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\[ \begin{aligned} \overset{\text{def.}}{\Longleftrightarrow} \end{aligned} \]

\[ \begin{aligned} ^\forall \epsilon \end{aligned} \]

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

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

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

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

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

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

\[(\Omega, \mathcal{F}, P)\]

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

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