# Set theory general questions

I'd be very thankful if you could help me with the following (I suppose basic) questions:

First of all, how is the union of a set defined: By ZF$4$, given a set $x$, there is a set consisting of all the elements of all the elements of $x$. Is this the union?

If yes, then if we take $X=\{\{1\}, \{1,2\}\}$, my notes claim that $\bigcup X=\{1,2\}$, but shouldn't it be $\{\{1\},\{2\},\{1,2\}\}$ since those are the elements of elements.

Another problem I have is the definition of brackets when using formulas - for example, if we take the definition of null set axiom we have the formula $∃x∀y(¬(y ∈ x))$ which I can't wrap my head around

Thanks!

Recall the definition:

$$\bigcup X = \{ y \mid \exists x \in X \colon y \in x \}$$

We have $1 \in \bigcup X$, since $y := 1 \in \{1\}$ and $x := \{1 \} \in X$.

Similarly $2 \in \bigcup X$, since $2 \in \{1,2 \}$ and $\{1,2\} \in X$.

It's now easy to verify that these are the only elements of $\bigcup X$ and hence that $\bigcup X = \{1,2\}$.

• So we don't actually care about sets, just about the elements and then taking them all to be a set?
– asdf
Jan 24, 2018 at 19:31
• Well elements are sets by definition. Therefore, it is efficient to use generalized unions and power sets to prove classes as sets.
– user468462
Jan 24, 2018 at 19:39
• @asdf It might be a bit confusion but if you carefully look at the definition of $\bigcup X$, you will see that it is the set of all sets which are elements of some set that itself is an element of $X$. I, however, find this verbal explanation more confusing than the formal definition.... If you want an 'intuitive' description of $\bigcup X$: It is the set that you obtain if you strip the outermost curcly brackets from every of its elements. Jan 24, 2018 at 19:49
• On the topic, how to visualise the union $x \bigcup \{x\}$? If we take x to be the set 1, then it is $1 \bigcup \{1\}$. Is $\{1\}=1?$
– asdf
Jan 24, 2018 at 20:42
• @asdf It's not clear to me what you are asking. If your question is: "What is $\bigcup \{ 1 \}$?", then you need to know what $1$ is -- as a set. In standard coding, $1 = \{ \emptyset \}$ and hence $\bigcup \{ 1\} = \bigcup \{\{\emptyset\}\} = \{\emptyset\}$. Jan 25, 2018 at 8:30

$\bigcup X = \{1\} \cup \{1,2\}=\{1,2\}$

• But isn't the definition "all elements of all elements". Doesn't this mean that $\{1,2\}$ has to be in the union as a set?
– asdf
Jan 24, 2018 at 19:23
• It is the union of all elements of X. Elements of X are sets. Therefore this is the union of sets of X. The union of sets is just the set of elements that belong to either of the sets. Jan 24, 2018 at 19:42