is $\:\: x \not\in S \:\:$ same as $\:\:\neg (x \in S)\:\:$ same as $\:\:x \in {S}^\mathsf{c}$? In the following list, $S$ is any set and ${S}^\mathsf{c}$ is its complement:

*

*$\:\: x \not\in S $

*$\:\: \neg (x \in S)\:\:$

*$\:\: x \in {S}^\mathsf{c}$
I am sure that 1 and 2 are equivalent but I have been pondering if 3 is also equivalent to the previous two. I have yet to come across a book that gives me a proof or disproof or say something about it. I cannot offer a attempt to prove it because I have no idea how to tackle it.
 A: Yes, those conditions are equivalent.  $S^c$ means the complement of $S$.
A: I figured I'd give a slightly more elaborate answer, in light of the discussion regarding the complement notation in the comments.
Strictly speaking, you can only take the complement of a set with respect to another set. If you have a set $X$ and a subset $S \subseteq X$, then the complement of $S$ in $X$, often denoted $X \backslash S$, is defined to be:
$$ X \backslash S = \{ x \in X : x \notin S \}. $$
When you use the notation $S^c$, there is an implicit assumption that all our elements and subsets are contained in some "ambient set" $X$. In this case, the notation $S^c$ means the same thing as $X \backslash S$. Then, when you write "$x \in S^c$", what you mean is $x \in X$ and $x \in X \backslash S$. Equivalently, $x \in X$ and $x \notin S$.
So, regarding your three equivalent statements, the confusion may be cleared up once we ask: what is $x$? If $x$ could be anything, then the notation $S^c$ is not well-defined. However, if we are considering $x$ to be an element of some ambient space $X$, then the notation $S^c$ means $X \backslash S$, and we get the equivalence you are after. I've restated a more explicit version below.

Let $X$ be any set, let $S \subseteq X$, and let $S^c$ be the complement of $S$ (in $X$). If $x \in X$, then the following are equivalent.

*

*$x \notin S$

*$\neg (x \in S)$

*$x \in S^c$
You've convinced yourself that 1. is equivalent to 2. To see that 1. is equivalent to 3., just note that $S^c = X \backslash S = \{ x \in X : x \notin S\}$ by definition. Since $x \in X$, $x \in S^c$ is equivalent to $x \notin S$.

