New LaTeX Formatting

I have hopefully found a new (and better!) way to format math and equations in my blog using $\LaTeX$. I'm going to try and integrate this as seemlessly as possible, but I can't really imagine it happening without some format hiccups, so try and bear with me. To demonstrate my newfound power I'm going to try and write out one of my old examples.

Example: Homomorphism

Choose $n \in \mathbf{N}$ and let $\phi:\mathbf{Z}\to \mathbf{Z}$ be given by $\phi(x)=x\; \text{mod}\; n$. Now, suppose $x,y \in \mathbf{Z}$. We can write $x = h n + p$ and $y = k n + q$ where $p,q \in \mathbf{Z}_n$ and $h,k \in \mathbf{Z}$. Then

\[ \phi(x + y) = x + y\; \text{mod}\; n = (h n + p) + (k n + q)\; \text{mod}\; n = p + q\; \text{mod}\; n \]

and also

\[ \phi(x) + \phi(y) = (h n + p\; \text{mod}\; n) + (k n + q\; \text{mod}\; n) = p + q\; \text{mod}\; n \]

so $\phi(x + y) = \phi(x) + \phi(y)$. Thus $\phi$ is operation preserving and is a homomorphism.

The image of $\phi$ should be fairly obvious. Indeed, $\mathbf{Z}_n \subseteq \mathbf{Z}$ and if $x \in \mathbf{Z}_n$ the $\phi(x) = x$ giving that $\mathbf{Z}_n \subseteq \text{Im}(\phi)$ but by the definition of modular division, nothing in $\text{Im}(\phi)$ can be outside of $\mathbf{Z}_n$ and so $\text{Im}(\phi) = \mathbf{Z}_n$. The kernel of $\phi$, however, is a little more interesting. Suppose that $x \in \text{Ker}(\phi)$. We can write $x = h n + p$ as before but then $\phi(x) = p = 0$ so $x = h n$ giving that the elements in the kernel of $\phi$ are the multiples of $n$, or $\text{Ker}(\phi) = n \mathbf{Z} = {k n : k \in \mathbf{Z}}$.

If you're wondering what's going on, I added some javascript (that I found thanks to WatchMath) that renders standard LaTeX into the images you see above. If you'd like to learn how to do it (its very easy), you can read about it in this journal entry on my personal blog.