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Let X be a topological space, and let a subset be negligible if it is of first category, that is, if it is a countable union of nowhere-dense sets (where a set is nowhere-dense if it is not dense in any open set). Then the negligible sets form a sigma-ideal. Let X be a directed set, and let a subset of X be negligible if it has an upper bound ...
In information theory, the source coding theorem (Shannon 1948) [2] informally states that (MacKay 2003, pg. 81, [3] Cover 2006, Chapter 5 [4]): N i.i.d. random variables each with entropy H(X) can be compressed into more than N H(X) bits with negligible risk of information loss, as N → ∞; but conversely, if they are compressed into fewer than N H(X) bits it is virtually certain that ...
The maximum of a subset of a preordered set is an element of which is greater than or equal to any other element of , and the minimum of is again defined dually. In the particular case of a partially ordered set , while there can be at most one maximum and at most one minimum there may be multiple maximal or minimal elements.
a finite directed graph G = (V, E), where V denotes the finite set of vertices and E ⊆ V×V is the set of directed edges; a source s ∈ V and a sink t ∈ V; a capacity function, which is a mapping : + denoted by c uv or c(u, v) for (u,v) ∈ E. It represents the maximum amount of flow that can pass through an edge.
(Coincidentally, since a generating set always exists, e.g. M itself, this shows that a module is a quotient of a free module, a useful fact.) A generating set of a module is said to be minimal if no proper subset of the set generates the module. If R is a field, then a minimal generating set is the same thing as a basis.
A set of real numbers (hollow and filled circles), a subset of (filled circles), and the infimum of . Note that for totally ordered finite sets, the infimum and the minimum are equal. A set of real numbers (blue circles), a set of upper bounds of (red diamond and circles), and the smallest such upper bound, that is, the supremum of (red diamond).
It is called a strongly minimal set if this is true even in all elementary extensions. A strongly minimal set, equipped with the closure operator given by algebraic closure in the model-theoretic sense, is an infinite matroid, or pregeometry. A model of a strongly minimal theory is determined up to isomorphism by its dimension as a matroid.
The axiom of constructibility implies the axiom of choice (AC), given Zermelo–Fraenkel set theory without the axiom of choice (ZF). It also settles many natural mathematical questions that are independent of Zermelo–Fraenkel set theory with the axiom of choice (ZFC); for example, the axiom of constructibility implies the generalized continuum hypothesis, the negation of Suslin's hypothesis ...