Discover a wealth of knowledge and get your questions answered on IDNLearn.com. Receive prompt and accurate responses to your questions from our community of knowledgeable professionals ready to assist you at any time.
Sagot :
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]
### 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]
We value your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. Find clear and concise answers at IDNLearn.com. Thanks for stopping by, and come back for more dependable solutions.
