Solve number 9-11

Given the sets:
$A = \{1, 2, 3\}$
$B = \{2, 3, 4\}$
$C = \{3, 4, 5\}$
Compute the number of elements in $A \cup B \cup C$, denoted as $|A \cup B \cup C|$.

First, find the union of all three sets:

$$A \cup B = \{1, 2, 3\} \cup \{2, 3, 4\} = \{1, 2, 3, 4\}$$ $$(A \cup B) \cup C = \{1, 2, 3, 4\} \cup \{3, 4, 5\}$$ $$A \cup B \cup C = \{1, 2, 3, 4, 5\}$$

Now, count the number of elements in the resulting set:

$$|A \cup B \cup C| = 5$$

The answer for Question 9 is:
$\boxed{5}$

Given:
Number of students enrolled in mathematics ($|M|$) = 100
Number of students enrolled in science ($|S|$) = 120
Number of students enrolled in both subjects ($|M \cap S|$) = 80
Find the number of students enrolled in at least one of the subjects, which is $|M \cup S|$.

Use the Principle of Inclusion-Exclusion for two sets:

$$|M \cup S| = |M| + |S| - |M \cap S|$$

Substitute the given values:

$$|M \cup S| = 100 + 120 - 80$$ $$|M \cup S| = 220 - 80$$ $$|M \cup S| = 140$$

The answer for Question 10 is:
$\boxed{140}$

Given the sets:
$A = \{1, 2\}$
$B = \{2, 3\}$
$C = \{3, 4\}$

a) Compute $(A \cup B) \cup C$
First, calculate $A \cup B$:

$$A \cup B = \{1, 2\} \cup \{2, 3\} = \{1, 2, 3\}$$

Next, find the union of this result with $C$:

$$(A \cup B) \cup C = \{1, 2, 3\} \cup \{3, 4\}$$ $$(A \cup B) \cup C = \{1, 2, 3, 4\}$$

b) Compute $A \cup (B \cup C)$
First, calculate $B \cup C$:

$$B \cup C = \{2, 3\} \cup \{3, 4\} = \{2, 3, 4\}$$

Next, find the union of $A$ with this result:

$$A \cup (B \cup C) = \{1, 2\} \cup \{2, 3, 4\}$$ $$A \cup (B \cup C) = \{1, 2, 3, 4\}$$

The answers for Question 11 are:

*a) $\boxed{\{1, 2, 3, 4\}}$

*b) $\boxed{\{1, 2, 3, 4\}}$