Given sets A, B, and C, compute their unions and intersections. Also, compute the symmetric difference between sets A and B.

Given the sets:
$A = \{1, 2, 3, 4\}$
$B = \{3, 4, 5, 6\}$
$C = \{2, 4, 6, 8\}$

a) Compute $A \cup B$
The union of two sets includes all elements present in either set.

$$A \cup B = \{1, 2, 3, 4\} \cup \{3, 4, 5, 6\}$$ $$A \cup B = \{1, 2, 3, 4, 5, 6\}$$

b) Compute $A \cap C$
The intersection of two sets includes only the elements common to both sets.

$$A \cap C = \{1, 2, 3, 4\} \cap \{2, 4, 6, 8\}$$ $$A \cap C = \{2, 4\}$$

c) Compute $(A \cup B) \cap C$
First, calculate $A \cup B$. From part (a), $A \cup B = \{1, 2, 3, 4, 5, 6\}$.
Next, find the intersection of this result with $C$.

$$(A \cup B) \cap C = \{1, 2, 3, 4, 5, 6\} \cap \{2, 4, 6, 8\}$$ $$(A \cup B) \cap C = \{2, 4, 6\}$$

The answers for Question 6 are:

*a) $\boxed{\{1, 2, 3, 4, 5, 6\}}$

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

*c) $\boxed{\{2, 4, 6\}}$

Given the sets:
$A = \{1, 2, 3\}$
$B = \{3, 4, 5\}$
Find the symmetric difference between A and B, denoted as $A \Delta B$.
The symmetric difference is defined as $A \Delta B = (A - B) \cup (B - A)$.

First, calculate $A - B$:

$$A - B = \{1, 2, 3\} - \{3, 4, 5\} = \{1, 2\}$$

Next, calculate $B - A$:

$$B - A = \{3, 4, 5\} - \{1, 2, 3\} = \{4, 5\}$$

Finally, find the union of these two results:

$$A \Delta B = \{1, 2\} \cup \{4, 5\}$$ $$A \Delta B = \{1, 2, 4, 5\}$$

The answer for Question 7 is:
$\boxed{\{1, 2, 4, 5\}}$

Given the sets:
$A = \{2, 4, 6, 8\}$
$B = \{4, 6, 10\}$

a) Compute $A - B$
The set difference $A - B$ contains elements that are in A but not in B.

$$A - B = \{2, 4, 6, 8\} - \{4, 6, 10\}$$ $$A - B = \{2, 8\}$$

b) Compute $B - A$
The set difference $B - A$ contains elements that are in B but not in A.

$$B - A = \{4, 6, 10\} - \{2, 4, 6, 8\}$$ $$B - A = \{10\}$$

The answers for Question 8 are:

*a) $\boxed{\{2, 8\}}$

*b) $\boxed{\{10\}}$

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