Let
A
=
{
0
,
1
,
2
,
.
.
.
,
8
}
and
B
=
{
2
,
3
,
4
,
.
.
.
,
26
}
.
What is the number of elements in
A
×
B
?

We are given two sets, \( A \) and \( B \), and the problem asks us to find the number of elements in the Cartesian product \( A \times B \).

### Step-by-step Solution:

1. **Set \( A \):**
   \[
   A = \{0, 1, 2, \dots, 8\}
   \]
   The elements in \( A \) are from 0 to 8, inclusive. The number of elements in \( A \) is:
   \[
   |A| = 9 \quad (\text{since there are 9 elements from 0 to 8})
   \]

2. **Set \( B \):**
   \[
   B = \{2, 3, 4, \dots, 26\}
   \]
   The elements in \( B \) are from 2 to 26, inclusive. The number of elements in \( B \) is:
   \[
   |B| = 26 - 2 + 1 = 25
   \]
   There are 25 elements in set \( B \).

3. **Cartesian Product \( A \times B \):**
   The Cartesian product \( A \times B \) consists of 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 \( B \):
   \[
   |A \times B| = |A| \times |B| = 9 \times 25 = 225
   \]

### Final Answer:
The number of elements in \( A \times B \) is \( \boxed{225} \).

---

Would you like further details or clarifications?

Here are 5 related questions to explore:
1. What is the general formula for the number of elements in the Cartesian product of two sets?
2. How would the result change if set \( B \) had fewer elements, say from 2 to 20?
3. What is the significance of Cartesian products in coordinate geometry?
4. Can Cartesian products be used to represent real-world relationships? If yes, how?
5. How do Cartesian products behave with infinite sets?

**Tip**: The Cartesian product of two sets gives you a way to pair elements, which is crucial in defining relations and functions in mathematics.

Let A = {0, 1, 2, ..., 8} and B = {2, 3, 4, ..., 26}. What is the number of elements in A × B?

Learn how to calculate the number of elements in the Cartesian product of two sets A = {0, 1, 2, ..., 8} and B = {2, 3, 4, ..., 26}. This problem explores set theory, Cartesian products, and cardinality. Suitable for students in Grades 9-12.

Math Problem Statement

Let A

{ 0 , 1 , 2 , . . . , 8 } and B

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