Let A = ℝ × ℝ . A relation R is defined on A by : (x_1, y_1) R...

For Statement I :
Reflexivity: $(x, y) R (x, y) \Rightarrow x^2 - y^2 = x^2 - y^2$, which is true.
Symmetry: $(x_1, y_1) R (x_2, y_2) \Rightarrow x_1^2 - y_1^2 = x_2^2 - y_2^2 \Rightarrow x_2^2 - y_2^2 = x_1^2 - y_1^2 \Rightarrow (x_2, y_2) R (x_1, y_1)$, which is true.
Transitivity: $(x_1, y_1) R (x_2, y_2)$ and $(x_2, y_2) R (x_3, y_3) \Rightarrow x_1^2 - y_1^2 = x_2^2 - y_2^2$ and $x_2^2 - y_2^2 = x_3^2 - y_3^2 \Rightarrow x_1^2 - y_1^2 = x_3^2 - y_3^2 \Rightarrow (x_1, y_1) R (x_3, y_3)$, which is true.
Thus, $R$ is an equivalence relation. Statement I is true.

For Statement II :
The equivalence class of $(2, 2)$ is the set of all points $(x, y)$ such that $(x, y) R (2, 2)$.
This gives $x^2 - y^2 = 2^2 - 2^2 = 0$.
$x^2 - y^2 = 0 \Rightarrow (x - y)(x + y) = 0$, which represents a pair of intersecting straight lines ($y = x$ and $y = -x$), not a rectangular hyperbola.
Thus, Statement II is false.

Answer: Statement I is true but Statement II is false