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The axioms we introduced so far, the axiom of existence, the axiom of empty set, the axiom of extensionality, and the axiom of separation do not suffice to justify the union operation $”\cup”$ on sets. Of course, for two given sets $A, B$, we could use the axiom of separation to justify the set union by
$$A\cup B:=\{z\mid x\in A\vee x\in B \},$$
like we did in the proof of justification of the set intersection. However, circumstances are quite different now. While in the case of set intersection the elements $z$ were already the common elements of both sets, in the case of a set union no common elements have to exist at all. In other words, we cannot just take it for granted that we can combine any elements of two arbitrary sets $X$ and $Y$ to form a new set $A\cup B$. In order to enforce this possibility, we need two new axioms, the axiom of pairing, and the axiom of union, which we will now introduce:

Axiom: Axiom of Pairing

For any two sets $X,Y$ there exists a set $Z:=\{X,Y\}$ containing both sets as its elements, formally

$$\forall X~\forall Y~\exists Z~\forall z~(z\in Z\Leftrightarrow z=X\vee z=Y)$$

| | | | | created: 2014-06-08 21:09:23 | modified: 2020-08-01 21:48:50 | by: bookofproofs

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