Null (mathematics)
Encyclopedia
In mathematics
, the word null (from German
null, which is from Latin
nullus, both meaning "zero", or "none") means of or related to having zero members in a set
or a value of zero. Sometimes the symbol ∅ is used to distinguish "null" from 0.
In a normed
vector space
the null vector is the zero vector; in a seminormed
vector space such as Minkowski space, null vectors are, in general, non-zero. In set theory
, the null set
is the set with zero elements; and in measure theory, a null set
is a set with zero measure.
A mathematical mapping
is said to be null potent (or nilpotent
) if repeated application can map the whole domain into the null element.
A null space
of a mapping is the part of the domain that is mapped into the null element of the image (the inverse image of the null element).
In statistics
, a null hypothesis
is a proposition presumed true unless statistical evidence indicates otherwise.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the word null (from German
German language
German is a West Germanic language, related to and classified alongside English and Dutch. With an estimated 90 – 98 million native speakers, German is one of the world's major languages and is the most widely-spoken first language in the European Union....
null, which is from Latin
Latin
Latin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient Proto-Indo-European language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...
nullus, both meaning "zero", or "none") means of or related to having zero members in a set
Empty set
In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...
or a value of zero. Sometimes the symbol ∅ is used to distinguish "null" from 0.
In a normed
Norm (mathematics)
In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector...
vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
the null vector is the zero vector; in a seminormed
Norm (mathematics)
In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector...
vector space such as Minkowski space, null vectors are, in general, non-zero. In set theory
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
, the null set
Empty set
In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...
is the set with zero elements; and in measure theory, a null set
Null set
In mathematics, a null set is a set that is negligible in some sense. For different applications, the meaning of "negligible" varies. In measure theory, any set of measure 0 is called a null set...
is a set with zero measure.
A mathematical mapping
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
is said to be null potent (or nilpotent
Nilpotent
In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n such that xn = 0....
) if repeated application can map the whole domain into the null element.
A null space
Null space
In linear algebra, the kernel or null space of a matrix A is the set of all vectors x for which Ax = 0. The kernel of a matrix with n columns is a linear subspace of n-dimensional Euclidean space...
of a mapping is the part of the domain that is mapped into the null element of the image (the inverse image of the null element).
In statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, a null hypothesis
Null hypothesis
The practice of science involves formulating and testing hypotheses, assertions that are capable of being proven false using a test of observed data. The null hypothesis typically corresponds to a general or default position...
is a proposition presumed true unless statistical evidence indicates otherwise.