We are required to find sets A, B and C such that \( A \cap B \), \( B \cap C \) and \( A \cap C \) are all non-empty, but \[ A \cap B \cap C = \varnothing. \]
Consider the following sets:
\[ A = \{1, 2\}, \quad B = \{2, 3\}, \quad C = \{1, 3\}. \]
Now, let us find their pairwise intersections:
\[ A \cap B = \{2\} \neq \varnothing \]
\[ B \cap C = \{3\} \neq \varnothing \]
\[ A \cap C = \{1\} \neq \varnothing \]
Now, find the intersection of all three sets:
\[ A \cap B \cap C = \varnothing \]
Hence, the sets \[ A = \{1, 2\}, \quad B = \{2, 3\}, \quad C = \{1, 3\} \] satisfy the given conditions.
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?