We are given two sets:

• $A = \{1, 2, 3, \dots, 13\}$
• $B = \{1, 2, 3, \dots, 24\}$

The tasks are:

1. The number of elements in $A$:

   Set $A$ starts from 1 and goes up to 13, so the total number of elements in $A$ is: $|A| = 13$

2. The number of elements in $B$:

   Set $B$ starts from 1 and goes up to 24, so the total number of elements in $B$ is: $|B| = 24$

3. The number of elements in $A \times B$:

   The Cartesian product $A \times B$ contains all ordered pairs $(a, b)$, where $a \in A$ and $b \in B$. The number of elements in the Cartesian product is the product of the number of elements in $A$ and $B$: $|A \times B| = |A| \times |B| = 13 \times 24 = 312$

Summary:

• The number of elements in $A$ is $13$.
• The number of elements in $B$ is $24$.
• The number of elements in $A \times B$ is $312$.

Would you like further explanation on any part of this?

Here are some related questions to consider:

1. What is the Cartesian product in set theory?
2. How do you calculate the number of elements in the union of two sets?
3. What would $A \cup B$ look like for the given sets?
4. How does set intersection work, and what is $A \cap B$?
5. What is the difference between Cartesian product and a power set?

Tip: When working with sets, always check if they have overlapping elements, as this is important for operations like unions or intersections.