Notation

Let p, q, and r be logical statements below and let A, B, C, E, and F all be subsets of a set S.

Logic

Truth Tables

p

\sim \! p

T

F

F

T

p

q

p \land q

p \lor q

p \to q

p \leftrightarrow q

T

T

T

T

T

T

T

F

F

T

F

F

F

T

F

T

T

F

F

F

F

F

T

T

Equivalent Statements

definition Two statements p and q are said to be equivalent, denoted p \equiv q, if the statement p \leftrightarrow q is a tautology.

The following equivalent statements are important to know:

Arguments

Four valid arguments that are important to know:

Sets

definition A is said to be a subset of B, denoted A \subseteq B, if the logical statement x \in A \to x \in B is a tautology.

definition The intersection of A and B, denoted A \cap B, is equal to the set \{x \in S | x \in A \land x \in B\}.

definition The union of A and B, denoted A \cup B, is equal to the set \{x \in S | x \in A \lor x \in B\}.

definition The complement of A with respect to S, denoted A', is equal to the set \{x \in S | x \not \in A\}.

definition Two sets A and B are said to be disjoint if A \cap B = \emptyset.

Set Laws

The following equalities of sets are important to know:

Probability

Basic Probability

definition A probability measure, P, on a set S is a function P: 2^S \rightarrow [0,1] such that:

  1. P(\emptyset) = 0

  2. P(S) = 1

  3. If A and B are disjoint, then P(A \cup B) = P(A) + P(B).

Addition Rule The Addition Rule for Probability is

P(E \cup F) + P(E \cap F) = P(E) + P(F)
Complement Rule The Complement Rule for Probability is
P(E) + P(E') = 1
Partition Rule 1 The Partition Rule 1 is
P(E \cup F) = P(E) + P(E' \cap F)
Partition Rule 2 The Partition Rule 2 is
P(E) = P(E \cap F) + P(E \cap F')

Conditional Probability

definition The conditional probability of E given F provided P(F) \neq 0 is:

P(E|F) = \frac{P(E \cap F)}{P(F)}

Product Rule The Product Rule for Probability is

P(E \cap F) = P(E|F) \cdot P(F) = P(F|E) \cdot P(E)

definition Two events E and F are said to be independent if any of the following equivalent conditions are true:

  1. P(E|F) = P(E)

  2. P(F|E) = P(F)

  3. P(E \cap F) = P(E) \cdot P(F)

Counting

Addition Rule The Addition Rule for Counting is

n(E \cup F) + n(E \cap F) = n(E) + n(F)
Complement Rule The Complement Rule for Counting is
n(E) + n(E') = n(S)

NUWiki: Math115 (last edited 2011-08-07 18:23:09 by JasonRibeiro)