Question

A subset S ⊆ \(\R^2\) is said to be bounded if there is an M > 0 such that |x| ≤ M and |y| ≤ M for all (x, y) ∈ S. Which of the following subsets of \(\R^2\) is/are bounded ?

• A. \({(x, y) ∈ \R^2 : e^{x^2} + y^2 \le 4}\)
• B. \({(x, y) ∈ \R^2 : x^4 + y^2 \le 4}\)
• C. \({(x, y) ∈ \R^2 : |x| + |y| \le 4}\)
• D. \({(x, y) ∈ \R^2 : e^{x^3} + y^2 \le 4}\)

Answer 1

The Correct Option is A, B, C

To determine which subsets of \(\R^2\) are bounded, we need to analyze each given subset with respect to the definition of boundedness. A subset \(S \subseteq \R^2\) is bounded if there exists some \(M > 0\) such that for all \((x, y) \in S\), the inequalities \(|x| \leq M\) and \(|y| \leq M\) hold. Let's evaluate each subset:

\({(x, y) \in \R^2 : e^{x^2} + y^2 \le 4}\) 

In this case, the term \(\displaystyle e^{x^2}\) increases rapidly as \(|x|\) increases, surpassing any constant bound. For the inequality to hold, \(\displaystyle e^{x^2}\) must remain small, leading to small values of \(|x|\). As a result, this set is bounded because both \(x\) and \(y\) are constrained by the inequality.

\({(x, y) \in \R^2 : x^4 + y^2 \le 4}\)

Here, the growth of \(\displaystyle x^4\) is similar to or faster than quadratic growth. Additionally, for \(x^4\) to remain less than or equal to 4, \(|x|\) must be limited. Similarly, \(y^2\\) being bounded implies \(|y|\) is limited as well. Therefore, this set is also bounded.

\({(x, y) \in \R^2 : |x| + |y| \le 4}\)

This set is clearly bounded since both \(|x|\) and \(|y|\) are explicitly limited by 4. The constraint guarantees that neither coordinate exceeds 4 in absolute value, thus bounding the set.

\({(x, y) \in \R^2 : e^{x^3} + y^2 \le 4}\)

The function \(\displaystyle e^{x^3}\) grows extremely fast, and unlike \(\displaystyle e^{x^2}\), there's no quadratic or linear behavior to counterbalance it when \(|x|\) is large. Therefore, it's possible for \(|x|\) to become arbitrarily large while still satisfying the inequality, especially for negative large values, making the set unbounded.

Based on the evaluations, the bounded subsets are:

• \({(x, y) \in \R^2 : e^{x^2} + y^2 \le 4}\)
• \({(x, y) \in \R^2 : x^4 + y^2 \le 4}\)
• \({(x, y) \in \R^2 : |x| + |y| \le 4}\)