Understanding f(y) in a Nonempty Set X of $\mathbb{R}^n$ - Bob19

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In summary, The conversation is about an assignment in Analysis that involves a nonempty subset of real numbers, X, and a function f that maps from real numbers to real numbers. The goal is to show that if y is in X, then the shortest distance between x and y is 0.
  • #1
Bob19
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Hi

i have this following assignment in Analysis

Given [tex]X \subseteq \mathbb{R}^n[/tex] which is a nonempty subset of [tex]\mathbb{R}^n[/tex]

The set [tex]\{ \| | x -y \| | \ | x \in X \}[/tex] has an infimum such that

[tex]f(y) = \{ \| | x -y \| | \ | x \in X \}[/tex]

where [tex]f: \mathbb{R}^n \rightarrow \mathbb{R}^n [/tex]

I need a hint on howto show that if [tex]y \in X[/tex] then f(y) = 0 ??

Regards,

Bob19
 
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  • #2
Fix x in X. What is the shortest distance between x and y if y is allowed to be in X (note that x is in X)?
 
  • #3
very similar to mathboy20s post in this subforum
 

1. What is the purpose of understanding f(y) in a nonempty set X of $\mathbb{R}^n$?

The purpose of understanding f(y) in a nonempty set X of $\mathbb{R}^n$ is to gain insight into the behavior and properties of a function f(y) defined over a set of real numbers. This understanding can help in making predictions, identifying patterns, and solving complex mathematical problems.

2. How is f(y) related to the set X of $\mathbb{R}^n$?

F(y) is related to the set X of $\mathbb{R}^n$ as it is defined over this set, meaning that the input values for the function come from the set X and the output values are also elements of $\mathbb{R}^n$. This relationship is important in understanding the behavior of the function and its effects on the set X.

3. What are some common techniques for understanding f(y) in a nonempty set X of $\mathbb{R}^n$?

Some common techniques for understanding f(y) include graphing the function, finding the domain and range, analyzing the critical points and inflection points, determining the concavity and convexity of the function, and calculating the derivatives and integrals of the function. These techniques provide valuable information about the behavior and properties of the function f(y).

4. Can f(y) be understood in a nonempty set X of $\mathbb{R}^n$ without using mathematical techniques?

No, understanding f(y) in a nonempty set X of $\mathbb{R}^n$ requires the use of mathematical techniques as these provide a systematic and rigorous approach to analyzing the function. Without these techniques, it would be difficult to gain a comprehensive understanding of the behavior and properties of the function.

5. How does understanding f(y) in a nonempty set X of $\mathbb{R}^n$ benefit scientific research?

Understanding f(y) in a nonempty set X of $\mathbb{R}^n$ benefits scientific research in many ways. It allows for the analysis and prediction of complex systems, the identification of relationships and patterns, and the development of mathematical models to describe real-world phenomena. This understanding also helps in solving problems and making decisions in various scientific fields such as physics, engineering, and economics.

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