Math Proof Notation question (Not vs Complement) I need some clarification for the following question:
Q: Prove that :If A and B are sets in a universal set U, then $(A ∪ B)^c = A^c ∩ B^c$.
The textbook solution started with: $$\begin{align}(A ∪ B)^c &= U − (A ∪  B) \\
&= \{x : (x ∈ U) ∧ (x ∉ A ∪ B)\} \\
&= \{x : (x ∈ U) ∧ ∼ ((x ∈ A)∨(x ∈ B))\}\end{align}$$
My question is that for "$A^c$", is this equivalent to "Not $A$" or $∼  A$. So that I can write out the above proof as the following:
$$\begin{align}(A ∪ B)^c &= \{x : x ∉ A ∪ B\} \\&= \{x : x ∈ ¬ (A ∪ B)\} \\&= \{x : (x ∈ ¬A) ∧ (x ∈ ¬B)\} \\&= \{x : (x ∈ ¬A) ∩ (x ∈ ¬B)\}
\\&= \{x : (x ∈ A^c) ∩ (x ∈ B^c)\} \\&= A^c ∩ B^c\end{align}$$
I feel like stating that $(x ∈ U)$ is a bit redundant? Like what exactly is the difference between the logic operators $∨$, $∧$ and $∪$, $∩$ ?
Thanks very much

You can't write $x \in \neg A$
$\neg$ is a logical operator that works on statements. So, with $x \in A$ being a statement, $\neg x \in A$ would mean that it is not true that $x \in A$. But $A$ is a set, not a statement, and so $\neg A$ doesn't make sense. 
You can, however, apply the complement operator on sets: $A^C$ is the set of all objects in $U$ that are not in $A$. 
Indeed, the $\neg$ operator for statements and the complement operator for sets are closely related: for any set $A$, we have that $x \in A^C$ if and only if $\neg x \in A$
