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One can also speak of "almost all" integers having a property to mean "all except finitely many", despite the integers not admitting a measure for which this agrees with the previous usage. For example, "almost all prime numbers are odd". There is a more complicated meaning for integers as well, discussed in the main article.
deg – degree of a polynomial, or other recursively-defined objects such as well-formed formulas. (Also written as ∂.) del – del, a differential operator. (Also written as.) det – determinant of a matrix or linear transformation. DFT – discrete Fourier transform. dim – dimension of a vector space.
Brauer's three main theorems (finite groups) Brauer–Cartan–Hua theorem (ring theory) Bregman–Minc inequality (discrete mathematics) Brianchon's theorem ; British flag theorem (Euclidean geometry) Brooks's theorem (graph theory) Brouwer fixed-point theorem ; Browder–Minty theorem (operator theory)
A rule of inference states that, given a particular formula (or set of formulas) with a certain property as a hypothesis, another specific formula (or set of formulas) can be derived as a conclusion. The rule is sound (or truth-preserving) if it preserves validity in the sense that whenever any interpretation satisfies the hypothesis, that ...
Replacement: (i) the formula to be replaced must be within a tautology, i.e. logically equivalent ( connected by ≡ or ↔) to the formula that replaces it, and (ii) unlike substitution its permissible for the replacement to occur only in one place (i.e. for one formula). Example: Use this set of formula schemas/equivalences: ( (a ∨ 0) ≡ a ).
[2] [3] The word equation and its cognates in other languages may have subtly different meanings; for example, in French an équation is defined as containing one or more variables, while in English, any well-formed formula consisting of two expressions related with an equals sign is an equation. [4]
For example, + and () = + define the function that associates to each number its square plus one. An expression with no variables would define a constant function . In this way, two expressions are said to be equivalent if, for each combination of values for the free variables, they have the same output, i.e., they represent the same function.
In mathematics, a formula generally refers to an equation or inequality relating one mathematical expression to another, with the most important ones being mathematical theorems. For example, determining the volume of a sphere requires a significant amount of integral calculus or its geometrical analogue, the method of exhaustion. [3]