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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!

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2 Answers 2

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

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  • $\begingroup$ So we don't actually care about sets, just about the elements and then taking them all to be a set? $\endgroup$
    – asdf
    Jan 24, 2018 at 19:31
  • $\begingroup$ Well elements are sets by definition. Therefore, it is efficient to use generalized unions and power sets to prove classes as sets. $\endgroup$
    – user468462
    Jan 24, 2018 at 19:39
  • $\begingroup$ @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. $\endgroup$ Jan 24, 2018 at 19:49
  • $\begingroup$ 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 $\{1\}=1?$ $\endgroup$
    – asdf
    Jan 24, 2018 at 20:42
  • $\begingroup$ @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 $\bigcup \{ 1\} = \bigcup \{\{\emptyset\}\} = \{\emptyset\}$. $\endgroup$ Jan 25, 2018 at 8:30
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$\bigcup X = \{1\} \cup \{1,2\}=\{1,2\}$

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  • $\begingroup$ 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? $\endgroup$
    – asdf
    Jan 24, 2018 at 19:23
  • $\begingroup$ 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. $\endgroup$
    – mandella
    Jan 24, 2018 at 19:42

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