Null (mathematics)
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In mathematics, the word null (from Latin nullus, meaning "none") may or may not have a meaning different from zero. Sometimes the symbol Ø is used to distinguish "null" from 0.
In a vector space the null vector is the zero vector; 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. Often it simply asserts the absence of something — for example it may say that a new drug being tested has no effect.
[edit] Null in set theory
The concept of null is ambiguous.
The symbol Ø can be used to symbolize true "nothingness" -- emptiness, non-existence -- as opposed to the zero-concept: four-letter word "zero", symbol 0, "empty set" { }. In this latter case the space between the braces could be symbolized by Ø. The empty braces themselves would symbolize the empty basket that is represented by symbol 0. For example, the intersection of two sets with no common elements could be thought of as either null/nothingness, or the empty set of symbols represented by "zero".
In set theory, Halmos uses and defines the symbol Ø (not 0!) as (the only) set with no elements (p. 8), thus to him the symbol Ø is synonomous with the empty basket called 0 (p. 45). His number 1 is in fact the set that "contains" the empty set: {Ø}. T
To define the unit, Suppes (1972) does not use the symbol Ø but rather symbol 0 in an otherwise basket of braces: {0} (cf page 129).
[edit] References
- Patrick Suppes(1972), Axiomatic Set Theory, Dover Publications, Inc., New York.
- Paul Halmos (1970), Naive Set Theory, Springer-Verlag New York,
