(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\...

Answer Details

(a) Finding ${\mathrm{<span\; style="font-family:\; Ubuntu;\; font-size:\; 16px;">A</span>}}^{\mathrm{<span\; style="font-family:\; Ubuntu;\; font-size:\; 16px;">\prime </span>}}<span\; style="font-family:\; Ubuntu;\; font-size:\; 16px;">\cap </span>{\mathrm{<span\; style="font-family:\; Ubuntu;\; font-size:\; 16px;">B</span>}}^{\mathrm{<span\; style="font-family:\; Ubuntu;\; font-size:\; 16px;">\prime </span>}}<span\; style="font-family:\; Ubuntu;\; font-size:\; 16px;">\cap </span>{\mathrm{<span\; style="font-family:\; Ubuntu;\; font-size:\; 16px;">C</span>}}^{\mathrm{<span\; style="font-family:\; Ubuntu;\; font-size:\; 16px;">\prime </span>}}$A' \cap B' \cap C'

Given:

• $\mu =\{x:1\le x\le 10\}=\{1,2,3,4,5,6,7,8,9,10\}\backslash mu\; =\; \backslash \{\; x\; :\; 1\; \backslash leq\; x\; \backslash leq\; 10\; \backslash \}\; =\; \backslash \{1,\; 2,\; 3,\; 4,\; 5,\; 6,\; 7,\; 8,\; 9,\; 10\backslash \}$
• $A=\{\text{multiples of 2}\}=\{2,4,6,8,10\}A\; =\; \backslash \{\; \backslash text\{multiples\; of\; 2\}\; \backslash \}\; =\; \backslash \{2,\; 4,\; 6,\; 8,\; 10\backslash \}$
• $B=\{\text{multiples of 3}\}=\{3,6,9\}B\; =\; \backslash \{\; \backslash text\{multiples\; of\; 3\}\; \backslash \}\; =\; \backslash \{3,\; 6,\; 9\backslash \}$
• $C=\{\text{factors of 6}\}=\{1,2,3,6\}C\; =\; \backslash \{\; \backslash text\{factors\; of\; 6\}\; \backslash \}\; =\; \backslash \{1,\; 2,\; 3,\; 6\backslash \}$

First, find the complements $A^{\mathrm{\prime }},B^{\mathrm{\prime }},A\text{'},\; B\text{'},$ and $C^{\mathrm{\prime }}C\text{'}$:

• $A^{\mathrm{\prime }}=\mu -A=\{1,3,5,7,9\}A\text{'}\; =\; \backslash mu\; -\; A\; =\; \backslash \{1,\; 3,\; 5,\; 7,\; 9\backslash \}$
• $B^{\mathrm{\prime }}=\mu -B=\{1,2,4,5,7,8,10\}B\text{'}\; =\; \backslash mu\; -\; B\; =\; \backslash \{1,\; 2,\; 4,\; 5,\; 7,\; 8,\; 10\backslash \}$
• $C^{\mathrm{\prime }}=\mu -C=\{4,5,7,8,9,10\}C\text{'}\; =\; \backslash mu\; -\; C\; =\; \backslash \{4,\; 5,\; 7,\; 8,\; 9,\; 10\backslash \}$

Now, find the intersection $A^{\mathrm{\prime }}\cap B^{\mathrm{\prime }}\cap C^{\mathrm{\prime }}A\text{'}\; \backslash cap\; B\text{'}\; \backslash cap\; C\text{'}$:

$A^{\mathrm{\prime }}\cap B^{\mathrm{\prime }}=\{1,3,5,7,9\}\cap \{1,2,4,5,7,8,10\}=\{1,5,7\}A\text{'}\; \backslash cap\; B\text{'}\; =\; \backslash \{1,\; 3,\; 5,\; 7,\; 9\backslash \}\; \backslash cap\; \backslash \{1,\; 2,\; 4,\; 5,\; 7,\; 8,\; 10\backslash \}\; =\; \backslash \{1,\; 5,\; 7\backslash \}$

$A^{\mathrm{\prime }}\cap B^{\mathrm{\prime }}\cap C^{\mathrm{\prime }}=\{1,5,7\}\cap \{4,5,7,8,9,10\}=\{5,7\}A\text{'}\; \backslash cap\; B\text{'}\; \backslash cap\; C\text{'}\; =\; \backslash \{1,\; 5,\; 7\backslash \}\; \backslash cap\; \backslash \{4,\; 5,\; 7,\; 8,\; 9,\; 10\backslash \}\; =\; \backslash \{5,\; 7\backslash \}$

So, $A^{\mathrm{\prime }}\cap B^{\mathrm{\prime }}\cap C^{\mathrm{\prime }}=\{5,7\}A\text{'}\; \backslash cap\; B\text{'}\; \backslash cap\; C\text{'}\; =\; \backslash \{5,\; 7\backslash \}$.

(b) Calculating the Savings Per Person

Given:

• Individual ticket price: $18.50
• Bulk purchase price for 5 tickets: $80.00

First, calculate the total cost of 5 individual tickets: $5\times 18.50=92.505\; \backslash times\; 18.50\; =\; 92.50$

The cost difference between individual tickets and the bulk purchase price: $92.50-80.00=12.5092.50\; -\; 80.00\; =\; 12.50$

Each person would save: $\frac{12.50}{5}=2.50\backslash frac\{12.50\}\{5\}\; =\; 2.50$

So, if the 4 gentlemen get a fifth person to join them and share the bulk purchase price equally, each person would save $2.50.