Union and Intersection

Q1: What is a union of sets?
A1: A union of sets is a a set whose elements are belong to at least one of the sets in the union.
Given: For 2 sets $A$ and $B$:
$A\cup B=\{ x: x\in A\ or\ x\in B\}$
Given: For countable collection of infinite sets $C_1, C_2, C_3, ...$:
$\bigcup\limits_{i=1}^{\infty}C_i=C_1 \cup C_2 \cup C_3 \cup ... =\{ x: x\in C_i\ for\ some\ i\}$

Q2: What is an intersection of sets?
A2: An intersection of sets is a set that contains only those elements that are shared by all sets that form the intersection.
Given: For 2 sets $A$ and $B$:
$A\cap B=\{ x: x\in A\ and\ x\in B\}$
Given: For countable collection of infinite sets $C_1, C_2, C_3, ...$:
$\bigcap\limits_{i=1}^{\infty}C_i=C_1 \cap C_2 \cap C_3 \cap ... =\{ x: x\in C_i\ for\ all\ i\}$

Q3: What is a disjoint collection of infinite sets $C_1, C_2, C_3, ...$?
A3: A disjoint collection of infinite sets is a collection of sets whose intersection is the empty set.
$\bigcap\limits_{i=1}^{\infty}C_i=\{\}$

Q4: What does it mean if a family of sets $P$ composed of cells $P_1, P_2, P_3, ...$ partition a set $A$?
A5: If $P$ partitions a set $A$, the following 3 conditions hold:

1. The partition does not contain the empty set
• $\{\}\not\in P$
2. The union of the cells is the set $A$. The partition covers $A$.
• $\bigcup\limits_{i=1}^{\infty}P_i=A$
3. No 2 cells share any elements. The partition is pairwise disjoint.
• $P_i\cap P_j=\{\}\ for\ all\ i\not =j$