Axiom of Union?

I'm reading Comprehensive Mathematics for Computer Scientists 1. On the second chapter: Axiomatic Set Theory.

He first states the axiom of the empty set, the axiom of equality and then he proceeds to the axiom of union:

Axiom 3 (Axiom of Union) If $a$ is a set, then there is a set:

$\{$$x$ | there exists an element $b\in a$ such that $x\in b$$\}$.

This set is denoted by $\bigcup a$ and is called the union of $a$.

Notation 2 If a = {b,c}. or a = {b,c,d}, respectively, one also writes b $\cup$ c, or b $\cup$ c $\cup$ d, respectively, instead of $\cup$a

I've learned the definition of Union while I was in school, but it wasn't with axioms, they just gave an intuitive example:

$a=\{1,2,3\}$

$b=\{4,5\}$

$a\bigcup b=\{1,2,3,4,5\}$

I can't see how the notion of this intuitive example happens on the axiom of union. In my example, it's easy to understand because there's a mention to another set, where's the mention in this axiom?

Solution 1:

The connection between your example and the more general definition is that $\bigcup\{a,b\}=a\cup b$. Written out in all its gory details, this is

$$\bigcup\Big\{\{1,2,3\},\{4,5\}\Big\}=\{1,2,3\}\cup\{4,5\}=\{1,2,3,4,5\}\;.$$

Let’s check that against the definition:

$$\begin{align*} &\bigcup\Big\{\{1,2,3\},\{4,5\}\Big\}\\ &\qquad=\left\{x:\text{there exists an element }y\in\Big\{\{1,2,3\},\{4,5\}\Big\}\text{ such that }x\in y\right\}\\ &\qquad=\Big\{x:x\in\{1,2,3\}\text{ or }x\in\{4,5\}\Big\}\\ &\qquad=\{1,2,3\}\cup\{4,5\}\\ &\qquad=\{1,2,3,4,5\}\;. \end{align*}$$

Take a slightly bigger example. Let $a,b$, and $c$ be any sets; then

$$\begin{align*} \bigcup\{a,b,c\}&=\Big\{x:\text{there exists an element }y\in\{a,b,c\}\text{ such that }x\in y\Big\}\\ &=\{x:x\in a\text{ or }x\in b\text{ or }x\in c\}\\ &=a\cup b\cup c\;. \end{align*}$$

One more, even bigger: for $n\in\Bbb N$ let $A_n$ be a set, and let $\mathscr{A}=\{A_n:n\in\Bbb N\}$. Then

$$\begin{align*} \bigcup\mathscr{A}&=\Big\{x:\text{there exists an }n\in\Bbb N\text{ such that }x\in A_n\Big\}\\ &=\{x:x\in A_0\text{ or }x\in A_1\text{ or }x\in A_2\text{ or }\dots\}\\ &=A_0\cup A_1\cup A_2\cup\dots\\ &=\bigcup_{n\in\Bbb N}A_n\;. \end{align*}$$

Solution 2:

Let $A=\{a,b\}$ (the set whose only elements are $a$ and $b$). Then the union of $a$ and $b$ that you described is what the Axiom of Union produces from $A$.

Remark: Informally, let $A$ be a set whose elements are a bunch of plastic bags with stuff in them (so $A$ is a set of sets). Then the set produced by the Axiom of Union from $A$ dumps the stuff contained in the bags into a single bag. (Duplicates are thrown away.)

Solution 3:

When we write $a\cup b$ we actually mean $\bigcup\{a,b\}$. This is a shorthand instead of writing long formulas every time we want to talk about the union of two sets.