Certainly! Let's express the given relation [tex]\(\{(5,3),(4,3),(3,3),(2,3),(1,3)\}\)[/tex] in three different ways: as a table, as a graph, and as a mapping diagram.
### 1. Table Representation
The table for the provided relation is already specified. Here it is again for clarity:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
5 & 3 \\
\hline
4 & 3 \\
\hline
3 & 3 \\
\hline
2 & 3 \\
\hline
1 & 3 \\
\hline
\end{array}
\][/tex]
### 2. Graph Representation
To create a graph of the relation, plot each of the points [tex]\((x, y)\)[/tex] on a coordinate plane.
1. The point [tex]\((5, 3)\)[/tex]
2. The point [tex]\((4, 3)\)[/tex]
3. The point [tex]\((3, 3)\)[/tex]
4. The point [tex]\((2, 3)\)[/tex]
5. The point [tex]\((1, 3)\)[/tex]
Since all the [tex]\(y\)[/tex] values are the same (i.e., 3), the points will lie on the line [tex]\(y = 3\)[/tex]. The [tex]\(x\)[/tex] values will be spread along the [tex]\(x\)[/tex]-axis.
#### Scatter Plot:
- Draw a horizontal line at [tex]\(y = 3\)[/tex].
- Mark the points at [tex]\( (5, 3) \)[/tex], [tex]\( (4, 3) \)[/tex], [tex]\( (3, 3) \)[/tex], [tex]\( (2, 3) \)[/tex], and [tex]\( (1, 3) \)[/tex].
The graph will look something like this:
```
y
|
4 ------
3 * ← All points lie on y = 3
2 ------
1 ------
1 2 3 4 5 x
```
### 3. Mapping Diagram
In a mapping diagram, we represent two columns: one for the [tex]\(x\)[/tex]-values and one for the [tex]\(y\)[/tex]-values. Then we draw arrows from each [tex]\(x\)[/tex]-value to its corresponding [tex]\(y\)[/tex]-value.
Mapping Diagram:
[tex]\[
\begin{array}{c|c}
x & y \\
\hline
5 & \\
4 & \\
3 & \\
2 & \\
1 & \\
\end{array}
\quad
\begin{array}{r}
\rightarrow 3 \\
\rightarrow 3 \\
\rightarrow 3 \\
\rightarrow 3 \\
\rightarrow 3 \\
\end{array}
\][/tex]
By representing it with arrows:
```
x y
5 → 3
4 → 3
3 → 3
2 → 3
1 → 3
```
In the mapping diagram, each element from the [tex]\(x\)[/tex]-set [tex]\(\{5, 4, 3, 2, 1\}\)[/tex] maps to the same element in the [tex]\(y\)[/tex]-set [tex]\(\{3\}\)[/tex].
In conclusion, we have expressed the given relation [tex]\(\{(5,3),(4,3),(3,3),(2,3),(1,3)\}\)[/tex] as a table, graph, and mapping diagram.
