From the image, the question involves Cartesian products of sets and identifying elements that belong to a particular Cartesian product. Let’s break it down step by step:

Sets Given:

• $A = \{1, 2\}$
• $B = \{1, 2, 3\}$
• $C = \{1, 10, 100\}$

Tasks:

1. Find the Cartesian Products:

   • $A \times B$
   • $B \times B$
   • $A \times B \times C$

2. **Identify an $n$-tuple that belongs to the product $(C \times B \times B \times A \times B)$.

Solution:

1. Cartesian Product $A \times B$:

• Each element of $A$ pairs with every element of $B$.
• $A \times B = \{(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3)\}$.

2. Cartesian Product $B \times B$:

• Each element of $B$ pairs with every element of $B$.
• $B \times B = \{(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)\}$.

3. Cartesian Product $A \times B \times C$:

• Each element of $A \times B$ pairs with every element of $C$.
• Result: $A \times B \times C = \{(1, 1, 1), (1, 1, 10), (1, 1, 100), \dots, (2, 3, 100)\}$, where $\dots$ indicates all combinations of elements from $A, B, C$.

4. Identify an $n$-tuple in $C \times B \times B \times A \times B$:

• Elements of $C$ pair with elements from $B, B, A,$ and $B$, forming 5-tuples.
• For example, a valid 5-tuple is $(10, 3, 2, 1, 3)$.

Would you like me to explain any part further or expand on Cartesian products? Here are some related questions to deepen your understanding:

1. What is the general formula for the size of a Cartesian product of two sets $X$ and $Y$?
2. Can you compute $B \times A$? How does it differ from $A \times B$?
3. How many elements are in $A \times B \times C$?
4. What is the role of order in Cartesian products? Does $A \times B$ equal $B \times A$?
5. How do you interpret a Cartesian product geometrically or visually?

Tip:

When working with Cartesian products, always ensure that the order of elements in the tuples matches the sets' sequence in the product!