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The axiom of constructibility is a possible axiom for set theory in mathematics that asserts that every set is constructible.The axiom is usually written as V = L.The axiom, first investigated by Kurt Gödel, is inconsistent with the proposition that zero sharp exists and stronger large cardinal axioms (see list of large cardinal properties).
A single valued version, called the principal value of the logarithm, can be defined which is discontinuous on the negative x axis, and is equal to the multivalued version on a single branch cut. Definitions
In this setting: = if, and only if, for all sequences x n (with, for all n, x n not equal to a) converging to a the sequence f(x n) converges to L. It was shown by Sierpiński in 1916 that proving the equivalence of this definition and the definition above, requires and is equivalent to a weak form of the axiom of choice .
In botanical nomenclature, the triple bar denotes homotypic synonyms (those based on the same type specimen), to distinguish them from heterotypic synonyms (those based on different type specimens), which are marked with an equals sign. [15] In chemistry, the triple bar can be used to represent a triple bond between atoms.
In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements.For example, in the conditional statement: "If P then Q", Q is necessary for P, because the truth of Q is guaranteed by the truth of P.
V is the volume of the gas; n is the amount of substance of the gas (measured in moles); k is a constant for a given temperature and pressure. This law describes how, under the same condition of temperature and pressure, equal volumes of all gases contain the same number of molecules. For comparing the same substance under two different sets of ...
In linear algebra, two rectangular m-by-n matrices A and B are called equivalent if = for some invertible n-by-n matrix P and some invertible m-by-m matrix Q.Equivalent matrices represent the same linear transformation V → W under two different choices of a pair of bases of V and W, with P and Q being the change of basis matrices in V and W respectively.
For each ordinal α of V, the set V B α is defined as follows. V B 0 is the empty set. V B α+1 is the set of all functions from V B α to B. (Such a function represents a subset of V B α; if f is such a function, then for any x ∈ V B α, the value f(x) is the membership degree of x in the set.) If α is a limit ordinal, V B α is the union ...