https://file.aiquickdraw.com/mathbot/file/1730811947335p6pcgb6z.jpg

Let's solve this step-by-step.

We are given three sets:

1. **Set A**: \( A = \{ x \mid x^2 - 1 = 0 \} \)
2. **Set B**: \( B = \{ x \mid x^2 + x - 2 = 0 \} \)
3. **Set C**: \( C = \{ x \mid x^3 - x^2 = 0 \} \)

### Step 1: Determine the elements of each set

#### Set A: \( x^2 - 1 = 0 \)
This can be factored as:
\[
x^2 - 1 = (x - 1)(x + 1) = 0
\]
So, \( x = 1 \) or \( x = -1 \).

Thus, \( A = \{ -1, 1 \} \).

#### Set B: \( x^2 + x - 2 = 0 \)
This can be factored as:
\[
x^2 + x - 2 = (x + 2)(x - 1) = 0
\]
So, \( x = -2 \) or \( x = 1 \).

Thus, \( B = \{ -2, 1 \} \).

#### Set C: \( x^3 - x^2 = 0 \)
This can be factored as:
\[
x^3 - x^2 = x^2(x - 1) = 0
\]
So, \( x = 0 \) or \( x = 1 \).

Thus, \( C = \{ 0, 1 \} \).

### Step 2: Calculate the required set operations

#### Part (a): \( A \cup (B \cap C) \)

1. Find \( B \cap C \):
   - \( B = \{ -2, 1 \} \)
   - \( C = \{ 0, 1 \} \)
   - The intersection \( B \cap C = \{ 1 \} \).

2. Now find \( A \cup (B \cap C) \):
   - \( A = \{ -1, 1 \} \)
   - \( B \cap C = \{ 1 \} \)
   - The union \( A \cup (B \cap C) = \{ -1, 1 \} \).

So, \( A \cup (B \cap C) = \{ -1, 1 \} \).

#### Part (b): \( (A \cup B) \cap C \)

1. Find \( A \cup B \):
   - \( A = \{ -1, 1 \} \)
   - \( B = \{ -2, 1 \} \)
   - The union \( A \cup B = \{ -2, -1, 1 \} \).

2. Now find \( (A \cup B) \cap C \):
   - \( A \cup B = \{ -2, -1, 1 \} \)
   - \( C = \{ 0, 1 \} \)
   - The intersection \( (A \cup B) \cap C = \{ 1 \} \).

So, \( (A \cup B) \cap C = \{ 1 \} \).

#### Part (c): \( (A - B) \cup C \)

1. Find \( A - B \):
   - \( A = \{ -1, 1 \} \)
   - \( B = \{ -2, 1 \} \)
   - The difference \( A - B = \{ -1 \} \) (since 1 is in both A and B).

2. Now find \( (A - B) \cup C \):
   - \( A - B = \{ -1 \} \)
   - \( C = \{ 0, 1 \} \)
   - The union \( (A - B) \cup C = \{ -1, 0, 1 \} \).

So, \( (A - B) \cup C = \{ -1, 0, 1 \} \).

### Summary of Results

- **(a)** \( A \cup (B \cap C) = \{ -1, 1 \} \)
- **(b)** \( (A \cup B) \cap C = \{ 1 \} \)
- **(c)** \( (A - B) \cup C = \{ -1, 0, 1 \} \)

Would you like further details on any part of this solution?

### Related Questions

1. How do you find the intersection and union of two sets?
2. What are some common methods for factoring polynomials?
3. How can you visualize set operations with Venn diagrams?
4. How does the complement of a set affect union and intersection results?
5. Can we apply similar methods to solve inequalities involving set notation?

### Tip

Always double-check your factorizations and solutions for each set to ensure accuracy in set operations.

Diketahui A = {x | x^2 - 1 = 0}, B = {x | x^2 + x - 2 = 0}, dan C = {x | x^3 - x^2 = 0}. Tentukan: a. A ∪ (B ∩ C); b. (A ∪ B) ∩ C; dan c. (A - B) ∪ C.

Solve the set theory problem involving the union, intersection, and difference of sets A, B, and C, with conditions given by quadratic and cubic equations. This solution covers key concepts in algebra and set theory, including factoring and set operations, suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation