# Mathematical object **Mathematical object** is anything that can be formally defined and used in deductive reasoning and *mathematical proof*. Common mathematical objects include permutations, combinations, partitions, relations, theorems, even proofs themselves. With set theory (ZFC) at the foundation of mathematics, mathematical objects are defined in terms of sets. **Discrete Structures** aid us in modelling the world in a way that enables us to think about it rigorously and computationally. **Modelling** is used to approximate objects and concepts by identifying and encoding patterns in the data. **Mathematical structure** on a set is an additional structural object that, in some manner, attaches (or relates) to that set, endowing it with some extra meaning or significance. Structure-preserving relations map structures in domain to equivalent structures in codomain: **homomorphisms** preserve algebraic structures, **homeomorphisms** preserve topological structures, **diffeomorphisms** preserve differential structures. Since computers operate on discrete bits of data, Discrete Mathematics overlaps with Computer Science in its study of: * algorithms * algorithmic complexity * proofs * formal language * automata * computability * correctness of programming languages * automated and assisted theorem proving Solving a problem computationally requires * modeling the world * devising an algorithm * determining its efficiency and correctness --- # Agent Instructions: Querying This Documentation If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question. Perform an HTTP GET request on the current page URL with the `ask` query parameter: ``` GET https://mandober.gitbook.io/math-debrief/100-fundamentals/terms/math-object.md?ask= ``` The question should be specific, self-contained, and written in natural language. The response will contain a direct answer to the question and relevant excerpts and sources from the documentation. Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.