$A$, $B$, $C$ and $D$ are subsets of $\mathbb{Z}$ .

$A = \{ \left. m \right|m{\text{ is a prime number less than 15}}\}$

$B = \{ \left. m \right|{m^4} = 8m\}$

$C = \{ \left. m \right|(m + 1)(m - 2) < 0\}$

$D = \{ \left. m \right|{m^2} < 2m + 4\}$

(a)     List the elements of each of these sets.

(b)     Determine, giving reasons, which of the following statements are true and which are false.

  (i)     $n(D) = n(B) + n(B \cup C)$

  (ii)     $D\backslash B \subset A$

  (iii)     $B \cap A' = \emptyset$

  (iv)     $n(B\Delta C) = 2$

(a)     by inspection, or otherwise,

A = {2, 3, 5, 7, 11, 13}     A1

B = {0, 2}     A1

C = {0, 1}     A1

D = {–1, 0, 1, 2, 3}     A1

[4 marks]

(b)     (i)     true     A1

$n(B) + n(B \cup C) = 2 + 3 = 5 = n(D)$     R1

(ii)     false     A1

$D\backslash B = \{  - 1,{\text{ }}1,{\text{ }}3\}  \not\subset A$     R1

(iii)     false     A1

$B \cap A' = \{ 0\}  \ne \emptyset$     R1

(iv)     true     A1     

$n(B\Delta C) = n\{ 1,{\text{ }}2\}  = 2$     R1

[8 marks]

Total [12 marks]

It was surprising and disappointing that many candidates regarded 1 as a prime number. One of the consequences of this error was that it simplified some of the set-theoretic calculations in part(b), with a loss of follow-through marks. Generally speaking, it was clear that the majority of candidates were familiar with the set operations in part(b).