Additional Mathematics 4037/22 — October/November 2011

Walkthrough

The four regions of a two-circle Venn diagram are: inside $A$ only, inside both (the intersection), inside $B$ only, and outside both circles.

First, $n(A' \cap B) = 6$ means exactly 6 elements lie in $B$ but not in $A$. Since all of $B$ contains 20 elements, the remaining elements of $B$ must be in the overlap:

$$n(A \cap B) = 20 - 6 = 14$$

Next, $A$ as a whole contains 15 elements, of which 14 are already accounted for in the overlap, so the part of $A$ outside $B$ holds just $15 - 14 = 1$ element.

Finally, the universal set holds 30 elements in total. Adding the three regions inside the circles gives $1 + 14 + 6 = 21$, so the region outside both circles contains $30 - 21 = 9$ elements.

Key Takeaways

• Each number given about a set pins down one or more regions by subtraction.
• $A' \cap B$ is read directly as "in $B$ but not in $A$".
• The universal-set total lets you find the outside region once every other region is known.
• Always check that the four region values sum to $n(\mathscr{E})$.

Common Mistakes

• Writing 6 in the intersection instead of the "$B$ only" region — $A' \cap B$ excludes $A$.
• Putting $n(A) = 15$ straight into the "$A$ only" region without subtracting the overlap.
• Forgetting to find the outside region at all, so losing the final B1FT mark.
• Mark scheme notes: each value must be correctly positioned — a correct number in the wrong region scores nothing.

Things to Be Careful About

• The last value ($9$) is marked B1FT: it follows from your own three earlier values, so an arithmetic slip earlier still earns this mark if your subtraction is consistent.
• Check the sum: $1 + 14 + 6 + 9 = 30 = n(\mathscr{E})$ confirms the diagram.

Walkthrough

Build the expression in two stages. First, $P \cup Q$ covers both upper circles completely, including their overlap. Second, $R'$ removes everything belonging to $R$, so intersecting with $R'$ deletes from that union every piece that falls inside the lower circle $R$. What remains — and what should be shaded — is the whole of $P$ and the whole of $Q$ apart from their portions inside $R$.

Key Takeaways

• Union means "or": shade everything in either set.
• Complement means "not": intersecting with $R'$ erases anything inside $R$.
• Overlap regions between $P$ and $Q$ stay shaded provided they do not touch $R$.

Common Mistakes

• Shading only $P$ and forgetting $Q$ (or vice versa).
• Leaving unshaded the $P \cap Q$ overlap because it looks like a separate region — it belongs to $P \cup Q$ and stays shaded unless it meets $R$.
• Shading inside $R$ by misreading the complement.

Things to Be Careful About

• This is a single B1 mark: the entire region must be shaded correctly; partial shading scores zero.