In a science lab the universal set \(\mathcal{E}\) is the 40 students doing practical work. Set \(C\) is those at a chemistry bench, and \(n(C) = 27\). Writ...
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In a science lab the universal set \(\mathcal{E}\) is the 40 students doing practical work. Set \(C\) is those at a chemistry bench, and \(n(C) = 27\).
- Write down \(n(\mathcal{E})\). (1)
- Find \(n(C')\). (1)
Answer Details
The universal set \(\mathcal{E}\) contains everything under discussion, here all the students doing practical work. The notation \(n(\ )\) means the number of members of a set, and the dash in \(C'\) means the complement of \(C\): everything in \(\mathcal{E}\) that is not in \(C\).
(a) The universal set is the \(40\) students, so
\[n(\mathcal{E}) = 40\][1]
(b) Every student is either at a chemistry bench or not, and no student can be both, so the two groups account for the whole universal set with no overlap:
\[n(C') = n(\mathcal{E}) - n(C) = 40 - 27 = 13\][1]
So \(13\) students are not at a chemistry bench.
The relationship \(n(C) + n(C') = n(\mathcal{E})\) always holds, whatever the set, which makes it a quick way to check any complement calculation. Note that the complement is measured against the universal set as defined in the question, the \(40\) students doing practical work, not against everyone in the school.
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