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Please recall the different basic possibilities to describe sets, in which the set-builder notation used the curly brackets, inside which we describe the definite properties of the set elements. The following corollary to the axiom of separation allows justification of this notation.

Corollary: Justification of the Set-Builder Notation

Let $p(z,X_1,\ldots,X_n)$ be an atomic formula in predicate logic, in which the $z$ is a free variable and in which $X_1,\ldots,X_n, X$ are sets. Let $Y$ be given, which fulfills the property of the axiom of separation, i.e. $$\forall X_1,\ldots,X_n \forall X~\exists Y~\forall z~(z\in Y \Leftrightarrow z\in X\wedge p(z,X_1,\ldots,X_n)).$$
Then $Y$ is unique. Therefore, we can therefore use the formula to define the set $Y,$, justifying the set-builder notation $$Y:=\{z\in X\mid p(z,X_1,\ldots,X_n)\}.$$

| | | | | Contributors: bookofproofs | References: [656], [983]

1.Proof: (related to "Justification of the Set-Builder Notation")

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Bibliography (further reading)

[983] Ebbinghaus, H.-D.: “Einf├╝hrung in die Mengenlehre”, BI Wisschenschaftsverlag, 1994, 3

[656] Hoffmann, Dirk W.: “Grenzen der Mathematik – Eine Reise durch die Kerngebiete der mathematischen Logik”, Spektrum Akademischer Verlag, 2011

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