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 .

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.

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

2.

Find an equation of the line that satisfies the given conditions.
3. Through ; slope 1

4. Through and

5. Slope 3; y-intercept -2

6. Through ; parallel to the -axis

7. Through ; parallel to the line

8. Through ; perpendicular to the line

Find the slope and y-intercept of the given line
9.

10.

Solution

Therefore is in the interval .

Solution

Multiply both sides by 5, we have
Now

So is in .

Solution

Divide both sides by 2, we have

So is in .

Solution

Add 4 to each term, we have

So is in .

Solution

Multiply by 5 we have . Subtract 4 to get . The last inequality can be simplify into
.Divide by -3 we have . Therefore the solution is .

Solve the nonlinear inequality. Express the solution using interval notation.
6.

Solution

Solution

Rewrite the inequality as

Hence is in the interval

Solution

Solution

10.

Solution

Solution

Solution

Write the following equations in the form (for example the equation can be written as )

4.

Solution

Solution

Find all real solutions to the equation.
6.

Solution

Hence or which implies that or .

Solution

Hence or which implies that or .

Solution

Hence the equation has no solution.
Remark: one can use determinant analysis to arrive at the some conclusion.

Solution

Note that the discriminant of this equation is
. Hence the equation has no solution.

10. A rectangular bedroom is 7ft longer than it is wide. Its area is . What is the width of the room?

Solution

Let be the width of the bedroom. Then the length of the other side is . Therefore the area of the room is

Now

Hence or . But since the width must be a positive number then the width is 12 ft.

Find the - and - intercepts of the graph of the equation.
3.
4.

Make a table of values and sketch the graph of the equation.
5.
6.
7.

Find an equation of the circle that satisfies the given conditions.
8. Center ; radius 8
9. Endpoints of a diameter are and

10. The equation is an equation of a circle. Find the center and radius of the circle.

Solve the nonlinear inequality. Express the solution using interval notation.
6.
7.
8.
9.
10. http://watchmath.com/vlog/?p=1019 Find all real solutions of the equation.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.