Equivalence relation and class, Proof. 
The relation $P$ on $ℝ$ is defined by $xPy$ iff $x^2=y^2.$
(a) Prove that the the relation $P$ is an equivalence relation.
(b) Describe the equivalence class of $3$ and $0$.

In order to solve this proof one must first understand what an equivalence relation is which means that a relation must be reflexive, symmetric and transitive. In order to do part (a) does one have to define what reflexive symmetric and transitive is in order to receive the right result? Does one prove this through example?
For part (b), I know that an equivalence class, the set of equivalence classes for the set relation $\Rightarrow $ m is denoted $Z_m$.
which is (by substituting) we get, $3P-3$ and $-3P3$ so $[3] \Rightarrow $ $\{-3,3\}$ and
$[0] = \{0\}$.

Yes, for part (a) to show that $P$ is an equivalence relation on $\mathbb{R}$ you need to check that it is reflexive, symmetric and transitive. Here is how you get started. 
Reflexivity: Take $a\in \mathbb{R}$ and clearly $aPa$ holds, since $a^2=a^2$ is true for all real numbers $a$. Hence $P$ is reflexive.
Symmetry: Take $a,b\in \mathbb{R}$. Now if you know that $aPb$, does that mean $bPa$? If you can show that, then $P$ is symmetric. 
Transitivity: Take $a,b,c \in \mathbb{R}$ and assume that $aPb$ and $bPc$ are both true. Can you show that also $aPc$ is true? 
Your answer for part (b) is correct. You are asked to describe the equivalence class of the values $3$ and $0$ in $\mathbb{R}$. Basically, which values $a\in \mathbb{R}$ satisfy $aP3$, i.e. $a^2=3^2$? The resulting set $\{-3,3\}$ is your equivalence class for $3$. Similarly which values $b\in \mathbb{R}$ satisfy $bP0$, i.e. $0^2=b^2$? The resulting set $\{0\}$ is your equivalence class for $0$.
