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More formally, proposition B is a corollary of proposition A, if B can be readily deduced from A or is self-evident from its proof. In many cases, a corollary corresponds to a special case of a larger theorem, [4] which makes the theorem easier to use and apply, [5] even though its importance is generally considered to be secondary to that of ...
A porism is a mathematical proposition or corollary. It has been used to refer to a direct consequence of a proof, analogous to how a corollary refers to a direct consequence of a theorem. In modern usage, it is a relationship that holds for an infinite range of values but only if a certain condition is assumed, such as Steiner's porism. [1]
The following corollary is also known as Nakayama's lemma, and it is in this form that it most often appears. [ 4 ] Statement 3 : If M {\displaystyle M} is a finitely generated module over R {\displaystyle R} , J ( R ) {\displaystyle J(R)} is the Jacobson radical of R {\displaystyle R} , and J ( R ) M = M {\displaystyle J(R)M=M} , then M = 0 ...
Ptolemy's Theorem yields as a corollary a pretty theorem [2] regarding an equilateral triangle inscribed in a circle. Given An equilateral triangle inscribed on a circle and a point on the circle. The distance from the point to the most distant vertex of the triangle is the sum of the distances from the point to the two nearer vertices.
In mathematics and other fields, [a] a lemma (pl.: lemmas or lemmata) is a generally minor, proven proposition which is used to prove a larger statement. For that reason, it is also known as a "helping theorem" or an "auxiliary theorem".
With the advent of algebraic logic, it became apparent that classical propositional calculus admits other semantics.In Boolean-valued semantics (for classical propositional logic), the truth values are the elements of an arbitrary Boolean algebra; "true" corresponds to the maximal element of the algebra, and "false" corresponds to the minimal element.
A proposition can be modeled equivalently with the inverse image of under the indicator function, which is sometimes called the characteristic set of the proposition. For instance, if w {\displaystyle w} and w ′ {\displaystyle w'} are the only worlds in which the sky is blue, the proposition that the sky is blue could be modeled as the set ...
In mathematics, the mean value theorem (or Lagrange's mean value theorem) states, roughly, that for a given planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints.