We are tasked with solving set theory problems based on the given sets:

• $A = \{1, 3, 4, 6, 7, 9\}$
• $B = \{4, 5, 6, 8\}$
• $C = \{4, 5, 8\}$

a) $A \cup B$ (Union of $A$ and $B$):

The union of two sets includes all elements that belong to either $A$ or $B$ (or both): $A \cup B = \{1, 3, 4, 6, 7, 9, 5, 8\}$

b) $A \cap B$ (Intersection of $A$ and $B$):

The intersection of two sets includes only the elements that belong to both $A$ and $B$: $A \cap B = \{4, 6\}$

c) $A \setminus B$ (Difference of $A$ and $B$):

The difference of $A$ and $B$ includes all elements of $A$ that are not in $B$: $A \setminus B = \{1, 3, 7, 9\}$

d) $(A \cup B) \setminus C$:

1. First, find $A \cup B$ (already computed in part (a)): $A \cup B = \{1, 3, 4, 6, 7, 9, 5, 8\}$
2. Subtract $C$ from $A \cup B$: $(A \cup B) \setminus C = \{1, 3, 6, 7, 9\}$ Explanation: $C = \{4, 5, 8\}$, so we remove $4, 5, 8$ from $A \cup B$.

e) $(B \cap C) \setminus A$:

1. First, find $B \cap C$: $B \cap C = \{4, 5, 8\}$ Explanation: These are the elements common to $B$ and $C$.
2. Subtract $A$ from $B \cap C$: $(B \cap C) \setminus A = \{5, 8\}$ Explanation: The elements $4$ is in $A$, so it is removed from $B \cap C$.

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Final Answers:

a) $A \cup B = \{1, 3, 4, 6, 7, 9, 5, 8\}$
b) $A \cap B = \{4, 6\}$
c) $A \setminus B = \{1, 3, 7, 9\}$
d) $(A \cup B) \setminus C = \{1, 3, 6, 7, 9\}$
e) $(B \cap C) \setminus A = \{5, 8\}$

Would you like further explanations or visualizations for any part?

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Here are 5 related questions for further exploration:

1. What is $A \cap C$ for the given sets?
2. How do we compute $B \setminus A$?
3. If $D = \{6, 7\}$, what is $A \cup D$?
4. How would the results change if $B = \{4, 5, 6, 7, 8\}$?
5. Is $C \subset B$ in the current setup?

Tip: When solving set problems, using a Venn diagram often helps visualize unions, intersections, and differences!