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