Only the ones I think is significant are recorded here

1.1.3 Properties of set operations

Significant equations:

• Distributive Laws: $(E \cup F) \cap G = (E \cap G) \cup (F \cap G)$, $(E \cap F) \cup G = (E \cup G) \cap (F \cup G)$

DeMorgan’s Law:

Convert $\cup$ and $\cap$, and put $^c$ into the brackets.

$$ (\bigcup \limits^n_{i=1}E_i)^c = \bigcap \limits^n_{i=1}E^c_i $$

$$ (\bigcap \limits^n_{i=1}E_i)^c = \bigcup \limits^n_{i=1}E^c_i $$

1.4 Some useful results on probability

Theorem:

1. $\mathbb P(A^c) = 1 - \mathbb P(A)$

2. Rule of total probability:

   If $A, B \subseteq \Omega$ are events in the sample space $\Omega$, then

   $\mathbb P(A) = \mathbb P(A\cap B) + \mathbb P(A \cap B^c)$

   $A = A \cap \Omega = A \cap(B\cup B^c) = (A \cap B) \cup (A \cap B^c)$

3. Generalization of the rule of total probability:

   More generally, if a finite collection of pairwise mutually exclusive events $\{C_1, C_2, \cdots, C_n\}$ satisfy

   $$ C_1 \cup C_2 \cup \cdots \cup C_n = \Omega $$

   then, for any event $A \subseteq \Omega$,

   $$ \mathbb P(A) = \mathbb P (A \cap C_1) + \mathbb P (A \cap C_2) + \cdots + P (A \cap C_n) $$

4. Monotonicity:

   If $A,B \subseteq \Omega$ are events in the sample space $\Omega$ and $A \subseteq B$, then

   $$ \mathbb P(A) \le \mathbb P(B) $$

5. Rule of addition:

   If $A, B \subseteq \Omega$ are events in the sample space $\Omega$, then

   $$ \mathbb P(A\cup B) = \mathbb P(A) + \mathbb P(B) - \mathbb P(A\cap B) $$