Definition of the complement of a set

Question

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?

Answer 1

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 $A$.