Represent the following set on a Venn diagram,

$A\Delta B$, the symmetric difference of the sets $A$ and $B$;

Represent the following set on a Venn diagram,

$A \cap (B \cup C)$.

For sets $P$, $Q$ and $R$, verify that $P \cup (Q\Delta R) \ne (P \cup Q)\Delta (P \cup R)$.

In the context of the distributive law, describe what the result in part (b)(i) illustrates.

     A1

[1 mark]

Note: Accept alternative set configurations

     A1

Note:     Accept alternative set configurations.

[1 mark]

LHS:

$Q\Delta R = \{ 1,{\text{ }}2,{\text{ }}4,{\text{ }}5\}$     (A1)

$P \cup (Q\Delta R) = \{ 1,{\text{ }}2,{\text{ }}3,{\text{ }}4,{\text{ }}5\}$     A1

RHS:

$P \cup Q = \{ 1,{\text{ }}2,{\text{ }}3,{\text{ }}4\}$ and $P \cup R = \{ 1,{\text{ }}2,{\text{ }}3,{\text{ }}5\}$     (A1)

$(P \cup Q)\Delta (P \cup R) = \{ 4,{\text{ }}5\}$     A1

hence $P \cup (Q\Delta R) \ne (P \cup Q)\Delta (P \cup R)$     AG

[4 marks]

the result shows that union is not distributive over symmetric difference     A1R1

Notes:     Award A1 for “union is not distributive” and R1 for “over symmetric difference”. Condone use of $\cup$ and $\Delta$.

[2 marks]