https://file.aiquickdraw.com/mathbot/file/1732075450744139vbmpj.jpg

Let's solve the problem given in the image.

### Problem
Given sets:
- \( A = \{5, 6, 7\} \)
- \( B = \{7, 8\} \)
- \( C = \{5, 8\} \)

Find:
1. \( (A \times B) \cap (A \times C) \)
2. \( (A \cup B \cup C)' \) (assuming the complement is taken with respect to the universal set, which we’ll assume later)

### Solution

#### Part (i) \( (A \times B) \cap (A \times C) \)
1. **Find \( A \times B \)**: This is the Cartesian product of sets \( A \) and \( B \), which includes all ordered pairs where the first element is from \( A \) and the second element is from \( B \).
   \[
   A \times B = \{ (5,7), (5,8), (6,7), (6,8), (7,7), (7,8) \}
   \]

2. **Find \( A \times C \)**: Similarly, the Cartesian product of sets \( A \) and \( C \).
   \[
   A \times C = \{ (5,5), (5,8), (6,5), (6,8), (7,5), (7,8) \}
   \]

3. **Intersection \( (A \times B) \cap (A \times C) \)**: We find common elements in \( A \times B \) and \( A \times C \).
   \[
   (A \times B) \cap (A \times C) = \{ (5,8), (6,8), (7,8) \}
   \]

So, the answer for part (i) is:
\[
(A \times B) \cap (A \times C) = \{ (5,8), (6,8), (7,8) \}
\]

#### Part (ii) \( (A \cup B \cup C)' \)
To find the complement \( (A \cup B \cup C)' \), we first need to determine \( A \cup B \cup C \).

1. **Union \( A \cup B \cup C \)**:
   \[
   A \cup B \cup C = \{5, 6, 7, 8\}
   \]

2. **Complement \( (A \cup B \cup C)' \)**: To find the complement, we need to assume a universal set. If no universal set is given, we can't determine this directly. However, if we assume the universal set \( U \) includes elements that are naturally within the range of numbers 1 to, say, 10, then:
   \[
   U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}
   \]
   Then the complement is:
   \[
   (A \cup B \cup C)' = U - (A \cup B \cup C) = \{1, 2, 3, 4, 9, 10\}
   \]

Thus, the answer for part (ii), assuming \( U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \), is:
\[
(A \cup B \cup C)' = \{1, 2, 3, 4, 9, 10\}
\]

Would you like further details or have any specific questions?

### Related Questions:
1. How would the answer change if the universal set \( U \) had different elements?
2. Can you explain how to find the Cartesian product of two sets?
3. What are some properties of set complements?
4. How does the intersection of Cartesian products differ from the union of Cartesian products?
5. How is the concept of set complement useful in probability?

### Tip:
When dealing with sets and Cartesian products, always start by clearly defining each set operation step by step to avoid errors.

If A = {5, 6, 7}, B = {7, 8}, and C = {5, 8}, then find (i) (A × B) ∩ (A × C) and (ii) (A ∪ B ∪ C)'.

Explore the solution to a set theory problem involving Cartesian products, intersections, unions, and complements. Learn how to compute (A × B) ∩ (A × C) and (A ∪ B ∪ C)' step by step with given sets A = {5, 6, 7}, B = {7, 8}, and C = {5, 8}. Suitable for Grades 9-12.

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