Graph of a set-valued mapping

Definition

The graph of a set-valued mapping is the collection of ordered pairs where each input is associated with a non-empty subset of outputs. This concept extends traditional functions, where instead of assigning a single output to each input, it assigns a set of possible outputs, allowing for more flexibility in modeling relationships. Understanding the graph of set-valued mappings is crucial for analyzing the properties and behaviors of these mappings in various mathematical contexts.

5 Must Know Facts For Your Next Test

1. The graph can be represented as a subset of the Cartesian product of the domain and the power set of the codomain.
2. Graphs of set-valued mappings can be more complex than graphs of single-valued functions since they may represent multiple outputs for a single input.
3. The properties of the graph, like closedness or compactness, can provide important insights into the nature of the set-valued mapping itself.
4. In many applications, understanding the graph helps in identifying optimal solutions or equilibrium points in variational problems.
5. The visual representation of the graph aids in intuitively grasping how inputs correspond to sets of outputs, which is especially useful in optimization contexts.

Review Questions

• How does the graph of a set-valued mapping differ from that of a traditional single-valued function?
  • The primary difference is that the graph of a set-valued mapping includes ordered pairs where each input can correspond to a set of possible outputs, rather than just one output as in traditional functions. This allows for representing relationships where multiple outcomes are possible for given inputs, adding complexity and richness to the analysis. Understanding this difference helps in modeling real-world situations where uncertainty or multiple choices exist.
• Discuss the implications of closedness and compactness in relation to the graph of a set-valued mapping.
  • Closedness and compactness are significant properties that can influence the behavior of set-valued mappings. A closed graph implies that if a sequence converges in the domain, then its images under the mapping also converge in a specific manner related to subsets. Compactness often leads to useful results such as ensuring the existence of limit points within compact subsets, which can provide guarantees for solutions in optimization problems related to set-valued mappings.
• Evaluate how understanding the graph of a set-valued mapping can impact decision-making in optimization problems.
  • Understanding the graph allows decision-makers to visualize and analyze all possible outcomes corresponding to different inputs, facilitating better assessments of potential solutions. In optimization contexts, this insight can help identify optimal points or strategies by revealing how adjustments in parameters might influence multiple outcomes. By analyzing the structure and properties of these graphs, practitioners can make more informed and effective choices, ultimately leading to improved results in various applications.

Related terms

Set-valued mapping: A mapping that assigns to each point in the domain a non-empty subset of the codomain rather than a single point.

Closed graph theorem: A principle that states under certain conditions, if the graph of a mapping is closed in the product space, then the mapping is continuous.

Continuity of set-valued mappings: A property describing how small changes in input lead to small changes in the output set, usually measured in terms of Hausdorff distance.