Axiom of Union

Set Theory · Axiom Academy

The union of all elements of a set exists For any set A, there exists a set that contains exactly those elements that belong to at least one member of A. In formal notation, we denote this union as ⋃A (the union of A): Let's work through an example. Consider the set: The Axiom of Union gives us ⋃A, which contains all elements that appear in at least one of these three sets: 3. Union Axiom vs. Binary Union The Axiom of Union is more general than the binary union you may already know: Binary union: A ∪ B (union of two sets) Union axiom: ⋃𝓕 (union of arbitrarily many sets) Binary union can be derived from the Union Axiom: A ∪ B = ⋃ A, B The Axiom of Union enables several important constructions: Flattening nested sets: Removing one level of nesting Building successors: n ∪ n in natural number construction Indexed unions: ⋃ Aᵢ : i ∈ I for families of sets

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