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How are the symbols $\cup$ and $\cap$ are used like that? Can someone please explain what the following does?

$$ \bigcup \left\{\{1\}, \{1,2\}, \bigcap\{\{2,3\}, \{3,4\}\}\right\} $$

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They are just the set union and intersection of the sets inside brackets. – Pedro Tamaroff Mar 1 at 2:25
    
@Pedro Tamaroff♦ what the above set is then? – MATH000 Mar 1 at 2:26
3  
You have a mismatched number of brackets. You have six occurrences of $\{$ and only five occurrences of $\}$. – JMoravitz Mar 1 at 2:31
    
@JMoravitz: Yes, I forgot to add a } at the end – MATH000 Mar 1 at 2:32
    
I added the brace at the end and made it bigger so that it's more obvious how the terms are grouped. – Théophile Mar 1 at 2:35
up vote 10 down vote accepted

When applied to collections of sets, they are the union of elements, and intersection of elements, of the collection.

$$\bigcup\{A, B, C\} = A\cup B\cup C \\ \bigcap\{A, B, C\} = A\cap B\cap C \\ \bigcup\{A, B, \bigcap\{C, D\}\} = A\cup B\cup(C\cap D)$$

And so forth.

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Given a set nonempty set $x$

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

and

$$ \bigcap x = \{ z \mid \forall y \in x \colon z \in y \}. $$

Now let us consider $\bigcap \{ \{2,3\}, \{3,4\} \}$. We have $\{2,3\} \in \{ \{2,3\}, \{3,4\} \}$ and $\{3,4\} \in \{ \{2,3\}, \{3,4\} \}$. $z = 3$ is the only element that is in both $\{2,3\}$ and $\{3,4\}$. This yields $\bigcap \{ \{2,3\}, \{3,4\} \} = \{3\}$. I'm pretty confident that you can take it from here.

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