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Section 3.1 Formula gallery

Formula 1
\[
  x \mapsto \{\, c \in C \mid c \leq x \,\}
\]

Formula 2
\[
  \left| \bigcup (\, I_{j} \mid j \in J \,) \right|
   < \mathfrak{m}
\]

Formula 3
\[
  A = \{\, x \in X \mid x \in X_{i},
        \mbox{ for some } i \in I \,\}
\]

Formula 4
\[
  \langle a_{1}, a_{2} \rangle \leq \langle a'_{1}, a'_{2}\rangle
   \qquad \mbox{if{f}} \qquad a_{1} < a'_{1} \quad  \mbox{or}
   \quad a_{1} = a'_{1} \mbox{ and } a_{2} \leq a'_{2}
\]

Formula 5
\[
  \Gamma_{u'} = \{\, \gamma \mid \gamma < 2\chi,
   \ B_{\alpha} \nsubseteq u', \ B_{\gamma} \subseteq u' \,\}
\]

Formula 6
\[
  A = B^{2} \times \mathbb{Z}
\]

Formula 7
\[
  \left( \bigvee (\, s_{i} \mid i \in I \,) \right)^{c} =
   \bigwedge (\, s_{i}^{c} \mid i \in I \,)
\]

Formula 8
\[
  y \vee \bigvee (\, [B_{\gamma}] \mid \gamma
   \in \Gamma \,) \equiv z \vee \bigvee (\, [B_{\gamma}]
   \mid \gamma \in \Gamma \,) \pmod{ \Phi^{x} }
\]

Formula 9
\[
  f(\mathbf{x}) = \bigvee\nolimits_{\!\mathfrak{m}}
   \left(\,
      \bigwedge\nolimits_{\mathfrak{m}}
      (\, x_{j} \mid j \in I_{i} \,) \mid i < \aleph_{\alpha}
   \,\right)
\]

Formula 10
\[
  \left. \widehat{F}(x) \right|_{a}^{b} =
     \widehat{F}(b) - \widehat{F}(a)
\]

Formula 11
\[
 u \underset{\alpha}{+} v \overset{1}{\thicksim} w
   \overset{2}{\thicksim} z
\]

Formula 12
\[
  f(x) \overset{ \text{def} }{=} x^{2} - 1
\]

Formula 13
\[
  \overbrace{a + b + \cdots + z}^{n}
\]

Formula 14
\[
  \begin{vmatrix}
     a + b + c & uv\\
     a + b & c + d
  \end{vmatrix}
  = 7
\]

\[
  \begin{Vmatrix}
     a + b + c & uv\\
     a + b & c + d
  \end{Vmatrix}
  = 7
\]

Formula 15
\[
  \sum_{j \in \mathbf{N}} b_{ij} \hat{y}_{j} =
  \sum_{j \in \mathbf{N}} b^{(\lambda)}_{ij} \hat{y}_{j} +
  (b_{ii} - \lambda_{i}) \hat{y}_{i} \hat{y}
\]

Formula 16
\[
  \left( \prod^n_{\, j = 1} \hat x_{j} \right) H_{c} =
   \frac{1}{2} \hat k_{ij} \det \hat{ \mathbf{K} }(i|i)
\]

\[
  \biggl( \prod^n_{\, j = 1} \hat x_{j} \biggr) H_{c} =
   \frac{1}{2} \hat{k}_{ij} \det \widehat{ \mathbf{K} }(i|i)
\]

Formula 17
\[
  \det \mathbf{K} (t = 1, t_{1}, \ldots, t_{n}) =
   \sum_{I \in \mathbf{n} }(-1)^{|I|}
   \prod_{i \in I} t_{i}
   \prod_{j \in I} (D_{j} + \lambda_{j} t_{j})
   \det \mathbf{A}^{(\lambda)} (\,\overline{I} | \overline{I}\,) = 0
\]

Formula 18
\[
  \lim_{(v, v') \to (0, 0)}
   \frac{H(z + v) - H(z + v') - BH(z)(v - v')}
        {\| v - v' \|} = 0
\]

Formula 19
\[
  \int_{\mathcal{D}} | \overline{\partial u} |^{2}
   \Phi_{0}(z) e^{\alpha |z|^2} \geq
   c_{4} \alpha \int_{\mathcal{D}} |u|^{2} \Phi_{0}
   e^{\alpha |z|^{2}} + c_{5} \delta^{-2} \int_{A}
   |u|^{2} \Phi_{0} e^{\alpha |z|^{2}}
\]

Formula 20
\[
  \mathbf{A} =
  \begin{pmatrix}
     \dfrac{\varphi \cdot X_{n, 1}}
           {\varphi_{1} \times \varepsilon_{1}}
     & (x + \varepsilon_{2})^{2} & \cdots
     & (x + \varepsilon_{n - 1})^{n - 1}
     & (x + \varepsilon_{n})^{n}\\[10pt]
     \dfrac{\varphi \cdot X_{n, 1}}
           {\varphi_{2} \times \varepsilon_{1}}
     & \dfrac{\varphi \cdot X_{n, 2}}
             {\varphi_{2} \times \varepsilon_{2}}
     & \cdots & (x + \varepsilon_{n - 1})^{n - 1}
     & (x + \varepsilon_{n})^{n}\\
     \hdotsfor{5}\\
     \dfrac{\varphi \cdot X_{n, 1}}
           {\varphi_{n} \times \varepsilon_{1}}
     & \dfrac{\varphi \cdot X_{n, 2}}
             {\varphi_{n} \times \varepsilon_{2}}
     & \cdots & \dfrac{\varphi \cdot X_{n, n - 1}}
                      {\varphi_{n} \times \varepsilon_{n - 1}}
     & \dfrac{\varphi\cdot X_{n, n}}
             {\varphi_{n} \times \varepsilon_{n}}
  \end{pmatrix}
   + \mathbf{I}_{n}
\]


Section 3.2. User-defined commands

Formula 20 with user-defined commands:

\newcommand{\quot}[2]{%
\dfrac{\varphi \cdot X_{n, #1}}%
{\varphi_{#2} \times \varepsilon_{#1}}}
\newcommand{\exn}[1]{(x+\varepsilon_{#1})^{#1}}

\[
  \mathbf{A} =
  \begin{pmatrix}
    \quot{1}{1} & \exn{2} & \cdots & \exn{n - 1}&\exn{n}\\[10pt]
    \quot{1}{2} & \quot{2}{2} & \cdots & \exn{n - 1} &\exn{n}\\
    \hdotsfor{5}\\
    \quot{1}{n} & \quot{2}{n} & \cdots &
    \quot{n - 1}{n} & \quot{n}{n}
  \end{pmatrix}
  + \mathbf{I}_{n}
\]

Section 3.3. Building a formula step-by-step

Step 1
$\left[ \frac{n}{2} \right]$

Step 2
\[
  \sum_{i = 1}^{ \left[ \frac{n}{2} \right] }
\]

Step 3
\[
  x_{i, i + 1}^{i^{2}} \qquad \left[ \frac{i + 3}{3} \right]
\]

Step 4
\[
  \binom{ x_{i,i + 1}^{i^{2}} }{ \left[ \frac{i + 3}{3} \right] }
\]

Step 5
$\sqrt{ \mu(i)^{ \frac{3}{2} } (i^{2} - 1) }$

$\sqrt{ \mu(i)^{ \frac{3}{2} } (i^{2} - 1) }$

Step 6
$\sqrt[3]{ \rho(i) - 2 }$  $\sqrt[3]{ \rho(i) - 1 }$

Step 7
\[
  \frac{ \sqrt{ \mu(i)^{ \frac{3}{2}} (i^{2} -1) } }
       { \sqrt[3]{\rho(i) - 2} + \sqrt[3]{\rho(i) - 1} }
\]

Step 8
\[
  \sum_{i = 1}^{ \left[ \frac{n}{2} \right] }
     \binom{ x_{i, i + 1}^{i^{2}} }
           { \left[ \frac{i + 3}{3} \right] }
     \frac{ \sqrt{ \mu(i)^{ \frac{3}{2}} (i^{2} - 1) } }
          { \sqrt[3]{\rho(i) - 2} + \sqrt[3]{\rho(i) - 1} }
\]

\[\sum_{i=1}^{\left[\frac{n}{2}\right]}\binom{x_{i,i+1}^{i^{2}}}
{\left[\frac{i+3}{3}\right]}\frac{\sqrt{\mu(i)^{\frac{3}
{2}}(i^{2}-1)}}{\sqrt[3]{\rho(i)-2}+\sqrt[3]{\rho(i)-1}}\]

%\[\sum_{i=1}^{\left[\frac{n}{2}\right]}\binom{x_{i,i+1}^{i^{2}}}
%{\left[\frac{i+3}{3}\right]}\frac{\sqrt{\mu(i)^{\frac{3}
%{2}}}(i^{2}-1)}}{\sqrt[3]{\rho(i)-2}+\sqrt[3]{\rho(i)-1}}\]


\end{document}

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