(i) {2,3,4}\( ⊂\) {1,2,3,4,5}
(ii) {a,b,c} \(⊄\) {b,c,d}
(iii) {x: x is a student of class XI of your school} \(⊂\) {x: x is student of your school}
(iv) {x: x is a circle in the plane} \(⊄\) {x: x is a circle in the same plane with radius 1 unit}
(v) {x: x is a triangle in a plane} \( ⊄\) {x: x is a rectangle in the plane}
(vi) {x: x is an equilateral triangle in a plane} \(⊂\) {x: x in a triangle in the same plane}
(vii) {x: x is an even natural number} ⊂ {x: x is an integer}
Consider the following subsets of the Euclidean space \( \mathbb{R}^4 \):
\( S = \{ (x_1, x_2, x_3, x_4) \in \mathbb{R}^4 : x_1^2 + x_2^2 + x_3^2 - x_4^2 = 0 \} \),
\( T = \{ (x_1, x_2, x_3, x_4) \in \mathbb{R}^4 : x_1^2 + x_2^2 + x_3^2 - x_4^2 = 1 \} \),
\( U = \{ (x_1, x_2, x_3, x_4) \in \mathbb{R}^4 : x_1^2 + x_2^2 + x_3^2 - x_4^2 = -1 \} \).
Then, which one of the following is TRUE?
Let the functions \( f: \mathbb{R} \to \mathbb{R} \) and \( g: \mathbb{R}^2 \to \mathbb{R} \) be given by \[ f(x_1, x_2) = x_1^2 + x_2^2 - 2x_1x_2, \quad g(x_1, x_2) = 2x_1^2 + 2x_2^2 - x_1x_2. \] Consider the following statements:
S1: For every compact subset \( K \) of \( \mathbb{R} \), \( f^{-1}(K) \) is compact.
S2: For every compact subset \( K \) of \( \mathbb{R} \), \( g^{-1}(K) \) is compact. Then, which one of the following is correct?