To determine which set represents the same relation as the table given, we need to analyze each option and compare it to the table.
The given table is:
[tex]\[
\begin{array}{|c|c|}
\hline
x & r(x) \\
\hline
0 & 5 \\
\hline
4 & 2 \\
\hline
6 & 9 \\
\hline
9 & 10 \\
\hline
\end{array}
\][/tex]
This table can be represented as a set of ordered pairs denoting the relation between [tex]\( x \)[/tex] and [tex]\( r(x) \)[/tex]:
[tex]\[
\{(0, 5), (4, 2), (6, 9), (9, 10)\}
\][/tex]
Now let's evaluate each given set:
1. [tex]\(\{(0, 5), (4, 2), (6, 9), (9, 10)\}\)[/tex]
This set precisely matches the table's representation. Each ordered pair [tex]\((x, r(x))\)[/tex] corresponds exactly to the pairs described in the table.
2. [tex]\(\{(5, 0), (2, 4), (9, 6), (10, 9)\}\)[/tex]
This set contains pairs where the elements seem reversed relative to the table's pairs. For example, the pair [tex]\((5, 0)\)[/tex] suggests that 5 maps to 0, but in the table, 0 maps to 5. Therefore, this set does not match the table's relation.
3. [tex]\(\{0, 2, 4, 5, 6, 9, 10\}\)[/tex]
This set appears to list the domain and range values from the table, but it does not form ordered pairs to represent the relationships between [tex]\( x \)[/tex] and [tex]\( r(x) \)[/tex]. Thus, it does not match the table's relation either.
4. [tex]\(\{2, 5, 9, 10\}\)[/tex]
This set includes some of the range values [tex]\( r(x) \)[/tex] from the table, but like the third set, it does not provide the complete ordered pairs necessary to represent the relation between [tex]\( x \)[/tex] and [tex]\( r(x) \)[/tex]. Therefore, this set is also unsuitable.
The set that correctly represents the same relation as the given table is:
[tex]\[
\{(0, 5), (4, 2), (6, 9), (9, 10)\}
\][/tex]
Thus, the correct answer is:
1.
