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$$
\begin{aligned}
\overset{\text{def.}}{\Longleftrightarrow}
\end{aligned}
$$

$$
\begin{aligned}
^\forall \epsilon
\end{aligned}
$$

$$
\begin{aligned}
&X : \text{set} \\
&\mathfrak{P}(X)
\coloneqq \{ A : \text{set} \mid A \subset X \} : \text{power set of } X \\
&\boldsymbol{B} \subset \mathfrak{P}(X) : \text{family of subsets of } X
\end{aligned}
$$

$$
\begin{aligned}
\boldsymbol{x} = \left(
\begin{array}{c}
x_{1} \\
\vdots \\
x_{j} \\
\vdots \\
x_{n}
\end{array}
\right)
\end{aligned}
$$

$$
\begin{aligned}
&A = \left(
\begin{array}{ccccc}
a_{11} & \cdots & a_{1j} & \cdots & a_{1n} \\
\vdots && \vdots && \vdots \\
a_{i1} & \cdots & a_{ij} & \cdots & a_{in} \\
\vdots && \vdots && \vdots \\
a_{m1} & \cdots & a_{mj} & \cdots & a_{mn} \\
\end{array}
\right) \in \mathbb{R} ^ {m \times n}
\end{aligned}
$$

$$
\begin{aligned}
&I = \left(
\begin{array}{ccccc}
    1 & 0 & \cdots & 0 & 0 \\
    0 & 1 & \ddots & & 0 \\
    \vdots & \ddots & \ddots & \ddots & \vdots \\
    0 & & \ddots & 1 & 0 \\
    0 & 0 & \cdots & 0 & 1
\end{array}
\right) \in \mathbb{R}^{n \times n}
\end{aligned}
$$

$$\int_{\mathbb{R}}{e^{-x^2}} dx = \sqrt{\pi}$$

$$\overset{\infty}{\underset{n=1}{\bigcup}} A_{n} \in \boldsymbol{B}$$

$$(\Omega, \mathcal{F}, P)$$

$$
\begin{aligned}
\left \Vert \quad
\begin{aligned}
&\textbf{min.} \quad
&&\sum_{j=1}^{n} c_{j} x_{j} \in \mathbb{R} \\
\\
&\textbf{s.t.} \quad
&&\sum_{j=1}^{n} a_{ij} x_{j} \geq b_{i} \, (i = 1, \cdots , m) \\
&&&x_{j} \geq 0 \, (j = 1, \cdots , n)
\end{aligned}
\right.
\end{aligned}
$$

$$
\begin{aligned}
D &= \left(
\begin{array}{c:c}
0 & C ^\mathrm{T} \\
\hdashline
-B & A
\end{array}
\right) \\
\\
&= \left(
\begin{array}{c:ccccc}
0 & c_{1} & \cdots & c_{j} & \cdots & c_{n} \\
\hdashline
-b_{1} & a_{11} & \cdots & a_{1j} & \cdots & a_{1n} \\
\vdots & \vdots && \vdots && \vdots \\
-b_{i}  & a_{i1} & \cdots & a_{ij} & \cdots & a_{in} \\
\vdots & \vdots && \vdots && \vdots \\
-b_{m} & a_{m1} & \cdots & a_{mj} & \cdots & a_{mn} \\
\end{array}
\right) \\
\end{aligned}
$$