Which set represents the same relation as the table below?

[tex]\[
\begin{tabular}{|c|c|}
\hline
$x$ & $r(x)$ \\
\hline
0 & 5 \\
\hline
4 & 2 \\
\hline
6 & 9 \\
\hline
9 & 10 \\
\hline
\end{tabular}
\][/tex]

A. [tex]\(\{(0,5), (4,2), (6,9), (9,10)\}\)[/tex]

B. [tex]\(\{(5,0), (2,4), (9,6), (10,9)\}\)[/tex]

C. [tex]\(\{0, 2, 4, 5, 6, 9, 10\}\)[/tex]

D. [tex]\(\{2, 5, 9, 10\}\)[/tex]



Answer :

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.