Let's break this problem down step by step. We are given three sets:

• $A = \{29, 30, 31\}$
• $B = \{30, 31, 32\}$
• $C = \{29, 31, 33\}$

The universe $U$ is the union of $A$, $B$, and $C$: $U = A \cup B \cup C = \{29, 30, 31, 32, 33\}$

We are asked to compute the following:

1. The sum of the elements of $A \cap B \cap C$

First, find the intersection of all three sets $A$, $B$, and $C$: $A \cap B \cap C = \{31\}$

So, the sum of the elements of $A \cap B \cap C$ is: $31$

2. The sum of the elements of $(A \cup B) \cap C$

First, compute $A \cup B$: $A \cup B = \{29, 30, 31, 32\}$ Now, find the intersection of $A \cup B$ with $C$: $(A \cup B) \cap C = \{29, 31\}$

So, the sum of the elements of $(A \cup B) \cap C$ is: $29 + 31 = 60$

3. The sum of the elements of $(B \cup C) \cap \neg A$

First, compute $B \cup C$: $B \cup C = \{29, 30, 31, 32, 33\}$ Next, find the complement of $A$ (i.e., $\neg A$) within the universe $U = \{29, 30, 31, 32, 33\}$: $\neg A = U - A = \{32, 33\}$ Now, find the intersection of $B \cup C$ with $\neg A$: $(B \cup C) \cap \neg A = \{32, 33\}$

So, the sum of the elements of $(B \cup C) \cap \neg A$ is: $32 + 33 = 65$

Final Answers:

1. The sum of the elements of $A \cap B \cap C$ is $31$.
2. The sum of the elements of $(A \cup B) \cap C$ is $60$.
3. The sum of the elements of $(B \cup C) \cap \neg A$ is $65$.

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Would you like further details or have any questions?

Here are some related questions to expand on the topic:

1. What is the union of $A \cap B \cap C$ with $A \cup B$?
2. How many elements are in the set $(A \cup B) \cap C$?
3. Can you describe the complement of $A$ in a different universal set?
4. What would happen if we replaced $B$ with another set in this problem?
5. How do the intersections of two sets differ from the intersection of three?

Tip: To better understand set operations, it can be helpful to draw Venn diagrams, as they visually represent intersections, unions, and complements.