Long Questions

10. If [tex]$A=\{2,5\}[tex]$[/tex] and [tex]$[/tex]B=\{3,4\}[tex]$[/tex], find [tex]$[/tex]A \times B$[/tex] and represent it in the following ways:
a) Tabulation method
b) Listing method
c) Set-builder method
d) Arrow diagram method



Answer :

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]