cartesian product

intersection and cartesian product commute

Given sets $A,B,C,D$, then $\left(A\times B\right)\cap \left(C\times D\right)=(A\cap C)\times (B\cap D)$

cartesian product distributes over intersection

Given sets $A,B,C$, then $(A\cap B)\times C=\left(A\times C\right)\cap \left(B\times C\right)$

Set Power

Suppose that $A$ is a set, and that $n\in {\mathbb{N}}_{1}$, then we define ${A}^{n}$ to be the set of all n-tuples of $A$, that is : $$A}^{n}:=\{({a}_{1},{a}_{2},...,{a}_{n}):\forall i\in [n],{a}_{i}\in A\$$