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]