LaTex Notes

Here are some notes I find useful for using LaTex

1. Force limits to the top and bottom side of inline sum : \(\sum\limits_{i=1}^{N} i\), use \sum\limits instead of \sum alone.
2. Force limits to the top and bottom side of inline integral : \(\int \limits_{x=1}^{3} xdx\), use \int\limits instead of \int alone.
3. Hat:
   1. Fitted value: \(\hat{y}\), use \hat{}
   2. Widehat: \(\widehat{\nabla f }\), 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: \(\nabla f\), use \nabla
7. Field representation: the complex field \(\mathbb{C}\), the real field \(\mathbb{R}\), use \mathbb{}
8. Logical operator
   • P and Q: \(P \land Q\), \land
   • P or Q: \(P \lor Q\), \lor
   • not P: \(\neg P\), \neg, or \(\bar{P}\), \bar{P}
   • P xor Q: \(p \oplus Q\), \oplus
   • if P then Q: \(P \rightarrow Q\), \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
   • \(\emptyset\): \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
   • \(\bigcup\limits_{i=1}^{n}\): \bigcup\limits_{i=1}^{n}
   • \(\bigcap\limits_{i=1}^{n}\): \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 \[\begin{align*} a &= b \\ &= c \\ &= d \end{align*}\]

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: \(\binom{n}{k_1,k_1,\dots,k_m} = \frac{n!}{k_1 ! k_2 !\dots k_m !}\), \binom {n}{k_1,k_1,\dots,k_m}, use binom{}{} for combinations
17. Multi conditions in summation: \(\sum\limits_{\substack{i=0 \\ i\neq j}}^{n}a_i\), \substack{i=0 \ \ i \neq j}, use substack{} for multi conditions, use \\to separate the lines
18. Arrows
    • \(\leftarrow\) \leftarrow, \(\Leftarrow\) \Leftarrow
    • \(\longleftarrow\) \longleftarrow, \(\Longrightarrow\) \Longrightarrow
    • \(\rightarrow\) \rightarrow, \(\Rightarrow\) \Rightarrow
    • \(\longrightarrow\) \longrightarrow, \(\Longrightarrow\) \Longrightarrow
    • \(\leftrightarrow\) \leftrightarrow, \(\Leftrightarrow\) \Leftrightarrow
    • \(\longleftrightarrow\) \longleftrightarrow, \(\Longleftrightarrow\) \Longleftrightarrow
19. The does not prove sign (appeared in Gödel's incompleteness theorems)
    • \(\nvdash\) \nvdash
    • \(\not\vdash\) \not\vdash