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Set-builder notation can be used to describe a set that is defined by a predicate, that is, a logical formula that evaluates to true for an element of the set, and false otherwise. [2] In this form, set-builder notation has three parts: a variable, a colon or vertical bar separator, and a predicate. Thus there is a variable on the left of the ...
In set-builder notation, it is used as a separator meaning "such that"; see { | }. 3. Restriction of a function : if f is a function , and S is a subset of its domain , then f | S {\displaystyle f|_{S}} is the function with S as a domain that equals f on S .
The algebra of sets is the set-theoretic analogue of the algebra of numbers. Just as arithmetic addition and multiplication are associative and commutative, so are set union and intersection; just as the arithmetic relation "less than or equal" is reflexive, antisymmetric and transitive, so is the set relation of "subset".
This article lists mathematical properties and laws of sets, involving the set-theoretic operations of union, intersection, and complementation and the relations of set equality and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations.
Set-builder notation makes use of predicates to define sets. In autoepistemic logic , which rejects the law of excluded middle, predicates may be true, false, or simply unknown . In particular, a given collection of facts may be insufficient to determine the truth or falsehood of a predicate.
A set for which membership can be decided by a recursive procedure or algorithm, also known as a decidable or computable set. recursively enumerable set A set for which there exists a Turing machine that will list all members of the set, possibly without halting if the set is infinite; also called "semi-decidable set" or "Turing recognizable set".
The 9-person Symbolab team, based in Tel Aviv, will join Course Hero . The platforms will live under independent branding for the near future, according to Andrew Grauer, CEO of Course Hero.
Intersection (set theory) – Set of elements common to all of some sets; Iterated binary operation – Repeated application of an operation to a sequence; List of set identities and relations – Equalities for combinations of sets; Naive set theory – Informal set theories; Symmetric difference – Elements in exactly one of two sets