Review of Basic Set Theory: 1.2

Sets are represented by capital letters like A, B, C.

Sets can be like {1, 2, 3, ...}, (0, 1), reals, R^k, etc.

    Operations

A \cup B - union of A and B, refers to a set such that x \in A \cup B
iff x\in A OR x\in B.

A \cap B - intersection, refers to a set such that x \in A \cup B iff
x\in A AND x\in B.

A \setminus B (written as \) - difference, x\in A \setminus B iff x\in A
and x\not\in B

B^c = \overbar{B} - complement, equivalent to \Omega \setminus B or
C \setminus B, where C or \Omega is the sample space of the problem.

Universe ex: all possible combinations of two coin flips.

Countable union:

 +\infty               n
   U     A_i    or     U
  i=1                 i=1

Represents union of a list of indexed sets.

Countable intersection is analagous.

UNcountable intersection/union is a union or intersection over an
uncountable set like the reals or (0, 1).

Symmetric difference: (A \union B) \setminus (A \cap B). (think of XOR)

    DeMorgan's Laws (and proofs)

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

Let x \in (A\cap B)^c \to x \not\in A \cap B.

If x\in A, x\not\in B \to x\in B^c. If x\in B, x\not\in A \to x\in A^c.
Therefore, x\in A^C \cup B^c.

(The proof the other way is similarly trivial)

(A \cup B)^C = A^C \cap B^c

These laws can be generalized to infinite unions (although not through
induction, which merely proves it for arbitrary large, finite sets).

Prove: Let B be the union of all A^c where A \in C. x\in B iff x \in
(infinite intersection of all A where A \in C)^c


[Proof omitted at present]

    Distributive laws

C_1 \cap (C_2 \cup C_3) = (C_1 \cap C_2) \cup (C_1 \cap C_3)
C_1 \cup (C_2 \cap C_3) = (C_1 \cup C_2) \cap (C_1 \cup C_3)

    Sample space and probability

Example sample space is n ordered Bernoulli trials [finite]
                     or (0,1) or (0,1)^2 or R      [continuous]

C = { w_1, w_2, ... } where w_i is an elementary event. ??MUST?? be
countable.

If A \subseteq C, A is an event (random event) in C.

    Probability set function

Assign to each w_j a number p(w_j) \in [0, 1].

\sum_{j=1}^\infty p(w_j) = 1.
p_j := p(w_j) also known as random weights

P(A) = \sum_{j : w_j \in A} p(w_j) [ the sum can be generalized to integral]

    Properties of P(A)

P : subsets of C --> [0, 1]

1) P(C) = 1
2) \forall A \subseteq C, 0 \leq P(A) \leq 1
3) If A_1, A_2 \subseteq C and A_1 \cap A_2 = \empty,
    P(A_1 \cup A_2) = P(A_1) + P(A_2)
4) \forall A_1, A_2 \subseteq C,
    P(A_1\cup A_2) = P(A_1) + P(A_2) - P(A_1 \cap A_2)
5) \forall A \subseteq C
    P(A) + P(A^c) = 1.
6) P(\empty) = 0
7) P(A) is nondecreasing: If A_1 \subseteq A_2 \subseteq C,
P(A_1) \leq P(A_2)

sigma-algebra backed by C is a set which contains C, and is closed under
countable union and complement w.r.t C. (these are not unique, e.g.
{C, \empty} and P(C) are both valid sigma-algebras).