The question pertains to Cartesian products and involves multiple sets $A$, $B$, and $C$ with their elements explicitly listed:

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

Analysis of the Questions in the Image:

1. Identifying Cartesian Products:

   • $A \times A$ produces all pairs of elements where the first and second elements are from $A$. For example: $A \times A = \{(1,1), (1,2), (2,1), (2,2)\}$
   • Similarly, $B \times B$, $C \times B \times A$, etc., are computed using the same concept.

2. Choosing an $n$-tuple for $C \times B \times B \times A \times B$:

   • The structure of the $n$-tuple corresponds to the product $(C \times B \times B \times A \times B)$, which means:
     • The first element comes from $C$,
     • The second and third elements come from $B$,
     • The fourth element comes from $A$,
     • The fifth element comes from $B$.

   You would need to pick one element from each set according to this sequence.

Would you like me to explain Cartesian products in detail or verify an $n$-tuple from the options provided? Let me know!

Expanding Your Knowledge

Here are 5 related questions to deepen your understanding:

1. What is the difference between Cartesian products and power sets?
2. How can you calculate the size (number of elements) of a Cartesian product like $A \times B$?
3. Can Cartesian products be extended to infinite sets? If so, how are they defined?
4. How do Cartesian products relate to relations and functions in mathematics?
5. How can Cartesian products be visualized graphically or geometrically?

Tip:

When working with Cartesian products, keep track of the sequence of sets, as the position in the $n$-tuple determines which set each element is chosen from.