Decrease the distance between these two formulas

Question

How can I decrease the distance between the following two formulas?

\documentclass[12pt,a4paper]{article}
\usepackage[utf8]{vietnam}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{graphicx}
\usepackage[margin=2cm]{geometry}
\usepackage{enumerate}
\usepackage{mathrsfs}
\usepackage{afterpage}
\begin{document}
\noindent\\
Ta có: $$ \begin{array}{c}
\forall y \in \alpha + [(-\alpha+E) \cap A] \Leftrightarrow y- \alpha \in (-\alpha +E) \cap A \Leftrightarrow
\text{$
    \begin{cases}
    y-\alpha \in -\alpha +E \\
    y- \alpha \in A
    \end{cases}$}\Leftrightarrow \text{$
    \begin{cases}
    y \in E \\
    y \in \alpha+A
    \end{cases}$}
\end{array}
$$
$$ \begin{array}{c}
\forall y \in \alpha + [(-\alpha+E) \setminus A] \Leftrightarrow y- \alpha \in (-\alpha +E) \setminus A \Leftrightarrow
\text{$
    \begin{cases}
    y-\alpha \in -\alpha +E \\
    y- \alpha \notin A
    \end{cases}$}\Leftrightarrow \text{$
    \begin{cases}
    y \in E \\
    y \notin \alpha+A
    \end{cases}$}
\end{array}
$$
Ta chứng minh $x+A$, $xA$ là $L$-đo được.
\end{document}

Answer 1

First of all: sorry, my editor cannot deal with your accented characters, so I had to drop them. Then

1. You should not put a cases environment in \text{$...$}, but you do not need cases here anyway. The case environment expects two columns, which is why it introduces the extra space here.
2. Please do not use $$...$$, but \[...\] instead.
3. You can use an array here.
4. I had a little help from David Carlisle.

\documentclass[12pt,a4paper]{article}
\usepackage[margin=2cm]{geometry}
\usepackage{amsmath}
\begin{document}

Ta co: \begin{gather*}
\forall y \in \alpha + [(-\alpha+E) \cap A]
 \Leftrightarrow y- \alpha \in (-\alpha +E) \cap A \Leftrightarrow
    \left\{\begin{array}{@{}l@{}}
    y-\alpha \in -\alpha +E \\ 
    y- \alpha \in A 
    \end{array}\right. \Leftrightarrow
    \left\{\begin{array}{@{}l@{}}
    y \in E \\ 
    y \in \alpha+A
    \end{array}\right.
\\
\forall y \in \alpha + [(-\alpha+E) \setminus A] \Leftrightarrow y- \alpha \in (-\alpha +E) \setminus A \Leftrightarrow
    \left\{\begin{array}{@{}l@{}} 
    y-\alpha \in -\alpha +E \\ 
    y- \alpha \notin A 
    \end{array}\right.\Leftrightarrow
    \left\{\begin{array}{@{}l@{}} 
    y \in E \\ 
    y \notin \alpha+A
    \end{array}\right.
\end{gather*}
Ta chung minh $x+A$, $xA$ la $L$-do duoc.
\end{document}

One may also use aligned.

\documentclass[12pt,a4paper]{article}
\usepackage[margin=2cm]{geometry}
\usepackage{amsmath}
\begin{document}

Ta co: 
\[\begin{aligned}
\forall y \in \alpha + [(-\alpha+E) \cap A]
 \Leftrightarrow y- \alpha \in (-\alpha +E) \cap A \Leftrightarrow
    &\left\{\begin{aligned}
    &y-\alpha \in -\alpha +E \\ 
    &y- \alpha \in A 
    \end{aligned}\right. \Leftrightarrow
    \left\{\begin{aligned}
    &y \in E \\ 
    &y \in \alpha+A
    \end{aligned}\right.
\\
\forall y \in \alpha + [(-\alpha+E) \setminus A] \Leftrightarrow y- \alpha \in (-\alpha +E) \setminus A \Leftrightarrow
    &\left\{\begin{aligned} 
    &y-\alpha \in -\alpha +E \\ 
    &y- \alpha \notin A 
    \end{aligned}\right.\Leftrightarrow
    \left\{\begin{aligned} 
    &y \in E \\ 
    &y \notin \alpha+A
    \end{aligned}\right.
\end{aligned}   
\]
Ta chung minh $x+A$, $xA$ la $L$-do duoc.
\end{document}

Answer 2

Hope it works as desired! Just use the setspace package and control the setstretch option. And try to use \[ ... \] instead of $$ ... $$.

\documentclass[12pt,a4paper]{article}
\usepackage[utf8]{vietnam}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{graphicx}
\usepackage[margin=2cm]{geometry}
\usepackage{enumerate}
\usepackage{mathrsfs}
\usepackage{afterpage}
\usepackage[nodisplayskipstretch]{setspace}
\setstretch{.75}

\begin{document}
\noindent\\
Ta có: $$ \begin{array}{c}
\forall y \in \alpha + [(-\alpha+E) \cap A] \Leftrightarrow y- \alpha \in (-\alpha+E) \cap A \Leftrightarrow
\text{$
\begin{cases} 
y-\alpha \in -\alpha +E \\ 
y- \alpha \in A 
\end{cases}$}\Leftrightarrow \text{$
\begin{cases} 
y \in E \\ 
y \in \alpha+A
\end{cases}$}
\end{array}
$$
$$ \begin{array}{c}
\forall y \in \alpha + [(-\alpha+E) \setminus A] \Leftrightarrow y- \alpha \in (-\alpha +E) \setminus A \Leftrightarrow
\text{$
\begin{cases} 
y-\alpha \in -\alpha +E \\ 
y- \alpha \notin A 
\end{cases}$}\Leftrightarrow \text{$
\begin{cases} 
y \in E \\ 
y \notin \alpha+A
\end{cases}$}
\end{array}
$$
Ta chứng minh $x+A$, $xA$ là $L$-đo được.
\end{document}