LaTex Notes

Here are some notes I find useful for using LaTex

1. Force limits to the top and bottom side of inline sum : $\sum _{i=1}^{N}i$, use \sum\limits instead of \sum alone.
2. Force limits to the top and bottom side of inline integral : $\underset{x=1}{\overset{3}{\int }}xdx$, use \int\limits instead of \int alone.
3. Hat:
   1. Fitted value: $\hat{y}$, use \hat{}
   2. Widehat: $\widehat{\mathrm{\nabla }f}$, use \widehat{}
4. Italic: $\mathcal{L}$, it only works for upper-case letters, use \mathcal{}
5. Underset: $\underset{x}{argmin}$, use \underset{x}{formula}
6. Gradient: $\mathrm{\nabla }f$, use \nabla
7. Field representation: the complex field $\mathbb{C}$, the real field $\mathbb{R}$, use \mathbb{}
8. Logical operator
   • P and Q: $P\wedge Q$, \land
   • P or Q: $P\vee Q$, \lor
   • not P: $\mathrm{\neg }P$, \neg, or $\overline{P}$, \bar{P}
   • P xor Q: $p\oplus Q$, \oplus
   • if P then Q: $P\to Q$, \rightarrow
   • P iff Q: $P\leftrightarrow Q$, \leftrightarrow
9. Set notation
   • $\mathbb{R}$, \mathbb{R}
   • $a\in A$: a \in A
   • $a\notin A$: a \notin A
   • $\mathrm{\varnothing }$: \emptyset
   • $A\cap B$: \cap
   • $A\cup B$: \cup
   • $\mathcal{P}(A)$ \mathcal{P} (power set)
   • $A\times B$: \times
   • $A\setminus B$: \setminus
   • $A\subseteq B$: \subseteq
   • $A\subset B$: \subset
   • $\bigcup _{i=1}^n$: \bigcup\limits_{i=1}^{n}
   • $\bigcap _{i=1}^n$: \bigcap\limits_{i=1}^{n}
10. Make equation smaller: \scriptstyle
11. Isomorphism and approximate equality

• $G_1\cong G_2$: \cong
• $a\approx b$: \approx
• $a\approxeq b$: \approxeq

12. Add label to equation: \tag{N}
13. How to write text inside equation: \text{text}
14. Multline equations $$\begin{array}{rl}a& =b\\ & =c\\ & =d\end{array}$$

allign* has no equation label, align has equation label, use &= to align equations marks, use \ \ to separate lines

15. function composition $g\circ f$, \circ
16. multinomial coefficient: $(\genfrac{}{}{0ex}{}{n}{k_1,k_1,\dots ,k_m})=\frac{n!}{k_1!k_2!\dots k_m!}$, \binom {n}{k_1,k_1,\dots,k_m}, use binom{}{} for combinations
17. Multi conditions in summation: $\sum _{\begin{array}{c}i=0\\ i\ne j\end{array}}^n{a}_i$, \substack{i=0 \ \ i \neq j}, use substack{} for multi conditions, use \\to separate the lines
18. Arrows
    • $\leftarrow$ \leftarrow, $\Leftarrow$ \Leftarrow
    • $⟵$ \longleftarrow, $⟹$ \Longrightarrow
    • $\to$ \rightarrow, $\Rightarrow$ \Rightarrow
    • $⟶$ \longrightarrow, $⟹$ \Longrightarrow
    • $\leftrightarrow$ \leftrightarrow, $\iff$ \Leftrightarrow
    • $⟷$ \longleftrightarrow, $⟺$ \Longleftrightarrow
19. The does not prove sign (appeared in Gödel's incompleteness theorems)
    • $⊬$ \nvdash
    • $⊬$ \not\vdash