Definition (Intersection of sets)

Given two subsets $U_1,U_2\subset V$, we define the intersection of $U_1$ and $U_2$ to be $$\begin{align*}U_1 \cap U_2 :=\{u\in V|u\in U_1 \text{ and } u\in U_2\} \end{align*}$$ More generally for $U_1,...\subset V$ are all subsets of $V$ then the intersection of $U_i$ is $$\begin{align*}\bigcap_{i=1}^{\infty}U_i:=\{u\in U_i, i=1,2,3,....\}\end{align*}$$

Definition (Union of sets)

Given two subsets $U_1, U_2 \subset V$, we define the union of $U_1$ and $U_2$ to be $$\begin{align*} U_1\cup U_2:=\{u\in V|u\in U_1 \text{ or } u\in U_2\} \end{align*}$$ More generally, for $U_1,....\subset V$ the union of all $U_i$ is $$\begin{align*} \bigcup_{i=1}^\infty U_i:=\{u\in V|u\in U_i, \text{ for some } i=1,2,3,\} \end{align*}$$

Definition (Sum of Subsets)

If $U_1,U_2\subset V$ where V is a vector space, then we define the sum $U_1+U_2\subset V$ to be $$\begin{align*} U_1+U_2:=\{u_1+u_2|u_1\in U_1 \text{ and } u_2\in U_2\} \end{align*}$$ More generally, for $U_1,....U_m\subset V$ then $$\begin{align*} U_1+...+U_m:\{u_1+...+u_m|u_i\in U_i, i=1,...,m\} \end{align*}$$