Question

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?

• A. S1 is TRUE and S2 is FALSE
• B. S2 is TRUE and S1 is FALSE
• C. both S1 and S2 are TRUE
• D. neither S1 nor S2 is TRUE

Hint:

For continuous functions, the preimage of a compact set is compact. However, not all continuous functions guarantee this property for their inverse images, so analyze the specific behavior of the function.

Answer 1

The Correct Option is B

To determine the correctness of the statements S1 and S2, we need to analyze the properties of the functions \( f \) and \( g \) regarding compactness and inverse images.

Statement S1: "For every compact subset \( K \) of \( \mathbb{R} \), \( f^{-1}(K) \) is compact."

The function \( f(x_1, x_2) = x_1^2 + x_2^2 - 2x_1x_2 = (x_1 - x_2)^2 \) is not a continuous function when considered from \(\mathbb{R}^2\) to \(\mathbb{R}\) with respect to compactness since it maps \(\mathbb{R}^2\) to \(\mathbb{R}\). For a function to have the property that the inverse image of every compact set is compact, it needs to be proper, which can be interpreted as the preimage of each compact set in the codomain is compact in the domain. Since \(f(x_1, x_2) = (x_1 - x_2)^2\) does not hold the Lipschitz condition over the entire domain, \(f^{-1}(K)\) need not be compact for every compact \(K\).

Hence, S1 is FALSE.

Statement S2: "For every compact subset \( K \) of \( \mathbb{R} \), \( g^{-1}(K) \) is compact."

The function \( g(x_1, x_2) = 2x_1^2 + 2x_2^2 - x_1x_2 \) is quadratic in its arguments. It can be expressed in matrix form as: \[ \begin{pmatrix} x_1 & x_2 \end{pmatrix} \begin{pmatrix} 2 & -\frac{1}{2} \\ -\frac{1}{2} & 2 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix}. \]

The matrix \(\begin{pmatrix} 2 & -\frac{1}{2} \\ -\frac{1}{2} & 2 \end{pmatrix}\) is positive definite, meaning that it has only positive eigenvalues. Because \(g\) is a continuous function from \(\mathbb{R}^2\) to \(\mathbb{R}\) and a continuous function on a compact set is bounded and achieves its bounds, the preimage \(g^{-1}(K)\) of any compact set \(K\) in \(\mathbb{R}\) is also compact, as the positive definiteness implies all level sets are bounded.

Hence, S2 is TRUE.

Given the analysis, the correct answer is: S2 is TRUE and S1 is FALSE.