null set
Noun: 1. A set with no elements: In mathematics, a "null set" is a set that contains no members or elements. It is the unique set of size zero. 2. A set of measure zero: In measure theory, a "null set" can also refer to a set which is considered negligible in size (having measure zero), even if it is not empty. For example, the set of rational numbers between 0 and 1 has measure zero within the set of all real numbers in that interval.
- As an empty set:
- The set of living dinosaurs is a null set.
- In the equation, the solution set was a null set, meaning there were no valid answers.
- As a set of measure zero:
- The probability of randomly selecting a specific point from a continuous distribution is zero, as the point is a null set.
- Symbol: The null set (empty set) is often denoted by the symbols ∅ or {}.
- Property: The cardinality (number of elements) of a null set (empty set) is 0.
- In Logic: A statement quantified over a null set is often considered vacuously true.
- Empty Set (∅): This is the most precise synonym for the first meaning of "null set" (a set with no members).
- Zero Set: Sometimes used interchangeably, especially in the context of measure theory.
- Empty set
- Void set
It is important to distinguish between the two primary mathematical meanings: 1. The empty set (absolutely no elements). 2. A set of measure zero (may have infinitely many elements but is considered "small" or negligible in a specific technical sense). The context usually clarifies which meaning is intended. In many introductory contexts, "null set" is synonymous with "empty set."
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a set that is empty; a set with no members
