The given problem involves sets and their Cartesian products and intersections. Let me break it down.

Problem

Given sets:

• $A = \{5, 6, 7\}$,
• $B = \{7, 8\}$,
• $C = \{5, 8\}$,

We are tasked to find:

1. $(A \times B) \cap (A \times C)$,
2. $A \times (B \cap C)$.

Step-by-Step Solution:

(i) $(A \times B) \cap (A \times C)$

1. Find $A \times B$: $A \times B = \{(a, b) \mid a \in A, b \in B\}$ Using the sets, $A \times B$ becomes: $A \times B = \{(5, 7), (5, 8), (6, 7), (6, 8), (7, 7), (7, 8)\}.$

2. Find $A \times C$: $A \times C = \{(a, c) \mid a \in A, c \in C\}$ Using the sets, $A \times C$ becomes: $A \times C = \{(5, 5), (5, 8), (6, 5), (6, 8), (7, 5), (7, 8)\}.$

3. Intersection $(A \times B) \cap (A \times C)$: Common elements between $A \times B$ and $A \times C$: $(A \times B) \cap (A \times C) = \{(5, 8), (6, 8), (7, 8)\}.$

(ii) $A \times (B \cap C)$

1. Find $B \cap C$: Intersection of $B$ and $C$: $B \cap C = \{8\}.$

2. Cartesian product $A \times (B \cap C)$: Using $A = \{5, 6, 7\}$ and $B \cap C = \{8\}$, we get: $A \times (B \cap C) = \{(5, 8), (6, 8), (7, 8)\}.$

Final Results:

1. $(A \times B) \cap (A \times C) = \{(5, 8), (6, 8), (7, 8)\}$,
2. $A \times (B \cap C) = \{(5, 8), (6, 8), (7, 8)\}$.

Would you like further explanation or examples? Here are five related questions to consider:

1. What are the key differences between a Cartesian product and an intersection of sets?
2. How would the results change if the sets $B$ and $C$ were disjoint?
3. Can you generalize the result for $(A \times B) \cap (A \times C)$ for any sets $A, B, C$?
4. How can these operations be visualized in a Venn diagram or grid representation?
5. What is the significance of the elements common between $B$ and $C$ in these calculations?

Tip: Cartesian products grow quickly in size as the sets increase; understanding their structure is crucial for simplifying problems like these!