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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$\begingroup$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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