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A set such as {{,,}} is a singleton as it contains a single element (which itself is a set, but not a singleton). A set is a singleton if and only if its cardinality is 1. In von Neumann's set-theoretic construction of the natural numbers, the number 1 is defined as the singleton {}.
The carrier (underlying set) associated with a unit type can be any singleton set. There is an isomorphism between any two such sets, so it is customary to talk about the unit type and ignore the details of its value. One may also regard the unit type as the type of 0-tuples, i.e. the product of no types.
Example of Kleene star applied to the empty set: ∅ * = {ε}. Example of Kleene plus applied to the empty set: ∅ + = ∅ ∅ * = { } = ∅, where concatenation is an associative and noncommutative product. Example of Kleene plus and Kleene star applied to the singleton set containing the empty string:
The original form of the pattern, appearing in Pattern Languages of Program Design 3, [2] has data races, depending on the memory model in use, and it is hard to get right. Some consider it to be an anti-pattern. [3] There are valid forms of the pattern, including the use of the volatile keyword in Java and explicit memory barriers in C++. [4]
Singleton pattern, a design pattern that allows only one instance of a class to exist; Singleton bound, used in coding theory; Singleton variable, a variable that is referenced only once; Singleton, a character encoded with one unit in variable-width encoding schemes for computer character sets
More specifically, the singleton pattern allows classes to: [2] Ensure they only have one instance; Provide easy access to that instance; Control their instantiation (for example, hiding the constructors of a class) The term comes from the mathematical concept of a singleton.
The domain of this function is the set of all real numbers. The image of this function is the singleton set {4}. The independent variable x does not appear on the right side of the function expression and so its value is "vacuously substituted"; namely y(0) = 4, y(−2.7) = 4, y(π) = 4, and so on.
The empty set is the unique initial object in Set, the category of sets. Every one-element set ( singleton ) is a terminal object in this category; there are no zero objects. Similarly, the empty space is the unique initial object in Top , the category of topological spaces and every one-point space is a terminal object in this category.