For a universal set $U$, I have two questions.
1) Is it guaranteed that any set $A$ is going to be a subset of $U$, because $U$ is the universal set?
2) In the context that $A$ is a subset of $U$, would it be valid to think of the absolute complement of $A$, as the logical negation of every element of $A$?
Best Answer
In elementary set theory, once we talk about a universal set $U$, every set will be a subset of $U$. The complement of $A$ is the set of all elements that are in $U$ but not in $A$. We do not speak of the logical negation of every element of $A$, because the elements of $A$ are not logical statements. Rather, an element of $U$ is in the complement of $A$ if and only if it is not in $A$.