Projection $\pi$
When normalizing a database that has a redundancy problem, it is often a good idea to split up a single table into multiple tables. In order to this we use the technique called projection. When projecting a database, we decompose a relation into multiple relations. If you let $\alpha_1, \alpha_2,…, \alpha_n \subseteq {A_1, …, A_3}$ be n subsets of a relation $R$’s attributes, we can define a new $R_i$ for any $\alpha_i$ such that:
$$R_i = \pi_{\alpha_i}R$$
In this case, $\alpha_1, \alpha_2,…, \alpha_n$ is a decomposition of the relation R. For this decomposition to be good, it needs to be lossless, meaning that the joining of all those projections ($R_1 \bowtie R_2 … \bowtie R_n$) must be equal to R.