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To solve these problems, we need to understand the concept of the Cartesian product of two sets. Let's analyze each part step by step.

### Part (a): Find \( A \times B \)

Given:
- Set \( A = \{n, r\} \)
- Set \( B = \{a, b, c\} \)

The Cartesian product \( A \times B \) consists of all ordered pairs \((x, y)\) where \(x \in A\) and \(y \in B\).

Calculating \( A \times B \):
1. Pairs involving \( n \): \((n, a), (n, b), (n, c)\)
2. Pairs involving \( r \): \((r, a), (r, b), (r, c)\)

So, \( A \times B = \{(n, a), (n, b), (n, c), (r, a), (r, b), (r, c)\} \).

Checking the provided options:
- Option **D** matches: \( \{(n, a), (n, b), (n, c), (r, a), (r, b), (r, c)\} \).

Therefore, the correct answer for part (a) is **D**.

### Part (b): Find \( B \times A \)

The Cartesian product \( B \times A \) consists of all ordered pairs \((x, y)\) where \(x \in B\) and \(y \in A\).

Calculating \( B \times A \):
1. Pairs involving \( a \): \((a, n), (a, r)\)
2. Pairs involving \( b \): \((b, n), (b, r)\)
3. Pairs involving \( c \): \((c, n), (c, r)\)

So, \( B \times A = \{(a, n), (a, r), (b, n), (b, r), (c, n), (c, r)\} \).

Checking the provided options:
- Option **B** matches: \( \{(a, n), (a, r), (b, n), (b, r), (c, n), (c, r)\} \).

Therefore, the correct answer for part (b) is **B**.

### Summary of Correct Answers:
- (a) **D**
- (b) **B**

Would you like further details or explanations on these problems? 

### Related Questions:
1. What is the Cartesian product of two sets and how is it calculated?
2. How does the order of sets affect the Cartesian product?
3. Can the Cartesian product of more than two sets be computed? If yes, how?
4. How is the concept of Cartesian products used in database management?
5. What are some applications of Cartesian products in real-world problems?

### Tip:
The Cartesian product is not commutative; that is, \( A \times B \neq B \times A \). The order of elements in each pair matters!

Let A = {n, r} and B = {a, b, c}. Use this information to complete parts (a) and (b) involving Cartesian products.

Learn how to calculate the Cartesian product of sets A = {n, r} and B = {a, b, c}. This problem covers key set theory concepts and helps develop an understanding of ordered pairs and set operations. Suitable for students in Grades 8-10.

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