Mr Okafor is teaching his class about combining sets. He draws a Venn diagram showing two sets, \(P\) and \(Q\), inside the universal set \(\mathscr{E}\), a...
Question 1 Report
Mr Okafor is teaching his class about combining sets. He draws a Venn diagram showing two sets, \(P\) and \(Q\), inside the universal set \(\mathscr{E}\), and shades one region in grey for the class to describe.
- Write down the shaded region using set notation. (2)
- Given that \(n(\mathscr{E}) = 25\), \(n(P) = 12\), \(n(Q) = 9\) and \(n(P \cap Q) = 4\), work out the number of elements in the shaded region. (2)
Answer Details
A Venn diagram represents sets as regions inside a rectangle (the universal set \(\mathscr{E}\)). Shading identifies a specific combination of set membership, which can be written using set notation such as intersection (\(\cap\)), union (\(\cup\)), or complement (\(\,'\)).
- The shading covers the part of \(P\) that does not overlap with \(Q\): every element in \(P\) that is not also in \(Q\). This region is written as:\[ P \cap Q' \][2] (1 for identifying the region as "in \(P\) but not \(Q\)", 1 for the correct notation \(P \cap Q'\))
- The number of elements in \(P\) only is the total in \(P\) minus those shared with \(Q\):\[ n(P \cap Q') = n(P) - n(P \cap Q) = 12 - 4 = 8 \][2] (1 for \(n(P) - n(P \cap Q)\), 1 for the value 8)
Whenever two circles overlap on a Venn diagram, the region belonging to one circle only must have the overlapping part subtracted out first; simply reading off \(n(P) = 12\) for the shaded region would wrongly include the 4 elements shared with \(Q\).
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