\(A\), \(B\), \(C\) 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).