Learn Math The Easiest

Let and be left -modules. We say that is a universal of the square diagram

if whenever we have another diagram

there exist a unique homomorphism such that the following diagram commutes

We want to show that the universal is unique of  to isomorphism, i.e., if we have two universal and then they are isomorphic. Since is a universal we have an that makes the above diagram commutes. Similarly, considering as a universal, we have that make the diagram

commmutes. Combining the two diagrams above we have the following commutative diagram

On the other hand we have the following obvious diagram

By uniqueness, it follows that and by similar argument, one can show that . Therefore and are isomorphisms and therefore is isomorphic to .

MMM #38:

Prove or disprove: The product of any four consecutive integers is always one less than a perfect square.
Don’t assume the integers are all positive.

Any four consecutive integers can be written as . The product of these numbers is

Therefore the statement always true.

This is the updated procedure of how to install latex on your Blogger.

What’s New?

• The script located in a more stable location (in google server) rather than using watchmath.com server (which sometimes down for some unknown reason).
• New script, adapting the preference of either using \textstyle of \displaystyle as the default mode.
• More flexibility. Customization on default size, color, adding latex packages and defining new commands.

How To Install

Use the instruction here, but in step one copy and use instead the following script.

<script src="http://latexonblogger.googlegroups.com/web/mathtex3.js?hl=en&amp;gda=R9ZVlD4AAABbAsbN4i7hi2ZFNZpVWCi9LFztQtuc529xoR6FBh351sAOXLBpscdVgFPClVlUxUzjsKXVs-X7bdXZc5buSfmx" type="text/javascript"></script>
<script type="text/javascript">window.mathPreamble =
 '\\newcommand{\\RR}{\\mathbb{R}}\\usepackage[usenames]{color}\\color{} \\gammacorrection{1.3}\\png \\normal ';replaceMath( document.body );</script>
 
<center>
<a href="http://www.watchmath.com">
<img alt="" src="http://www.watchmath.com/images/formula.png" width="100"></a>
<br>
<a href="http://watchmath.com/vlog/?p=1244">
Install LaTeX?</a></center>

Usage

• Use $\LaTeX$ in order to get an inline math in \textstyle mode. For example if you type $\sum_{i=1}^nx_i$ (in blogger) you get .
• Use $$\LaTeX $$ to gen an inline math in \displaystyle  mode. For example if you type $$\sum_{i=1}^nx_i$$  (in blogger) you get .
• Use \[\LaTeX\] to get a math symbol written in a new line and aligned center. For example if you type \[\sum_{i=1}^nx_i\]  (in blogger) you get .

Customization

Modify

window.mathPreamble =
 '\\newcommand{\\RR}{\\mathbb{R}}\\usepackage[usenames]{color}\\color{} \\gammacorrection{1}\\png \\normal ';

in the script above. All extra command will be written inside the quotation marks.

• Color: To change the default color, replace \\color{} by \\color{colorname} where colorname is one of the entries in the following table
• Size: by default the latex output is in normal size. To change the default size, change \\normal by one of the following: \\tiny, \\scriptsize,\\large,\\Large,\\huge.
• Latex Package: by default the package \\usepackage[usenames]{color} is included. You can add more package by adding  \\usepackage{packagename}. To test whether a particular package is supported, type \usepackage{packagename} (use only one backslash this time) in an inline command. For example, fancybox package is supported by mathtex because if you type $\usepackage{fancybox}\shadowbox{This is fancy}$ you will get . Here is a list of package that I know work with mathtex: xypic, calrsfs, calligra, fancybox,epic,eepic(?). Let me know if you find another package that works with mathtex.
• Newcommand: we have included an example how to define a new command in the script above. We use \\newcommand{\\RR}{\\mathbb{R}} to define a command \RR. Typing $\RR$ (in blogger) will give you the same result as typing $\mathbb{R}$, i.e., you get .
• Darkness: \\gammacorrection{1.3} controls the darkness of the output. Do  experiments to get the best result by changing 1.3 into any number you like.

Demo?

Visit http://www.watchmath.blogspot.com