Tao Analysis I - 3.4.8

Exercise 3.4.8

Show that Axiom 3.5 can be deduced from Axiom 3.1, Axiom 3.4, and Axiom 3.12.

Let's remind ourselves of these axioms.

Axiom 3.5 (Pairwise union). Given any two sets $A$, $B$, there exists a set $A \cup B$, called the union of $A$ and $B$, which consists of all the elements which belong to $A$ or $B$ or both. In other words, for any object $x$,

$$ x \in A \cup B \iff (x \in A \lor x \in B)$$

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 3.4 (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 3.12 (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 $x$

$$x \in \bigcup A \iff (x \in S \text{ for some } S \in A)$$

Thoughts

The Axiom of Union 3.12 is a generalisation of Axiom of Pairwise Union 3.5. 

This suggests we should apply the given constraints to Axiom 3.12 to yield 3.5.

Solution

Consider two sets, $A$ and $B$. 

Axiom 3.1 tells us these sets are objects. As such they can be considered elements of other sets.

Axiom 3.4 for pair sets tells us that two objects A and B, then there exists a s set $C = \{A, B\}$ whose only elements are $A$ and $B$. In our case, $A$ and $B$ are also sets themselves.

Axion 3.12 tells us that there exists a set $\bigcup C$ whose elements are precisely those objects which are elements of the elements of $C$. That is,  the elements of $\bigcup C$ are the elements of $A$ or $B$, or both.

$$ x \in \bigcup \{A,B\} \iff (x \in A \lor x \in B)$$

This is the definition of a pairwise set of $A$ and $B$, this Axiom 3.5 follows from Axions 3.1, 3.4 and 3.12. $\square$