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We have just shown how the axiom of power set ensures the existence and uniqueness of the power set. The following diagram illustrates the power set $\mathcal P(X)$ for a set $X$ containing three elements:

Using the axiom of separation, we can now separate a subset of $\mathcal P(X)$ containing exactly $X$ as its single element:

Moreover, the resulting set is unique by the axiom of extensionality. This motivates the following definition:

Definition: Singleton

For every set $X$ the set $\{X\}$ is well-defined and is called the singleton of $X.$ Formally, using the power set, we have $$\{X\}:=\{z\in\mathcal P(X)\mid z=X\}.$$

| | | | | created: 2019-01-13 11:13:00 | modified: 2019-01-13 11:17:08 | by: bookofproofs | references: [656], [983]

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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