a reasonable start to the proof. Once again a number of candidates were let down by poor explanation.