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"Universal set is the set that all the sets in study ares subsets"

Ok, so let´s consider this set:

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$$U = \{A\, , \,B\, ;\, 1 \}$$

$$U = \{\,\{2\,;\,3\,\}\,;\,\{\,3\,;\,4\,\}\,;\,1\,\}$$

So $$\{2\,;\,3\,\} \in\,U$$ and $$\{2\,;\,3\,\}$$ is not a subset of $$U$$

Then, $$\{\,\{2\,;\,3\,\} \notin U$$ wich is wrong

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In this case you should have $U=\{1,2,3,4\}$. –  copper.hat Oct 31 '13 at 17:51
    
It's unclear what you're asking, in part because of the botched first sentence of your question. Do you wonder what "universe" means, in some context? –  Magdiragdag Oct 31 '13 at 17:52

2 Answers 2

up vote 1 down vote accepted

Your $U$ is a universal set only if the sets that you’re studying are limited to the following eight sets: $\varnothing$; $\{\{2,3\}\}$; $\{\{3,4\}\}$; $\{\{1\}\}$; $\{\{2,3\},\{3,4\}\}$; $\{\{2,3\},\{1\}\}$; $\{\{3,4\},\{1\}\}$; $\{\{2,3\},\{3,4\},\{1\}\}$. These eight sets are the only sets that are subsets of $U$, so if you’re studying just these sets, then you can use $U$ as your universal set. If $\{2,3\}$ is one of the sets that you’re studying, you need it to be a subset of $U$, so you need $2$ and $3$ to be elements of $U$: $U=\{2,3,...\}$.

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So the universal set is the set that contains all the elements of the power sets of all the sets in study? –  Voyager Oct 31 '13 at 19:39
    
@Voyager: The smallest usable universal set in a given context is the union of all the sets that you want to study: that’s the smallest set that has each of them as a subset. –  Brian M. Scott Oct 31 '13 at 19:41
    
Brian, just one more question. If i draw a venn diagram of the natural numbers and separate inside between two sets, the even numbers and the odd numbers. So, what is the correct way to write: 2 belongs to U or {2} belongs to U? Thank you very much for your help!! –  Voyager Oct 31 '13 at 20:07
    
@Voyager: It sounds like you want $\Bbb N$, the set of natural numbers, to be your universal set, in which case it’s $2\in U$ or $\{2\}\subseteq U$: the two statements say the same thing. –  Brian M. Scott Oct 31 '13 at 20:09
    
yes, but in our diagram, the 2 is inside in the set of the even numbers (E), so 2 belongs to E and {2} is a subset of U, right? –  Voyager Oct 31 '13 at 20:13

$U$ is the set of all elements in $U$, which includes the elements in $A$ and in $B$.

Hence, $U = \{1, 2, 3, 4\}$, and so, although $2 \in U$ and $3 \in U$, it is not the case that $\{2, 3\} \in U$, but rather, $\{2, 3\} \subset U$.

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