Write down all the elements of \(A\) and all the elements of \(B\).

Determine the symmetric difference, \(A\Delta B\), of the sets \(A\) and \(B\).

Write down all the elements of \(A \cap B,{\text{ }}A \cap C\) and \(B \cup C\).

Hence by considering \(A \cap (B \cup C)\), verify that in this case the operation \( \cap \) is distributive over the operation \( \cup \).

the elements of \(A\) are: 3, 7, 11, 15, 19     A1

the elements of \(B\) are 2, 3, 5, 7, 11, 13, 17, 19     A1

Note:     Accept \(A = \{ 3,{\text{ }}7,{\text{ }}11,{\text{ }}15,{\text{ }}19\} \) and \(B = \{ 2,{\text{ }}3,{\text{ }}5,{\text{ }}7,{\text{ }}11,{\text{ }}13,{\text{ }}17,{\text{ }}19\} \)

[2 marks]

attempt to determine \(A\backslash B \cup B\backslash A\) or \((A \cup B) \cap (A \cap B)'\)     (M1)

symmetric difference \( = \{ 2,{\text{ }}5,{\text{ }}13,{\text{ }}15,{\text{ }}17\} \)     A1

Note:     Allow (M1)A1FT.

[2 marks]

the elements of \(A \cap B\) are 3, 7, 11 and 19     A1

the elements of \(A \cap C\) are 7 and19     A1

the elements of \(B \cup C\) are 2, 3, 5, 7, 9, 11, 13, 17 and 19     A1

Note:     Accept \(A \cap B = \{ 3,{\text{ }}7,{\text{ }}11,{\text{ }}19\} ,{\text{ }}A \cap C = \{ 7,{\text{ }}19\} \) and \(B \cup C = \{ 2,{\text{ }}3,{\text{ }}5,{\text{ }}7,{\text{ }}9,{\text{ }}11,{\text{ }}13,{\text{ }}17,{\text{ }}19\} \).

[3 marks]

we need to show that

\(A \cap (B \cup C) = (A \cap B) \cup (A \cap C)\)     (M1)

\(A \cap (B \cup C) = \{ 3,{\text{ }}7,{\text{ }}11,{\text{ }}19\} \)     A1

\((A \cap B) \cup (A \cap C) = \{ 3,{\text{ }}7,{\text{ }}11,{\text{ }}19\} \)     A1

hence showing the required result

Note:     Allow (M1)A1FTA1FT.

[3 marks]