Sets 

*A*

, 

*B*

, and 

*C*

 are subsets of the universal set 

*U*

.
These sets are defined as follows.

*U*

= *f*, *g*, *h*, *p*, *q*, *r*, *x*, *y*, *z* 

*A*

= *f*, *g*, *h*, *q*, *r* 

*B*

= *q*, *r*, *x*, *z* 

*C*

= *h*, *p*, *q*, *z* 

Find 

∪∩*B*′*AC*

.
Write your answer in roster form or as 

∅

.

=∪∩*B*′*AC*

To solve for \( \left( A \cap C \cap B' \right) \cup B \), we'll go through the problem step by step.

### Given Sets:
- Universal set \( U = \{f, g, h, p, q, r, x, y, z\} \)
- Set \( A = \{f, g, h, q, r\} \)
- Set \( B = \{q, r, x, z\} \)
- Set \( C = \{h, p, q, z\} \)

### Step 1: Find the complement of \( B \), denoted as \( B' \)

The complement of \( B \) contains all elements in the universal set \( U \) that are not in \( B \):

\[
B' = U \setminus B = \{f, g, h, p, y\}
\]

### Step 2: Find the intersection of \( A \cap C \)

We now find the common elements between sets \( A \) and \( C \):

\[
A = \{f, g, h, q, r\}
\]
\[
C = \{h, p, q, z\}
\]

The intersection \( A \cap C \) is:

\[
A \cap C = \{h, q\}
\]

### Step 3: Find the intersection of \( A \cap C \cap B' \)

Now we intersect \( A \cap C = \{h, q\} \) with \( B' = \{f, g, h, p, y\} \):

\[
A \cap C \cap B' = \{h\}
\]

### Step 4: Find the union of \( (A \cap C \cap B') \cup B \)

Now we take the union of \( \{h\} \) and \( B = \{q, r, x, z\} \):

\[
(A \cap C \cap B') \cup B = \{h\} \cup \{q, r, x, z\} = \{h, q, r, x, z\}
\]

### Final Answer:

\[
\left( A \cap C \cap B' \right) \cup B = \{h, q, r, x, z\}
\]

This is the result in roster form.

Would you like further clarification or details on any of the steps?

---

Here are 5 related questions to deepen understanding:
1. What is the result of \( A \cup B \)?
2. How do you compute \( A \cap B' \)?
3. What is the complement of set \( C \) in the universal set \( U \)?
4. Can you find the intersection \( B \cap C \)?
5. What is \( (A \cup B) \cap (A \cup C) \)?

**Tip**: When working with set operations, drawing a Venn diagram can help visualize intersections and unions more clearly.

Sets A, B, and C are subsets of the universal set U. These sets are defined as follows. U = {f, g, h, p, q, r, x, y, z} A = {f, g, h, q, r} B = {q, r, x, z} C = {h, p, q, z} Find (A ∩ C ∩ B') ∪ B. Write your answer in roster form or as ∅.

Learn how to solve a set theory problem involving the universal set U = {f, g, h, p, q, r, x, y, z} and subsets A, B, and C. We explore union, intersection, and complement operations step-by-step. Suitable for Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation