Show set (which is a subset of R^n) is bounded

In summary, the set D is bounded because each coordinate is bounded by a certain value and can be contained within a sphere with radius equal to the square root of the sum of these values.
  • #1
Berrius
19
0

Homework Statement


Show that [itex]D = { (x,y,z) \in \mathbb{R}^{3} | 7x^2+2y^2 \leq 6, x^3+y \leq z \leq x^2y+5y^3}[/itex] is bounded.

Homework Equations


Definition of bounded:[itex]D \subseteq \mathbb{R}^{n}[/itex] is called bounded if there exists a M > 0 such that [itex]D \subseteq \{x \in \mathbb{R}^{n} | ||x|| \leq M\}[/itex]

The Attempt at a Solution


I have to find a M such that [itex]D \subseteq \{(x,y,z) \in \mathbb{R}^{3} | x^2 + y^2 + z^2 \leq M\}[/itex]. I thought of just picking a very high M, say 999999. But how do I show it works?
 
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  • #2
It's often easier to show that each coordinate is bounded, say x,y and z are all smaller than 10 maybe. Then ||(x,y,z)|| < ||(10,10,10)|| = M
 
  • #3
So if I say something like: [itex]7x^2+2y^2 \leq 6 \Rightarrow y^2 \leq 3 \lt 4 \Rightarrow y \lt 2[/itex]
and [itex]7x^2+2y^2 \leq 6 \Rightarrow x^2 \leq \frac{6}{7} \lt 1 \Rightarrow x \lt 1[/itex]
and [itex]z \leq x^2y+5y^3 \Rightarrow z \lt (2+5*8)=42[/itex]
So choose M = 4+1+42^2 = 1769. And this M will do.
 
  • #4
Careful, M=||(1,2,42)|| is the square root of the number you put up. As long as M is larger than 1 that won't matter, but if the norm happens to be smaller than 1 failing to take the square root can give a value of M that doesn't work
 

Related to Show set (which is a subset of R^n) is bounded

What does it mean for a show set to be bounded?

A show set is considered bounded if there exists a finite number that serves as an upper bound for the absolute value of all its elements. This means that the elements of the set do not grow infinitely large.

How can you prove that a show set is bounded?

To prove that a show set is bounded, you can show that there exists a finite number that is an upper bound for all the absolute values of its elements. This can be done through various mathematical techniques, such as using the definition of a bounded set or using the Bolzano-Weierstrass theorem.

What is the significance of a show set being bounded?

A bounded show set is significant because it ensures that the elements of the set do not grow infinitely large, making it easier to analyze and work with mathematically. It also allows for the application of various theorems and properties that are only applicable to bounded sets.

Can a show set be both bounded and unbounded?

No, a show set cannot be both bounded and unbounded. It is either one or the other. If a set is bounded, it means that there exists a finite upper bound for its elements. If a set is unbounded, it means that its elements have no finite upper bound, and they can grow infinitely large.

What are some examples of bounded and unbounded show sets?

An example of a bounded show set is the set of all real numbers between 0 and 1. An example of an unbounded show set is the set of all positive real numbers.

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