nLab
discrete mathematics

A good part of mathematics refers to continuous or even smooth spaces of various kinds. Discrete mathematics is about discrete structures: logic and combinatorics are typical fields in this domain. Basic algebra and set theory are discrete, but more advanced material deals with continuous structures. There is a good deal of overlap between discrete mathematics and finite mathematics, and not every author distinguishes them. While modern number theory mixes ideas of discrete and continuous, in its original problematics most of the number theory may be considered discrete. However socially, and often in methods, modern number theory is for many closer to algebraic geometry and analysis than to combinatorics.

In nnlab many items are dedicated to the foundational issues in set theory (starting with standard basic notions like set, proper class, empty set, singleton, subset, relation, reflexive relation, transitive relation, symmetric relation, function, partial function, injection, surjection, bijection, union, disjoint union, complement, intersection, symmetric difference, power set, pointed set along with more foundational items like axiom of choice, axiom of extensionality, axiom of foundation, axiom of infinity, Zorn's lemma, well-ordering theorem, decidable subset, material set theory, inhabited set, COSHEP, Markov's principle, Grothendieck universe, choice operator, pure set, hereditarily finite set). Logic also has some entries; among them are truth value or 0-poset, predicate, excluded middle, implication. We pay attention to the alternative approaches to logic and set theory, like the internal language of a topos; cf. Mitchell-Benabou language, Lawvere’s ETCS, Nelson’s internal set theory etc.

Some other basic discrete structures which have their own entry in nnlab are preorder, order, partial order, total order, well-order, equivalence relation, quotient set, poset, proset, cyclic order, semilattice, lattice, distributive lattice, meet, join, supremum, infimum, suplattice, multiset, graph, quiver, tree, filter, span, multispan, natural number, sequence, …

Of course, a plain old category is a discrete structure per se.

Revised on April 8, 2010 17:53:26 by Zoran Škoda (193.55.10.104)