18 examples
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- A more powerful mechanism for denoting a set in terms of its elements is set-builder notation.
- For these examples, formal languages are specified using set-builder notation.
- Consider the following example in set-builder notation.
- In the 1960s, set-builder notation was developed for describing a set by stating the properties that its members must satisfy.
- There is some effort in providing C++ with list-comprehension constructs/syntax similar to the set builder notation.
- Mathematicians find this approach of providing building rules to be convenient and important so they have extended and formalized the set builder notation as further described in this article.
- An extension of set-builder notation replaces the single variable x with a term T that may include one or more variables, combined with functions acting on them.
- The canonical restriction on set builder notation asserts that X is a set only if A is already known to be a set.
- It follows the form of the mathematical set-builder notation (set comprehension) as distinct from the use of map and filter functions.
- Versions of set builder notation are also available in Z which allow for terms more complicated than a single variable, using a bullet to indicate the form of members of the set.
- It follows the form of the mathematical set-builder notation (set comprehension.) as distinct from the use of map and filter functions.
- In set theory and its applications to logic, mathematics, and computer science, set-builder notation is a mathematical notation for describing a set by stating the properties that its members must satisfy.
- Thus it is true when x is in R, and it is false when x is not in R. This follows from our definition of set builder notation and the extension here to allow expressions.
- That is, for sets A and B, the Cartesian product is the set of all ordered pairs where and, Products can be specified using set-builder notation, e.g.
- A set denoted in formal set builder notation has three parts: a variable, a colon or vertical bar separator, and a logical predicate.
- The set builder notation and list comprehension notation are both instances of a more general notation known as monad comprehensions, which permits map/filter-like operations over any monad with a zero element.
- The notation of list comprehensions is similar to the set-builder notation, but sets can't be made into a monad, since there's a restriction on the type of computation to be comparable for equality, whereas a monad does not put any constraints on the types of computations.
- Quine's (1982) semantics for each predicate functor are stated below in terms of abstraction (set builder notation), followed by either the relevant axiom from Kuhn (1983), or a definition from Quine (1976).