(a) If A = {multiples of 2}, B = {multiples of 3} and C = {factors of 6} are subsets of \(\mu\) = {x: \(1 \leq x \leq 10\)} find A′ \(\cap\) B′ \(\cap\) C′ ...
Question 1 Report
(a) If A = {multiples of 2}, B = {multiples of 3} and C = {factors of 6} are subsets of \(\mu\) = {x: \(1 \leq x \leq 10\)} find A′ \(\cap\) B′ \(\cap\) C′
(b) Tickets for a movie premiere cost $18.50 each while the bulk purchase price for 5 tickets is $80.00. If 4 gentlemen decide to get a fifth person to join them so that they can share the bulk purchase price equally, how much would each person save?
Answer Details
(a) The universal set is \(\mu = \{1,2,3,4,5,6,7,8,9,10\}\).
- \(A = \{\text{multiples of }2\} = \{2,4,6,8,10\}\)
- \(B = \{\text{multiples of }3\} = \{3,6,9\}\)
- \(C = \{\text{factors of }6\} = \{1,2,3,6\}\)
By De Morgan's law, \(A' \cap B' \cap C' = (A \cup B \cup C)'\).
\[A \cup B \cup C = \{1,2,3,4,6,8,9,10\}.\]
Therefore the elements of \(\mu\) not in this union are:
\[A' \cap B' \cap C' = \{5, 7\}.\]
(b) Buying singly, one ticket costs \(\$18.50\). The bulk price for 5 tickets is \(\$80.00\), shared equally among the 5 people:
\[\text{Cost per person} = \frac{80.00}{5} = \$16.00.\]
Each person's saving compared with buying a single ticket:
\[18.50 - 16.00 = \$2.50.\]
Each person would save \(\mathbf{\$2.50}\).
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