We are given that \[ A \subset B. \]
We are required to show that \[ C - B \subset C - A. \]
Let \[ x \in C - B. \]
Then by definition of set difference,
\[ x \in C \quad \text{and} \quad x \notin B. \]
Since \[ A \subset B, \] every element of \( A \) is also an element of \( B \).
Hence, if \[ x \notin B, \] then \[ x \notin A. \]
Therefore, we have
\[ x \in C \quad \text{and} \quad x \notin A. \]
This implies
\[ x \in C - A. \]
Hence,
\[ C - B \subset C - A. \]
Thus, it is proved that if \[ A \subset B, \] then \[ C - B \subset C - A. \]
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?