1  Set Theory

1.1 Sets

A set is a collection of objects, and these objects are called the elements of the set.

We can either describe a set by listing all the elements; for example, \(S = \{1,2,3,4,5,6\}\). For large or infinite sets, the set-builder notation is helpful. For example, the set of all positive real numbers may be written as: \[ S=\{x : x \text{ is a real number and } x > 0\} \]

1.2 Set Operations

1.2.1 Negation

The event not \(E\) is defined as \(E^c=\{x \in \Omega, x\not\in E\}\). This set is also called the complement of \(E\).

Example: You buy a stock for $100 and wish to evaluate the probability of selling it for a higher price \(x\) in one year. The event that no profit is made on the sale can be written as \(E^c = \{x : 0 \ge x \ge 100, x\in Q\} = \Omega \setminus E\) .

1.2.2 Union and Intersection

The union of \(A\) and \(B\) is defined as \(A \cup B = \{x : x\in A \text{ or } x \in B\}\). The intersection of \(A\) and \(B\) is defined as \(A\cap B = \{x : x \in A \text{ and } x \in B\}\).

Example: Consider the insurance company which has written 100 individual life insurance policies and is interested in the number of claims which will occur in the next year. The sample space is \(\Omega = \{0,1,2,...,100\}\). The company is interested in the following two events:

• \(A = \{0,1,2,...,8\}\): there are at most 8 claims
• \(B = \{5,6,...,11,12\}\): the number of claims is between 5 and 12, inclusive Then the events \(A \cup B\) and \(A\cap B\) are given by:
• \(A \cup B = \{0,1,2,3,...,10,11,12\}\)
• \(A \cap B = \{5,6,7,8\}\)

Countable union and intersection of a sequence of sets \(A_1, A_2, ...\) is defined as: \[ \begin{align}\bigcup_{n\ge 1} A_n &= \{x : \exists n \ge 1 \text{ s.t. } x \in A_n\} \\ \bigcap_{n\ge 1} A_n &= \{x : x \in A_n \ \forall n \ge 1\}\end{align} \]

1.3 Set Identities

1.3.1 Distributive Laws for Sets

Two distributive laws for set operations are the following: \[ A \cap (B \cup C) = (A \cap B)\cup(A \cap C) \\ A \cup (B \cap C) = (A \cup B)\cap(A \cup C) \]

1.3.2 De Morgan’s Law

Two other useful set identities are the following: \[ (A \cup B)^c = A^c \cap B^c \\ (A \cap B)^c = A^c \cup B^c \]