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My book defines the complement of a set as, "Let $U$ be the universal set. The complement of the set $A$, denoted by $\bar{A}$, is the complement of $A$ with respect to $U$. Therefore, the complement of the set $A$ is $U−A$."

To me, it seems like it would be important to add that $A \subseteq U$; and you could possibly have $U \subseteq A$, so $A - U = \varnothing$. Is this a valid and important point; furthermore, should it have been added to the definition?

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The universal set contains all the elements by definition, e.g. in probability theory it is the whole probability space $\Omega$. –  dtldarek Feb 27 '13 at 21:17

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When a set $U$ is designated as the universal set, $U$ is understood to be the "universe", so any set $A$ in the universe is necessarily a subset of $U$; that is, in this context, it is implicitly understood that $A \subseteq U$. It may be that $A = U$, in which case, $A - U = U - A = \varnothing$.

But even so, it seems to me that when working with a universal set $U$, it is still a good idea to explicitly state the relation of $A$ with respect to $U$: i.e. $A \subseteq U$.

A universal set $U$ is important to have and to state explicitly if the complement of a set is to have any meaning: we need to know with respect to "which set" $A^c = \bar{A}$ is defined: all elements of $U$ not belonging to $A$.

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Okay, I understand. One more question, in my book, the mathematical definition is given by $\bar{A} = \{x \in U | x \neg \in A \}$. Could I rewrite it as $\bar{A} = \{x| x \in U \wedge x \neg \in A \}$ Because it would most helpful to write it in that way for a proof I am working on. –  Mack Feb 27 '13 at 21:23
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Yes, indeed you may. That's essentially equivalent. –  amWhy Feb 27 '13 at 21:25
    
Yes. I do believe so. Thank you! –  Mack Feb 27 '13 at 21:39
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The notation $x\lnot \in A$ is somewhat unusual. Much more common is $x\not\in A$, or, if you want to use symbols of logic, $\lnot(x\in A)$. –  André Nicolas Feb 27 '13 at 21:51
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\notin = $\notin$ = \nin –  amWhy Feb 27 '13 at 22:07

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