Formal definition of a complement of a language.

Question

I am in the midst of learning decidable and undecidable language and I came across the following theorem:

A language $A \subseteq \Sigma^*$ is decidable if and only if $A$ as well as $\Sigma^* \setminus A$ are semi-decidable.

I'm not sure what $\Sigma^* \setminus A$, the complement of the language $A$, is supposed to mean. A language is a set of strings, so would its complement then be the set of all strings that are not in $A$? Can anyone provide a more formal explanation and definition?

Answer 1

$\Sigma^*\setminus A = \{x\mid x\in\Sigma^*\wedge x\not\in A\}$.

Answer 2

$\Sigma$ is supposed to denote the alphabet.

A string is just an arbitrary ordering of some (or all) elements of the alphabet.

$\Sigma^n$ is the set of all possible n-sized strings.

$\Sigma^{*}$ = $\bigcup_{i}\Sigma^{i}$

You have a language A,defined over a subset of $\Sigma^{*}$.
$\Sigma^{*}\backslash A$ is simply all strings in $\Sigma^{*}$ but not in $A$