Let \[R = \left. {\{ (P,Q)} \right|P\]and \[Q\]are at the same distance from the origin\[\} \] be a relation, then the equivalence class of \[(1, - 1)\]is the set:
A. \[S = \left. {\{ (x,y)} \right|{x^2} + {y^2} = 1\} \]
B. \[S = \left. {\{ (x,y)} \right|{x^2} + {y^2} = 4\} \]
C. \[S = \left. {\{ (x,y)} \right|{x^2} + {y^2} = \sqrt 2 \} \]
D. \[S = \left. {\{ (x,y)} \right|{x^2} + {y^2} = 2\} \]

Hint: We are given a relation \[R = \{ P,Q\} \] where \[P\] and \[Q\] are at the same distance from the origin. We have to find the set of the equivalent class \[(1, - 1)\]. We know that the origin is given by \[(0,0)\]. We will be using the distance formula to find the required set.

Formula Used:
Following a formula will be useful for solving this question
\[d = \sqrt {{{({x_2} - {x_1})}^2} + {{({y_2} - {y_1})}^2}} \]

Complete step-by-step solution:
Let us assume the coordinates of \[P\] and \[Q\] as \[({x_1},{y_1})\] and \[({x_2},{y_2})\]
We are given that \[OP = OQ\]
By using distance formula, we get
\[
  \sqrt {{{({x_1} - 0)}^2} + {{({y_1} - 0)}^2}} = \sqrt {{{({x_2} - 0)}^2} + {{({y_2} - 0)}^2}} \\
  \sqrt {{{({x_1})}^2} + {{({y_1})}^2}} = \sqrt {{{({x_2})}^2} + {{({y_2})}^2}} \\
 \]
On squaring both sides we get
\[{({x_1})^2} + {({y_1})^2} = {({x_2})^2} + {({y_2})^2}\]

Similarly for the equivalence class \[(1, - 1)\],
Let us consider set \[R\] as \[(x,y)\] and \[S\] as \[(1, - 1)\]
On considering the equivalence class, we have \[OR = OS\]
By using the distance formula, we get
\[
  \sqrt {{{(x - 0)}^2} + {{(y - 0)}^2}} = \sqrt {{{(1 - 0)}^2} + {{( - 1 - 0)}^2}} \\
  \sqrt {{{(x)}^2} + {{(y)}^2}} = \sqrt {{{(1)}^2} + {{( - 1)}^2}} \\
 \]
On squaring both sides we get
\[
  {(x)^2} + {(y)^2} = {(1)^2} + {( - 1)^2} \\
   = 1 + 1 \\
   = 2 \\
 \]

Hence, option (D) is correct

Additional Information Distance is a numerical dimension of how far piecemeal objects or points are. In drugs or everyday operations, the distance may relate to a physical length or an estimation grounded on other criteria. A set is a fine model for a collection of different effects; a set contains rudiments or members, which can be found in objects of any kind figures, symbols, points in space, lines, other geometrical shapes, variables, or indeed other sets.

Note: Students may make mistakes while finding out the set of the equivalence class. It is very essential for students to use the distance formula properly in order to find the settings for the equivalence class.