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.