EDB — 026

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[026] The axiom of union 1 says that for each set \(A\) there is a set \(B\) that contains all the elements of the elements of \(A\); in symbols,

\[ ∀ A ∃ B, ∀ x, (x∈ B \iff (∃ y,y∈ A ∧ x∈ y))~ ~ . \]

This implies that this set is unique, by the axiom of extensionality [1Y8]; we indicate this set \(B\) with \({\underline⋃} A\) (so as not to confuse it with the symbol already introduced before).

For example if

\[ A = \{ \{ 1,3,\{ 5,2\} \} , \{ 7,19\} \} \]

then

\[ {\underline⋃}A = \{ 1,3,\{ 5,2\} , 7,19 \} \quad . \]

Given \(A_ 1,\ldots A_ k\) sets, let \(D=\{ A_ 1,\ldots A_ k\} \) 2 we define

\[ A_ 1∪ A_ 2\ldots ∪ A_ k {\stackrel{.}{=}}{\underline⋃}D \quad . \]

  1. This is the ”official” version of Zermelo–Fraenkel. However, the simplified version [1Y2] is often used
  2. The existence of this set can be proven, see [029]
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  • axiom, of union
  • union of sets
  • \( \underline \bigcup \)
  • \(\cup \)
  • formal set theory
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