Let S=𝐑+^𝐧. Prove that S-S=𝐑^𝐧.

Name: Problem 5, Chapter 2: Competitive Equilibrium: Theory and Applications
Uploaded: 2020-10-25T17:21:03-08:00
Duration: 2 min 46 s
Channel: Subham Jyoti Mishra
Description: Let S=𝐑+^𝐧. Prove that S-S=𝐑^𝐧.

Key Concepts

Nonnegative Orthant

The nonnegative orthant refers to the subset of n-dimensional real space where each component of a vector is nonnegative. This concept is pivotal in many areas of mathematics, particularly in optimization and analysis, as it specifies a region of the space with special algebraic and geometric properties which often simplifies analysis.

Set Difference Operation

The set difference operation, in this context, means forming a new set by taking the difference of every pair of elements from a given set. When applied to a subset of a vector space, this operation can reveal structural properties of the original set, such as whether it spans the entire space or forms a subgroup.

Vector Space Structure

Understanding the vector space structure of R^n is crucial, as it implies that linear combinations of vectors, including differences, can cover the entire space. This property is used to deduce that certain operations (like the difference of elements) applied to a subset may yield the complete space.

Decomposition into Positive and Negative Parts

A key technique in proving that the difference of two nonnegative vectors can yield any vector in R^n is the decomposition of any arbitrary vector into its positive and negative parts. By expressing a vector as the difference between its componentwise maximum with zero and the negative of its componentwise minimum with zero, one can show constructively that every vector in the space can be attained from elements of the nonnegative orthant.

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