# The axiom of union, combined with the axiom of pair set, implies the axiom of pair pairwise union.

Axiom 3.1: (Sets are objects). If $$A$$ is a set, then $$A$$ is also an object. In particular, given two sets $$A$$ and $$B$$, it is meaningful to ask whether $$A$$ is also an element of $$B$$.

Axiom 3.3 (Singleton sets and pair sets). If $$a$$ is an object, then there exists a set $$\{a\}$$ whose only element is $$a$$, i.e., for every object $$y$$, we have $$y \in \{a\}$$ if and only if $$y=a$$; we refer to $$\{a\}$$ as the singleton set whose element is $$a$$. Furthermore, if $$a$$ and $$b$$ are objects, then there exists a set $$\{a , b\}$$ whose only elements are $$a$$ and $$b$$; i.e., for every object $$y$$, we have $$y \in \{a, b\}$$ if and only if $$y=a$$ or $$y=b$$; we refer to this set as the pair set formed by $$a$$ and $$b$$.

Axiom 3.4 (Pairwise union). Given any two sets $$A$$, $$B$$, there exists a set $$A \cup B$$, called the union $$A \cup B$$ of $$A$$ and $$B$$, whose elements consists of all the elements which belong to $$A$$ or $$B$$ or both. In other words, for any object $$x$$, $$x \in A \cup B \iff (x \in A \text{or} x \in B).$$

Axiom 3.11 (Union). Let $$A$$ be a set, all of whose elements are themselves sets. Then there exists a set $$\bigcup A$$ whose elements are precisely those objects which are elements of the elements of $$A$$, thus for all objects $$x$$ $$x \in \bigcup A \iff (x \in S \text{for some} S \in A).$$

Exercise 3.4.8. Show that Axiom 3.4 can be deduced from Axiom 3.1, Axiom 3.3 and Axiom 3.11.

I am struggling with this exercise in Analysis 1 by Tao. My attempt is as follows: Let $$C = \{A , B\}$$. Then, $$x \in \bigcup C \iff x \in A \text{or} B$$ by the axiom of the union, and $$\bigcup C$$ is a set. I think that $$\bigcup C$$ is equal to $$A \cup B$$, but I don't know how to complete the proof. I appreciate if you give some help.

Yes, $$\bigcup C=A\cup B$$, because $$x\in\bigcup C\iff (x\in A) \text{ or } (x\in B) \iff x\in A\cup B \,.$$
• @shk910 to guarantee that $\{A,B\}$ is a set – Alessandro Codenotti Feb 23 at 11:55