Suppose A={29,30,31}A={29,30,31}, B={30,31,32}B={30,31,32}, and C={29,31,33}C={29,31,33} are sets. The universe UU is the union of AA, BB and CC

Give the sum of the elements of the set A∩B∩CA∩B∩C  

Give the sum of the elements of the set (A∪B)∩C(A∪B)∩C  

Give the sum of the elements of the set (B∪C)∩−A(B∪C)∩−A  

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 \( 65 \).

---

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.

Suppose A={29,30,31}, B={30,31,32}, and C={29,31,33} are sets. The universe U is the union of A, B, and C. Give the sum of the elements of the set A∩B∩C. Give the sum of the elements of the set (A∪B)∩C. Give the sum of the elements of the set (B∪C)∩−A.

Solve set theory problems with detailed solutions, including the sum of elements in A∩B∩C, (A∪B)∩C, and (B∪C)∩-A. Learn about set operations like union, intersection, and complements. Ideal for Grades 10-12.

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