What's a proposition?

Proposition:

a statement or assertion that expresses a judgment or opinion. (Google.com)

Proposition:

a declarative sentence (that is, a sentence that declares a fact) that is either true or false, but not both.

Some propositions

• George is a lecturer
• Honolulu is the capital of Hawaii
• The temperature outside is 90 degrees

What isn't a proposition?

• Watch this screencast
• How much time is left in this screencast?
• x > 5
• This statement is false

Propositional Variables

• We typically use the letters p, q, r, s ...
• Each value can either be true or false (T/F)
• We'll see what this has to do with computers later

Compound Propositions

• Each of the letters are "atoms"
• Make more "interesting" propositions with atoms and logical operators.

Logical Operators

Negation

Negation:

If p is a proposition, then ¬p is the opposite of the truth value of p.

Disjunction:

Let p and q be propositions. The disjunction p v q is equivalent to logical OR. The disjunction is false if both p and q are false and true otherwise.

Conjunction:

Let p and q be propositions. The conjunction p Λ q is equivalent to logical AND. The conjunction is true if both p and q are true and false otherwise.

Exclusive Or:

Let p and q be propositions. The Exclusive Or of p and q is p ⊕ q and is equivalent to logical XOR. Exclusive or is true if either p or q are true but not both.

Conditionals

Conditional:

Let p and q be propositions. The conditional statement p → q is the proposition "if p then q". The conditional is false when p is true and q is false and true otherwise.

Converse:

Let p and q be propositions. The converse of the proposition p → q is q → p.

Inverse:

Let p and q be propositions. The inverse of the proposition p → q is ¬p → ¬q.

Contrapositive:

Let p and q be propositions. The contrapositive of the proposition p → q is ¬q → ¬p.

Equivalence:

A conditional statement is equivalent to its contrapositive. The converse of a conditional statement is equivalent to the inverse.

Biconditional:

Let p and q be propositions. The biconditional statement p ↔ q is the proposition "p if and only if q". The conditional is true when p and q have the same truth value and false otherwise.

Truth Tables

// One Truth Variable

p
--
T
F

// Two truth variables

p | q
-----
T | T
T | F
F | T
F | F

// Three variables

p | q | r
---------
T | T | T
T | T | F
T | F | T
T | F | F
F | T | T
F | T | F
F | F | T
F | F | F

The number of rows is 2^n

Logic and Bit Operations

Computers use binary

• 1's and 0's (T and F)
• A chain of 1's and 0's are a bitstring
• Bitstrings are typically split into groups of 4 (Hexadecimal)
• Read 1.2 to see some applications