Set (abstract Data Type) - Type Theory

Type Theory

In type theory, sets are generally identified with their indicator function: accordingly, a set of values of type may be denoted by or . (Subtypes and subsets may be modeled by refinement types, and quotient sets may be replaced by setoids.) The characteristic function F of a set S is defined as: F(x) = \begin{cases} 1, & \mbox{if } x \in S \\ 0, & \mbox{if } x \not \in S \end{cases}

In theory, many other abstract data structures can be viewed as set structures with additional operations and/or additional axioms imposed on the standard operations. For example, an abstract heap can be viewed as a set structure with a min(S) operation that returns the element of smallest value.

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