Scaffolding pallets on a building site are tagged with the numbers 1 to 30, so \(\mathcal{E} = \{x : x \text{ is an integer}, 1 \le x \le 30\}\). Let \(A = ...
Question 1 Report
Scaffolding pallets on a building site are tagged with the numbers 1 to 30, so \(\mathcal{E} = \{x : x \text{ is an integer}, 1 \le x \le 30\}\). Let \(A = \{x : x \text{ is a multiple of } 3\}\) and \(B = \{x : x \text{ is a multiple of } 4\}\).
- List the elements of \(A \cap B\) (2)
- Find \(n(A \cup B)\) (2)
Answer Details
Both \(A\) and \(B\) here are described by a rule applied to the range 1 to 30, so they must be listed before combining, and the counting formula avoids listing the full union.
(a) The multiples of 3 that are also multiples of 4 are the multiples of 12: within 1 to 30 these are 12 and 24, so \(A \cap B = \{12, 24\}\). [2 marks]
(b) Counting the multiples of 3 up to 30 gives \(n(A) = 10\), and the multiples of 4 give \(n(B) = 7\), with the overlap \(n(A \cap B) = 2\). So \(n(A \cup B) = n(A) + n(B) - n(A \cap B) = 10 + 7 - 2 = 15\). [2 marks]
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