# 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.

# 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 and 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 and are true and false otherwise.

# Exclusive Or:

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

# Conditional:

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

# Converse:

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

# Inverse:

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

# Contrapositive:

Let p and q be propositions. The contrapositive of the proposition → 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 ↔ is the proposition "p if and only if q". The conditional is true when p and 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``````

# 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

By George Lee

# ics141-propositional-logic

A presentation on logics.

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