Certainly! Let's analyze the given sets \( A = \{2, 5\} \) and \( B = \{3, 4\} \) and find their Cartesian product \( A \times B \). We will represent this product in various formats: tabulation method, listing method, set-builder method, and arrow diagram method.
### a) Tabulation Method
In the tabulation method, we arrange the elements of \( A \) and \( B \) in a table to show all possible pairs \((a, b)\) where \( a \in A \) and \( b \in B \).
| \(A\) | \(B\) |
|------|------|
| 2, | 3 |
| 2, | 4 |
| 5, | 3 |
| 5, | 4 |
So the tabulation method gives us:
[tex]\[
\begin{array}{c|c}
A & B \\
\hline
2 & 3 \\
2 & 4 \\
5 & 3 \\
5 & 4 \\
\end{array}
\][/tex]
Which corresponds to the pairs: \((2, 3)\), \((2, 4)\), \((5, 3)\), and \((5, 4)\).
### b) Listing Method
In the listing method, we write out all ordered pairs of the Cartesian product explicitly as a list.
So, we list the pairs for \( A \times B \):
[tex]\[
A \times B = \{(2, 3), (2, 4), (5, 3), (5, 4)\}
\][/tex]
### c) Set-Builder Method
In the set-builder method, we describe the Cartesian product using a mathematical notation that shows the relationship between the elements of the sets without listing every element.
For the sets \( A \) and \( B \):
[tex]\[
A \times B = \{ (a, b) \mid a \in A, b \in B \}
\][/tex]
This translates to:
[tex]\[
A \times B = \{(a, b) \mid a \in \{2, 5\}, b \in \{3, 4\}\}
\][/tex]
### d) Arrow Diagram Method
In the arrow diagram method, we illustrate the Cartesian product as a diagram where elements of \( A \) are connected to elements of \( B \) with arrows, representing paired relations.
The arrow diagram for \( A \times B \) is:
```
2 ---> 3
2 ---> 4
5 ---> 3
5 ---> 4
```
Or represented more compactly as a mapping:
- \( 2 \to \{3, 4\} \)
- \( 5 \to \{3, 4\} \)
So, the arrow diagram method provides:
[tex]\[
\{ 2: [3, 4], 5: [3, 4] \}
\][/tex]
### Summary
Thus, given \( A = \{2, 5\} \) and \( B = \{3, 4\} \), we can represent \( A \times B \) in the following ways:
a) Tabulation method:
| \(A\) | \(B\) |
|------|------|
| 2 | 3 |
| 2 | 4 |
| 5 | 3 |
| 5 | 4 |
b) Listing method:
[tex]\[ A \times B = \{(2, 3), (2, 4), (5, 3), (5, 4)\} \][/tex]
c) Set-builder method:
[tex]\[ A \times B = \{(a,b) \mid a \in A, b \in B\} \][/tex]
d) Arrow diagram method:
[tex]\[ \{ 2: [3, 4], 5: [3, 4] \} \][/tex]
