Answer
See explanation
Work Step by Step
We are asked to prove the set identity:
\[
A \times (B \cap C) = (A \times B) \cap (A \times C)
\]
using an **element argument**.
Assume all sets are subsets of a universal set \(U\).
---
### ✅ Goal:
Prove both inclusions:
1. \(A \times (B \cap C) \subseteq (A \times B) \cap (A \times C)\)
2. \((A \times B) \cap (A \times C) \subseteq A \times (B \cap C)\)
---
### 🔹 Part 1: \(A \times (B \cap C) \subseteq (A \times B) \cap (A \times C)\)
Let \((a, x) \in A \times (B \cap C)\).
Then:
- \(a \in A\),
- \(x \in B \cap C\) → \(x \in B\) and \(x \in C\)
So:
- \((a, x) \in A \times B\),
- \((a, x) \in A \times C\)
Therefore:
\[
(a, x) \in (A \times B) \cap (A \times C)
\]
✅ So:
\[
A \times (B \cap C) \subseteq (A \times B) \cap (A \times C)
\]
---
### 🔹 Part 2: \((A \times B) \cap (A \times C) \subseteq A \times (B \cap C)\)
Let \((a, x) \in (A \times B) \cap (A \times C)\).
Then:
- \((a, x) \in A \times B\) → \(a \in A\), \(x \in B\)
- \((a, x) \in A \times C\) → \(a \in A\), \(x \in C\)
Therefore:
- \(x \in B \cap C\),
- and \(a \in A\)
So:
\[
(a, x) \in A \times (B \cap C)
\]
✅ Thus:
\[
(A \times B) \cap (A \times C) \subseteq A \times (B \cap C)
\]
---
### ✅ Final Conclusion:
Since both inclusions hold, we conclude:
\[
\boxed{A \times (B \cap C) = (A \times B) \cap (A \times C)}
\]
✔️ This completes the proof using an element argument.
