Let $$R = \{(P, Q) | P \text{ and } Q \text{ are at the same distance from the origin}\}$$ be a relation, then the equivalence class of $$(1, -1)$$ is the set:

The relation $$R$$ is defined on points in the plane such that $$(P, Q) \in R$$ if and only if $$P$$ and $$Q$$ are at the same distance from the origin. This means $$P$$ and $$Q$$ lie on the same circle centered at the origin.

The equivalence class of a point $$(1, -1)$$ consists of all points $$(x, y)$$ that are at the same distance from the origin as $$(1, -1)$$.

The distance of $$(1, -1)$$ from the origin is $$\sqrt{1^2 + (-1)^2} = \sqrt{2}$$.

Therefore the equivalence class is the set of all points $$(x, y)$$ satisfying $$\sqrt{x^2 + y^2} = \sqrt{2}$$, which gives $$x^2 + y^2 = 2$$.

Hence $$S = \{(x, y) \mid x^2 + y^2 = 2\}$$.