We clearly see that each cell, $R_{i}$ is disjoint $$R_{i}\bigcap R_{j}=\emptyset\text{ if }i\not=j$$also we see that their union form a partition over the real line $$\bigcup_{i=1}^{N}R_{i}=\mathbb{R}$$

We also see that the scalar quantizer $Q$ is completely described by its codebook, $\mathcal{C}$, and its quantizer cells $\{ R_{1},\dots,R_{N} \}$ since $$Q(x)=y_{i}\iff x\in R_{i}$$