To solve the problem, we need to calculate the number of elements in the Cartesian product $A \times B$.

Given sets:

• $A = \{4, 5, 6, \dots, 19\}$
• $B = \{8, 9, 10, \dots, 23\}$

Step 1: Calculate the number of elements in set $A$

Set $A$ contains all integers from 4 to 19, inclusive. The number of elements in $A$ is: $|A| = 19 - 4 + 1 = 16$

Step 2: Calculate the number of elements in set $B$

Set $B$ contains all integers from 8 to 23, inclusive. The number of elements in $B$ is: $|B| = 23 - 8 + 1 = 16$

Step 3: Calculate the number of elements in $A \times B$

The number of elements in the Cartesian product $A \times B$ is the product of the number of elements in $A$ and $B$: $|A \times B| = |A| \times |B| = 16 \times 16 = 256$

Thus, the number of elements in $A \times B$ is 256.

Would you like more details or have any other questions?

Related Questions:

1. What is the Cartesian product, and why is its size the product of the sizes of the sets?
2. How does the concept of set cardinality apply to infinite sets?
3. Can Cartesian products be applied to more than two sets? How does that affect the number of elements?
4. How would this problem change if $A$ and $B$ had overlapping elements?
5. What would the Cartesian product look like visually or in matrix form?

Tip: The Cartesian product pairs each element from one set with every element from another, making it an important concept in both set theory and relational databases.