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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 ...
8 Ways of defining sets/Relation to descriptive set theory. 9 More general objects still called sets. ... Download as PDF; Printable version; In other projects ...
Here, the list [0..] represents , x^2>3 represents the predicate, and 2*x represents the output expression.. List comprehensions give results in a defined order (unlike the members of sets); and list comprehensions may generate the members of a list in order, rather than produce the entirety of the list thus allowing, for example, the previous Haskell definition of the members of an infinite list.
An alternative notation for [] used in mathematical logic and set theory is ″. [6] [7] Some texts refer to the image of f {\displaystyle f} as the range of f , {\displaystyle f,} [ 8 ] but this usage should be avoided because the word "range" is also commonly used to mean the codomain of f . {\displaystyle f.}
Universe set and complement notation The notation L ∁ = def X ∖ L . {\displaystyle L^{\complement }~{\stackrel {\scriptscriptstyle {\text{def}}}{=}}~X\setminus L.} may be used if L {\displaystyle L} is a subset of some set X {\displaystyle X} that is understood (say from context, or because it is clearly stated what the superset X ...
As is also common, one makes use set builder notation for classes, which, in most contexts, are not part of the object language but used for concise discussion. In particular, one may introduce notation declarations of the corresponding class via " A = { z ∣ Q ( z ) } {\displaystyle A=\{z\mid Q(z)\}} ", for the purpose of expressing any Q ( a ...
Expressions definable in set-builder notation make sense in both ZFC and NFU: it may be that both theories prove that a given definition succeeds, or that neither do (the expression {} fails to refer to anything in any set theory with classical logic; in class theories like NBG this notation does refer to a class, but it is defined differently ...
The reason is as follows: The intersection of the collection is defined as the set (see set-builder notation) = {:,}. If M {\displaystyle M} is empty, there are no sets A {\displaystyle A} in M , {\displaystyle M,} so the question becomes "which x {\displaystyle x} 's satisfy the stated condition?"
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