From Aluffi's Algebra: Chapter $0$
Describe generators and relations for all dihedral groups $D_{2n}$. (Hint: Let $x$ be the reflection about a line through the center of a regular n-gon and a vertex, and let $y$ be the counterclockwise rotation by $\frac{2\pi}{n}$. The group $D_{2n}$ will be generated by $x$ and $y$, subject to three relations. To see that these relations really determine $D_{2n}$, use them to show that any product $x^{i_1}y^{i_2}x^{i_3}y^{i_4} \cdots$ equals $x^iy^j$ for some $i$, $j$ with $0 \leq i \leq 1$, $0 \leq j < n$)
The generators form the set $\{x, y\}$ where $x$ is a reflection and $y$ is the rotation by $\frac{2\pi}{n}$. The $3$ relations are: $x^2 = e$, $y^n=e$ and $(xy)^2=e$.
From this information I can show that $x^{i_1}y^{i_2}x^{i_3}y^{i_4} \cdots=x^iy^j$ for $0 \leq i \leq 1$, $0 \leq j < n$. However, I'm not sure why exactly this shows that the relations determine $D_{2n}$. In addition, how would I show that every element in the group can be written as a product of $x$ and $y$?