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.