Given the following​ sets, find the set

left parenthesis Upper A union Upper B right parenthesis prime intersect Upper C(A ∪ B)′ ∩ C.

Uequals=StartSet 1 comma 2 comma 3 comma . . . comma 9 EndSet{1, 2, 3, . . . , 9}

Aequals=StartSet 1 comma 2 comma 3 comma 5 EndSet{1, 2, 3, 5}

Bequals=StartSet 3 comma 7 comma 9 EndSet{3, 7, 9}

Cequals=StartSet 1 comma 2 comma 4 comma 5 comma 7 EndSet{1, 2, 4, 5, 7}

Question content area bottom

Part 1

Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.

A.

left parenthesis Upper A union Upper B right parenthesis prime intersect Upper C(A ∪ B)′ ∩ Cequals=StartSet nothing EndSetenter your response here

​(Use a comma to separate answers as​ needed.)

B.

left parenthesis Upper A union Upper B right parenthesis prime intersect Upper C(A ∪ B)′ ∩ C

is the empty set.

We are tasked with finding \( (A \cup B)' \cap C \), where the universal set \( U = \{1, 2, 3, \dots, 9\} \), and the sets \( A \), \( B \), and \( C \) are given as follows:

- \( A = \{1, 2, 3, 5\} \)
- \( B = \{3, 7, 9\} \)
- \( C = \{1, 2, 4, 5, 7\} \)

### Step 1: Find \( A \cup B \)
The union of sets \( A \) and \( B \) includes all the elements that are in either \( A \) or \( B \):

\[
A \cup B = \{1, 2, 3, 5\} \cup \{3, 7, 9\} = \{1, 2, 3, 5, 7, 9\}
\]

### Step 2: Find \( (A \cup B)' \)
The complement of \( A \cup B \), denoted as \( (A \cup B)' \), consists of the elements in the universal set \( U \) that are **not** in \( A \cup B \):

\[
(A \cup B)' = U - (A \cup B) = \{1, 2, 3, 4, 5, 6, 7, 8, 9\} - \{1, 2, 3, 5, 7, 9\} = \{4, 6, 8\}
\]

### Step 3: Find \( (A \cup B)' \cap C \)
Now, we find the intersection of \( (A \cup B)' \) and \( C \). This consists of the elements that are in both \( (A \cup B)' \) and \( C \):

\[
(A \cup B)' \cap C = \{4, 6, 8\} \cap \{1, 2, 4, 5, 7\} = \{4\}
\]

### Final Answer:
The set \( (A \cup B)' \cap C = \{4\} \).

Thus, the correct answer is:
\[
\boxed{\{4\}}
\]

Would you like further details or have any questions?

Here are five related questions to deepen your understanding:

1. What is the difference between the union and intersection of sets?
2. How do you compute the complement of a set with respect to a universal set?
3. If \( A \cap B = \emptyset \), what does that imply about \( A \cup B \)?
4. How do Venn diagrams help in visualizing set operations like union and intersection?
5. What is the relationship between the complement of a union and the complement of an intersection?

**Tip:** Always remember that the complement of a set \( A \) includes all elements in the universal set that are not in \( A \).

Given the following sets, find the set (A ∪ B)′ ∩ C. U = {1, 2, 3, ... , 9}, A = {1, 2, 3, 5}, B = {3, 7, 9}, C = {1, 2, 4, 5, 7}.

Learn how to find the set (A ∪ B)′ ∩ C using the universal set U = {1, 2, 3, ..., 9}, and the given sets A = {1, 2, 3, 5}, B = {3, 7, 9}, and C = {1, 2, 4, 5, 7}. This problem involves applying the union, complement, and intersection of sets, and is suitable for students in Grades 9-12.

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