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Today, we will work in groups to talk about different relations. For each of the following relations, you should answer two questions:
\begin{itemize}
\item Prove that it is an equivalence relation. 
\item Describe it's equivalence classes
\end{itemize}

Some are easier than others, and they are presented in what I feel is increasing difficulty.

\section{Complex Magnitude}

Recall the complex numbers are numbers of the form $a + b i$ where $i = \sqrt{-1}$. The \df{magnitude} of a complex number $a+bi$ is
\[
	| a + bi | = \sqrt{ a^2 + b^2 } 
\]

Define the relation $\circ$ on $\mathbb{C}$ by : If $z_1,z_2\in\mathbb{C}$ then $z_1 \circ z_2$ if and only if $|z_1| = |z_2|$

\section{Periodic Sine}

Recall that $\sin(x)$ is a function on the real numbers that takes in a real numbers, reads it as an angle, and then returns the $y$-coordinate of the point on the unit circle corresponding to that angle. This is a periodic function that has period $2\pi$. But besides its periodicity, it is still not 1-1, as $\sin(0) = \sin(\pi)$. 

So, we make a realtion $\iso$ on real numbers and say $x\iso y$ if and only if $\sin(x) = \sin(y)$. 

\section{Differing by a Rational}

The real numbers $\R$ contain both rationals and irrationals. But, the rationals have nice closure properties (the sum and difference of two rationals is rational) where as the irrationals do not. 

So, define a relation on $\R$ by relating two numbers if they differ by a rational; that is define $\cong$ on $\R$ by $x\cong y$ if and only if $x - y \in \Q$. 

\section{Odd Factors}

Consider the positive integers (that is: natural numbers, without $0$). We can write every positive integer $n$  uniquely as some odd number times a power of $2$. This odd number is the largest odd number that divides $n$. For example, for $18$, the largest odd number that divides it is $9$. 

So define the relation $\equiv$ on $\Z^+$ by $n \equiv m$ if and only if $n$ and $m$ have the same largest odd number factor (for example: $9\equiv 18$). 

\section{Powers of Two}

Similar as above, we have that every positive integer $n$ can be unique wirtten as some odd numbers times a power of $2$. 


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