Search results
Results from the WOW.Com Content Network
set is smaller than its power set; uncountability of the real numbers; Cantor's first uncountability proof. uncountability of the real numbers; Combinatorics; Combinatory logic; Co-NP; Coset; Countable. countability of a subset of a countable set (to do) Angle of parallelism; Galois group. Fundamental theorem of Galois theory (to do) Gödel number
This article lists mathematical properties and laws of sets, involving the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations.
Modern proof theory treats proofs as inductively defined data structures, not requiring an assumption that axioms are "true" in any sense. This allows parallel mathematical theories as formal models of a given intuitive concept, based on alternate sets of axioms, for example axiomatic set theory and non-Euclidean geometry.
As an example, the back-and-forth method can be used to prove Cantor's isomorphism theorem, although this was not Georg Cantor's original proof. This theorem states that two unbounded countable dense linear orders are isomorphic. [1] Suppose that (A, ≤ A) and (B, ≤ B) are linearly ordered sets;
Lubell's proof of this result on set families is a double counting argument on permutations, used to prove an inequality rather than an equality. Erdős–Ko–Rado theorem, an upper bound on intersecting families of sets, proven by Gyula O. H. Katona using a double counting inequality. [3] Proofs of Fermat's little theorem.
Linear congruence theorem (number theory, modular arithmetic) Linear speedup theorem (computational complexity theory) Linnik's theorem (number theory) Lions–Lax–Milgram theorem (partial differential equations) Liouville's theorem (complex analysis, entire functions) Liouville's theorem (conformal mappings) Liouville's theorem (Hamiltonian ...
Considering in particular the set in R n where the v-directional derivative of u fails to exist (which must be proved to be measurable), the latter condition is met due to the one-dimensional case of Rademacher's theorem. The second step of Morrey's proof establishes the linear dependence of the v-directional derivative of u upon v. This is ...
In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication.This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible, with the identity matrix as the identity element of the group.