Additional Mathematics 4037/12 — May/June 2017

Walkthrough

This question tests reading set expressions onto a three-circle Venn diagram. Each diagram earns one mark.

For $(A \cup B) \cap C$: build the inner set first. $A \cup B$ covers every point in either circle. The intersection with $C$ then acts like a filter: keep only what lies inside $C$ as well. So the shaded area is exactly the portions of circle $C$ that touch $A$ or touch $B$ — two lens shapes meeting at the central triple-overlap. The bottom sliver of $C$, which belongs to neither $A$ nor $B$, is excluded.

For $(A \cap B) \cup C$: again start inside. $A \cap B$ is the single lens where $A$ and $B$ cross. The union with $C$ merges that lens with the whole of circle $C$, so effectively everything except the 'only $A$' and 'only $B$' crescents is shaded.

For $(A' \cap B') \cap C$: the complement notation flips things. $A'$ is outside $A$, $B'$ is outside $B$, so $A' \cap B'$ is the region belonging to neither. Intersecting with $C$ traps us inside $C$, leaving just the crescent of $C$ that avoids both other circles.

Key Takeaways

• Work compound set expressions from the inside out, exactly like nested brackets in algebra.
• Union = shade everything in either set; intersection = shade only the common part; complement = everything outside.
• On a three-circle diagram there are eight distinct regions; each expression selects a specific combination of them.

Common Mistakes

• Shading all of $C$ for $(A \cup B) \cap C$ instead of only the parts of $C$ meeting $A$ or $B$ — forgetting the intersection filter.
• For $(A \cap B) \cup C$, shading only the $A \cap B$ lens and forgetting to include the whole of $C$.
• Misreading $A' \cap B'$ as '$A$ and $B$' rather than 'outside both $A$ and $B$', which would wrongly shade the triple overlap.
• Including the triple-overlap region when it should be excluded (or vice versa) — check each of the eight regions against the expression before finishing.

Things to Be Careful About

• The mark scheme awards B1 for each correct diagram (B3 total), so each of the three shadings must be complete and exact — a partially correct shading scores zero for that diagram.
• The boundary between shaded and unshaded must follow the circle edges precisely; shading that spills into a neighbouring region loses the mark.
• Remember $A'$ means the complement (everything outside $A$), not the set $A$ itself.