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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
Cantor–Bernstein–Schroeder theorem (set theory, cardinal numbers) Cantor's intersection theorem (real analysis) Cantor's isomorphism theorem (order theory) Cantor's theorem (set theory, Cantor's diagonal argument) Carathéodory–Jacobi–Lie theorem (symplectic topology) Carathéodory's existence theorem (ordinary differential equations)
One can obtain explicit formulas for the above expressions in the form of determinants, by considering the first n of Newton's identities (or it counterparts for the complete homogeneous polynomials) as linear equations in which the elementary symmetric functions are known and the power sums are unknowns (or vice versa), and apply Cramer's rule ...
In the field of representation theory in mathematics, a projective representation of a group G on a vector space V over a field F is a group homomorphism from G to the projective linear group = /, where GL(V) is the general linear group of invertible linear transformations of V over F, and F ∗ is the normal subgroup consisting of nonzero scalar multiples of the identity transformation (see ...
Every set representing an ordinal number is well-founded, the set of natural numbers is one of them. Applied to a well-founded set, transfinite induction can be formulated as a single step. To prove that a statement P(n) holds for each ordinal number: Show, for each ordinal number n, that if P(m) holds for all m < n, then P(n) also holds.
De Morgan's laws represented with Venn diagrams.In each case, the resultant set is the set of all points in any shade of blue. In propositional logic and Boolean algebra, De Morgan's laws, [1] [2] [3] also known as De Morgan's theorem, [4] are a pair of transformation rules that are both valid rules of inference.
(For the example of x + y − z, the subgroup H in S 3 contains the identity and the transposition (x y).) So the size of H divides n!. With the later development of abstract groups, this result of Lagrange on polynomials was recognized to extend to the general theorem about finite groups which now bears his name.
In general, the transpose of a continuous linear map : is the linear map : ′ ′ (′):= ′, or equivalently, it is the unique map satisfying ′, = (′), for all and all ′ ′ (the prime symbol in ′ does not denote a derivative of any kind; it merely indicates that ′ is an element of the continuous dual space ′).