Let's solve this step by step:

Given:

• $A = \{5, 8\}$
• $B = \{p, q\}$
• $C = \{r, v\}$

(a) How many elements are in $A \times B$?

The Cartesian product $A \times B$ consists of all ordered pairs where the first element is from set $A$ and the second element is from set $B$.

• Set $A$ has 2 elements.
• Set $B$ has 2 elements.

Thus, the number of elements in $A \times B$ is: $2 \times 2 = 4$

(b) How many elements are in $B \times C$?

The Cartesian product $B \times C$ consists of all ordered pairs where the first element is from set $B$ and the second element is from set $C$.

• Set $B$ has 2 elements.
• Set $C$ has 2 elements.

Thus, the number of elements in $B \times C$ is: $2 \times 2 = 4$

(c) How many elements are in $A \times B \times C$?

The Cartesian product $A \times B \times C$ consists of all ordered triples where the first element is from set $A$, the second element is from set $B$, and the third element is from set $C$.

• Set $A$ has 2 elements.
• Set $B$ has 2 elements.
• Set $C$ has 2 elements.

Thus, the number of elements in $A \times B \times C$ is: $2 \times 2 \times 2 = 8$

(d) Determine $A \times B \times C$

We need to list all the ordered triples in the form $(x, y, z)$, where $x \in A$, $y \in B$, and $z \in C$.

The elements are: $(5, p, r), (5, p, v), (5, q, r), (5, q, v), (8, p, r), (8, p, v), (8, q, r), (8, q, v)$

Summary:

1. (a) Number of elements in $A \times B$ is 4.
2. (b) Number of elements in $B \times C$ is 4.
3. (c) Number of elements in $A \times B \times C$ is 8.
4. (d) The elements of $A \times B \times C$ are: $\{(5, p, r), (5, p, v), (5, q, r), (5, q, v), (8, p, r), (8, p, v), (8, q, r), (8, q, v)\}$

Would you like a detailed explanation of any part? Here are some related questions you might find helpful:

1. What is the definition of a Cartesian product in set theory?
2. How can Cartesian products be visualized using diagrams?
3. What are some applications of Cartesian products in mathematics?
4. How does the concept of Cartesian products extend to higher dimensions?
5. Can Cartesian products be used with infinite sets?

Tip: The size of the Cartesian product is always the product of the sizes of the individual sets. For example, $|A \times B \times C| = |A| \times |B| \times |C|$.