Math 441 - Abstract Algebra - Main

Please Login to access more options.

You can see the code used to create any symbol on this website by right clicking over the symbol. Here are some commonly used symbols.

$\sim$, $\wedge$, $\vee$, $\cup$, $\cap$, $\bigcup$, $\bigcap$, $\forall$, $\exists$, $\Rightarrow$, $\in$, $\notin$, $\subseteq$, $\not\subseteq$, $\mathcal{P}$, $\mathcal{A}$, $\mathcal{B}$, $\mathcal{C}$, $\mathbb{R}$, $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{W}$, $\square$, $\blacksquare$.

If you want an equation centered with the equal signs aligned, use the align environment. Double click to see the code. Use & to align, and \\ to start a new line. The double dollar signs is what centers the code. The {rll} tell the computer to make 3 columns, aligning them left, right, right. REMEMBER, right click the math below to see the code used to create it. Then you can copy/paste the code to use it in your own work. As an example, you might type $$ \begin{align} \underbrace{1+2+3+\cdots + n} + (n+1) &=\frac{n(n+1)}{2}+(n+1) && \text{ (replace using the assumption) }\\ &=(n+1)\left[\frac{n}{2}+1\right] &&\text{ (factor) }\\ &=(n+1)\frac{n+2}{2} &&\text{(get a common denominator).} \end{align} $$

$$ \begin{pmatrix} 1&2&3\\ 2&3&1 \end{pmatrix} \begin{bmatrix} 1&2&3\\ 2&3&1 \end{bmatrix} \begin{vmatrix} 1&2&3\\ 2&3&1 \end{vmatrix} \begin{matrix} 1&2&3\\ 2&3&1 \end{matrix} \left\{ \begin{matrix} 1&2&3\\ 2&3&1 \end{matrix} \right\} $$

Why $\LaTeX$?

To enter mathematics on the web, we will use the language of LaTeX. LaTeX is a powerful language which typesets mathematics. Some publishers require that you submit work in LaTeX. The exams, quizzes, and other materials that I provide throughout the semester are created using $\LaTeX$.

This page is designed as a place to quickly find the commands you need to get a certain symbol to appear on the web. You can find additional commands at http://www.artofproblemsolving.com/LaTeX/AoPS_L_GuideSym.php. Note that not every LaTeX command will display properly on this website (displaying mathematics on the web is currently under development accross the nation). If you are trying to get a certain symbol to work, please email me and I may be able to get it to work.

Another way to find a symbol that is not on this page is to use a web search engine. For example, if you want to find out how to create a large sigma for using in sigma notation, search for "Latex sigma sum". Most of the time you will find a page that quickly shows you the command you use to produce a sigma.

Currently I use a program called MathJax to display mathematics on the web. One nice feature of MathJax is that you can right click on a math symbol and it will show you the syntax used to create that symbol.

Syntax Examples

We enclose all math symbols inside of $ $ or $$ $$. This tells the computer that we have entered math mode. LaTeX itself just requires the dollar signs, but in order to get math on the web, we add the braces. One easy way to place $ $ around some text is to highlight the text and then select the sqrt{n} button above. Double dollar signs center the mathematics on its own line.

$A$ $A$ (most mathematics symbols are italicized.)

$A=B$ $A=B$

$|P_1 P_2|$ $|P_1 P_2|$

$\frac{x}{y}$ $\frac{x}{y}$

$A_1$ $A_1$

$A_{n+1}$ $A_{n+1}$ (You have to put braces around subscripts and superscripts if they are longer than one character.)

$x_{1}^{3}$ $x_{1}^{3}$

$x_{n+1}^{m-2}$ $x_{n+1}^{m-2}$

$x<y$ $x<y$

$x\leq y$ $x\leq y$

$x\geq y$ $x\geq y$

$x\neq y$ $x\neq y$

$\{x|P(x)\}$ $\{x|P(x)\}$ (to get a bracket to show, you must precede it with a slash)

$\emptyset$ $\emptyset$

$\sum$ $\sum$

$\sum_{n=1}^{\infty}$ $\sum_{n=1}^{\infty}$

$\displaystyle\sum_{n=1}^{\infty}$ $\displaystyle\sum_{n=1}^{\infty}$

$$\sum_{n=1}^{\infty}$$ $$\sum_{n=1}^{\infty}$$ (notice how there are two dollar signs)

${\mathbb{R}}$ ${\mathbb{R}}$ The real numbers

${\mathbb{Z}}$ ${\mathbb{Z}}$ The integers

$\alpha$ $\alpha$

$\beta$ $\beta$

$\gamma$ $\gamma$

$\lambda$ $\lambda$

$\Lambda$ $\Lambda$

$\delta$ $\delta$

$\Delta$ $\Delta$

$f\circ g: A\to B$ $f\circ g: A\to B$

$\hat{x}$ $\hat{x}$

$\bar{x}$ $\bar{x}$

$A\times B$ $A\times B$

$A\cdot B$ $A\cdot B$

$x_1+x_2+\cdots+x_{n-1}+x_n$ $x_1+x_2+\cdots+x_{n-1}+x_n$

$A\approx B$ $A\approx B$

$\sin x \cos x \tan x \cot x \sec x \csc x$ $\sin x \cos x \tan x\ cot x \sec x \csc x$

$(\frac{1}{n})$ $(\frac{1}{n})$

$\left(\frac{1}{n}\right)$ $\left(\frac{1}{n}\right)$ (use \left( \right) when you need bigger parenthesis)

$|x|=\left\{ \begin{array}{c} x \text{ if } x\geq 0\\ -x \text{ if } x<0 \end{array} \right.$

$|x|=\left\{ \begin{array}{c} x \text{ if } x\geq 0\\
 -x \text{ if } x<0 \end{array}\right.$

$ \left<1,2,4 \right> $ $ left<1,2,4 \right> $}

$1\vec{i} + 1\vec{j} +1 \vec{k} $ $1\vec{i} + 1\vec{j} +1 \vec{k} $

$f^\prime$ $f^\prime$

$ \int f(x) dx $ $ \int f(x) dx $

$ \int_a^b f(x) dx $ $ \int_a^b f(x) dx $

$ \displaystyle\int_a^b f(x) dx $ $ \displaystyle\int_a^b f(x) dx $

$$ \int_a^b f(x) dx $$ $$ \int_a^b f(x) dx $$

$ \frac{d}{dx}[f(x)] $ $ \frac{d}{dx}[f(x)] $

$$ \frac{d}{dx}[f(x)] $$ $$ \frac{d}{dx}[f(x)] $$

Lining up calculations accross multiple lines.

The following examples show you how to line up an equation.

Find the derivative of $f(x)=x^2$ using the definition of the derivative.

We use the definition of the derivative as follows. $$ \begin{array}{rl} \lim_{h\to 0} \frac{f(x+h)-f(x)}{h} &= \lim_{h\to 0} \frac{(x+h)^2-x^2}{h}\\ &= \lim_{h\to 0} \frac{x^2+2xh+h^2-x^2}{h}\\ &= \lim_{h\to 0} \frac{2xh+h^2}{h}\\ &= \lim_{h\to 0} 2x+h\\ &= 2x \end{array} $$ This shows that the derivative is $f^\prime(x) = 2x$.

We use the definition of the derivative as follows. $$ \begin{array}{rl}\displaystyle \lim_{h\to 0} \frac{f(x+h)-f(x)}{h} &= \lim_{h\to 0} \frac{(x+h)^2-x^2}{h}\\ &= \lim_{h\to 0} \frac{x^2+2xh+h^2-x^2}{h}\\ &= \lim_{h\to 0} \frac{2xh+h^2}{h}\\ &= \lim_{h\to 0} 2x+h\\ &= 2x \end{array} $$ This shows that the derivative is $f^\prime(x) = 2x$.

The & symbols line up the equation, and \\ means create a new line. You have to start the array environment, and end it. The command {rl} says that there are two columns, with the first column right aligned and the second left aligned. Feel free to copy the syntax and take it to your homepage ( and play with the options there. Please don't edit this page.

Big View Home Login Edit History Recent Changes
HomePage Current Problems All Problems Pictures ProblemsByWeek List of Problems List of Definitions List of Theorems List of Conjectures All Recent changes (use this to find a page that may have been lost)	Main La Te X Commands Please Login to access more options. You can see the code used to create any symbol on this website by right clicking over the symbol. Here are some commonly used symbols. \(\sim\), \(\wedge\), \(\vee\), \(\cup\), \(\cap\), \(\bigcup\), \(\bigcap\), \(\forall\), \(\exists\), \(\Rightarrow\), \(\in\), \(\notin\), \(\subseteq\), \(\not\subseteq\), \(\mathcal{P}\), \(\mathcal{A}\), \(\mathcal{B}\), \(\mathcal{C}\), \(\mathbb{R}\), \(\mathbb{N}\), \(\mathbb{Z}\), \(\mathbb{W}\), \(\square\), \(\blacksquare\). If you want an equation centered with the equal signs aligned, use the align environment. Double click to see the code. Use & to align, and \\ to start a new line. The double dollar signs is what centers the code. The {rll} tell the computer to make 3 columns, aligning them left, right, right. REMEMBER, right click the math below to see the code used to create it. Then you can copy/paste the code to use it in your own work. As an example, you might type $$ \begin{align} \underbrace{1+2+3+\cdots + n} + (n+1) &=\frac{n(n+1)}{2}+(n+1) && \text{ (replace using the assumption) }\\ &=(n+1)\left[\frac{n}{2}+1\right] &&\text{ (factor) }\\ &=(n+1)\frac{n+2}{2} &&\text{(get a common denominator).} \end{align} $$ $$ \begin{pmatrix} 1&2&3\\ 2&3&1 \end{pmatrix} \begin{bmatrix} 1&2&3\\ 2&3&1 \end{bmatrix} \begin{vmatrix} 1&2&3\\ 2&3&1 \end{vmatrix} \begin{matrix} 1&2&3\\ 2&3&1 \end{matrix} \left\{ \begin{matrix} 1&2&3\\ 2&3&1 \end{matrix} \right\} $$ Why $\LaTeX$? To enter mathematics on the web, we will use the language of LaTeX. LaTeX is a powerful language which typesets mathematics. Some publishers require that you submit work in LaTeX. The exams, quizzes, and other materials that I provide throughout the semester are created using $\LaTeX$. This page is designed as a place to quickly find the commands you need to get a certain symbol to appear on the web. You can find additional commands at http://www.artofproblemsolving.com/LaTeX/AoPS_L_GuideSym.php. Note that not every LaTeX command will display properly on this website (displaying mathematics on the web is currently under development accross the nation). If you are trying to get a certain symbol to work, please email me and I may be able to get it to work. Another way to find a symbol that is not on this page is to use a web search engine. For example, if you want to find out how to create a large sigma for using in sigma notation, search for "Latex sigma sum". Most of the time you will find a page that quickly shows you the command you use to produce a sigma. Currently I use a program called MathJax to display mathematics on the web. One nice feature of MathJax is that you can right click on a math symbol and it will show you the syntax used to create that symbol. Syntax Examples We enclose all math symbols inside of $ $ or `$$ $$`. This tells the computer that we have entered math mode. LaTeX itself just requires the dollar signs, but in order to get math on the web, we add the braces. One easy way to place $ $ around some text is to highlight the text and then select the sqrt{n} button above. Double dollar signs center the mathematics on its own line. \(A\) $A$ (most mathematics symbols are italicized.) \(A=B\) $A=B$ \(\|P_1 P_2\|\) $\|P_1 P_2\|$ \(\frac{x}{y}\) $\frac{x}{y}$ \(A_1\) $A_1$ \(A_{n+1}\) $A_{n+1}$ (You have to put braces around subscripts and superscripts if they are longer than one character.) \(x_{1}^{3}\) $x_{1}^{3}$ \(x_{n+1}^{m-2}\) $x_{n+1}^{m-2}$ \(x<y\) $x<y$ \(x\leq y\) $x\leq y$ \(x\geq y\) $x\geq y$ \(x\neq y\) $x\neq y$ \(\{x\|P(x)\}\) $\{x\|P(x)\}$ (to get a bracket to show, you must precede it with a slash) \(\emptyset\) $\emptyset$ \(\sum\) $\sum$ \(\sum_{n=1}^{\infty}\) $\sum_{n=1}^{\infty}$ \(\displaystyle\sum_{n=1}^{\infty}\) $\displaystyle\sum_{n=1}^{\infty}$ $$\sum_{n=1}^{\infty}$$ `$$\sum_{n=1}^{\infty}$$` (notice how there are two dollar signs) \({\mathbb{R}}\) ${\mathbb{R}}$ The real numbers \({\mathbb{Z}}\) ${\mathbb{Z}}$ The integers \(\alpha\) $\alpha$ \(\beta\) $\beta$ \(\gamma\) $\gamma$ \(\lambda\) $\lambda$ \(\Lambda\) $\Lambda$ \(\delta\) $\delta$ \(\Delta\) $\Delta$ \(f\circ g: A\to B\) $f\circ g: A\to B$ \(\hat{x}\) $\hat{x}$ \(\bar{x}\) $\bar{x}$ \(A\times B\) $A\times B$ \(A\cdot B\) $A\cdot B$ \(x_1+x_2+\cdots+x_{n-1}+x_n\) $x_1+x_2+\cdots+x_{n-1}+x_n$ \(A\approx B\) $A\approx B$ \(\sin x \cos x \tan x \cot x \sec x \csc x\) $\sin x \cos x \tan x\ cot x \sec x \csc x$ \((\frac{1}{n})\) $(\frac{1}{n})$ \(\left(\frac{1}{n}\right)\) $\left(\frac{1}{n}\right)$ (use \left( \right) when you need bigger parenthesis) \(\|x\|=\left\{ \begin{array}{c} x \text{ if } x\geq 0\\ -x \text{ if } x<0 \end{array} \right.\) $\|x\|=\left\{ \begin{array}{c} x \text{ if } x\geq 0\\ -x \text{ if } x<0 \end{array}\right.$ \( \left<1,2,4 \right> \) `$ left<1,2,4 \right> $}` \(1\vec{i} + 1\vec{j} +1 \vec{k} \) $1\vec{i} + 1\vec{j} +1 \vec{k} $ \(f^\prime\) $f^\prime$ \( \int f(x) dx \) $ \int f(x) dx $ \( \int_a^b f(x) dx \) $ \int_a^b f(x) dx $ \( \displaystyle\int_a^b f(x) dx \) $ \displaystyle\int_a^b f(x) dx $ $$ \int_a^b f(x) dx $$ `$$ \int_a^b f(x) dx $$` \( \frac{d}{dx}[f(x)] \) $ \frac{d}{dx}[f(x)] $ $$ \frac{d}{dx}[f(x)] $$ `$$ \frac{d}{dx}[f(x)] $$` Lining up calculations accross multiple lines. The following examples show you how to line up an equation. Find the derivative of \(f(x)=x^2\) using the definition of the derivative. We use the definition of the derivative as follows. $$ \begin{array}{rl} \lim_{h\to 0} \frac{f(x+h)-f(x)}{h} &= \lim_{h\to 0} \frac{(x+h)^2-x^2}{h}\\ &= \lim_{h\to 0} \frac{x^2+2xh+h^2-x^2}{h}\\ &= \lim_{h\to 0} \frac{2xh+h^2}{h}\\ &= \lim_{h\to 0} 2x+h\\ &= 2x \end{array} $$ This shows that the derivative is \(f^\prime(x) = 2x\). We use the definition of the derivative as follows. $$ \begin{array}{rl}\displaystyle \lim_{h\to 0} \frac{f(x+h)-f(x)}{h} &= \lim_{h\to 0} \frac{(x+h)^2-x^2}{h}\\ &= \lim_{h\to 0} \frac{x^2+2xh+h^2-x^2}{h}\\ &= \lim_{h\to 0} \frac{2xh+h^2}{h}\\ &= \lim_{h\to 0} 2x+h\\ &= 2x \end{array} $$ This shows that the derivative is \(f^\prime(x) = 2x\). The & symbols line up the equation, and \\ means create a new line. You have to start the array environment, and end it. The command {rl} says that there are two columns, with the first column right aligned and the second left aligned. Feel free to copy the syntax and take it to your homepage ( and play with the options there. Please don't edit this page. Edit Page History Source Attach File Backlinks List Group Page last modified on November 27, 2019, at 01:48 PM
skin config pmwiki-2.2.142

La Te X Commands

Why $\LaTeX$?

Syntax Examples

Lining up calculations accross multiple lines.