Logic

Statements
• Logic is the tool for reasoning about
the truth or falsity of statements.
– Propositional logic is the study of
Boolean functions
– Predicate logic is the study of
quantified Boolean functions (first
order predicate logic)

Arithmetic vs. Logic
Arithmetic Logic
0 false
1 true
Boolean variable statement variable
form of function statement form
value of function truth value of statement
equality of function equivalence of statements

Notation
Word Symbol
and v
or w
implies 6
equivalent ]
not ~
not 5
parentheses ( ) used for grouping terms

Notation Examples
English Symbolic
A and B A v B
A or B A w B
A implies B A 6 B
A is equivalent to B A ] B
not A ~A
not A 5A

Statement Forms
• (p v q) and (q v p) are different as statement
forms. They look different.
• (p v q) and (q v p) are logically equivalent. They
have the same truth table.
• A statement form that represents the constant 1
function is called a tautology. It is true for all
truth values of the statement variables.
• A statement form that represents the constant 0
function is called a contradiction. It is false for
all truth values of the statement variables.

Truth Tables - NOT
P 5P
T F
F T

Truth Tables - AND
P Q PvQ
T T T
T F F
F T F
F F F

Truth Tables - OR
P Q PwQ
T T T
T F T
F T T
F F F

Truth Tables - EQUIVALENT
P Q P]Q
T T T
T F F
F T F
F F T

Truth Tables - IMPLICATION
P Q P6Q
T T T
T F F
F T T
F F T

Truth Tables - Example
P true means rain
P false means no rain
Q true means clouds
Q false means no clouds

Truth Tables - Example
P Q P6Q P6Q
rain clouds rainclouds T
rain no clouds rainno clouds F
no rain clouds no rainclouds T
no rain no clouds no rainno clouds T

Algebraic rules for statement forms
• Associative rules:
(p v q) v r ] p v (q v r)
(p w q) w r ] p w (q w r)
• Distributive rules:
p v (q w r) ] (p v q) w (p v r)
p w (q v r) ] (p w q) v (p w r)
• Idempotent rules:
p v p ] p
p w p ] p

Rules (continued)
• Double Negation:
55p ] p
• DeMorgan’s Rules:
5(p v q) ] 5p w 5q
5(p w q) ] 5p v 5q
• Commutative Rules:
p v q ] q v p
p w q ] q w p

Rules (continued)
• Absorption Rules:
p w (p v q) ] p
p v (p w q) ] p
• Bound Rules:
p v 0 ] 0
p v 1 ] p
p w 0 ] p
p w 1 ] 1
• Negation Rules:
p v 5p ] 0
p w 5p ] 1

A Simple Tautology
P  Q is the same as 5Q 5P
This is the same as asking: PQ ] 5Q  5P
How can we prove this true?
By creating a truth table!
P Q
T T
T F
F T
F F

A Simple Tautology (continued)
Add appropriate columns
P Q 5P 5Q
T T F F
T F F T
F T T F
F F T T

Remember your implication table!
P Q 5P 5Q PQ
T T F F T
T F F T F
F T T F T
F F T T T

P Q 5P 5Q PQ 5Q5P
T T F F T T
T F F T F F
F T T F T T
F F T T T T

P Q 5P 5Q PQ 5Q5P PQ ] 5Q5P
T T F F T T T
T F F T F F T
F T T F T T T
F F T T T T T
Since the last column is all true, then the original
statement:
PQ ] 5Q5P is a tautology
Note: 5Q5P is the contrapositive of PQ

Translation of English
If P then Q: PQ
P only if Q: 5Q5P or
PQ
P if and only if Q: P ] Q
also written as P iff Q

Translation of English
P is sufficient for Q: PQ
P is necessary for Q: 5P5Q or
QP
P is necessary and sufficient for Q:
P ] Q
P unless Q: 5QP or
5PQ

Predicate Logic
• Consider the statement: x2
> 1
• Is it true or false?
• Depends upon the value of x!
• What values can x take on (what is the
domain of x)?
• Express this as a function: S(x) = x2
> 1
• Suppose the domain is the set of reals.
• The codomain (range) of S is a set of
statements that are either true or false.

Example
• S(0.9) = 0.92
> 1 is a false statement!
• S(3.2) = 3.22
> 1 is a true statement!
• The function S is an example of a
predicate.
• A predicate is any function whose
codomain is a set of statements that are
either true or false.

Note
• The codomain is a set of statements
• The codomain is not the truth value of the
statements
• The domain of predicate is arbitrary
• Different predicates can have different domains
• The truth set of a predicate S with domain D is
the set of the x ε D for which S(x) is true:
{x ε D | S(x) is true}
• Or simply: {x | S(x)}

Quantifiers
• The phrase “for all” is called a universal
quantifier and is symbolically written as œ
• The phrase “for some” is called an existential
quantifier and is written as ›
Notations for set of numbers:
Reals Integers
Rationals Primes
Naturals (nonnegative integers)

Goldbach’s conjecture
• Every even number greater than or equal
to 4 can be written as the sum of two
primes
• Express it as a quantified predicate
• It is unknown whether or not Goldbach’s
conjecture is true. You are only asked to
make the assertion
• Another example: Every sufficiently large
odd number is the sum of three primes.

Negating Quantifiers
• Let D be a set and let P(x) be a predicate
that is defined for x ε D, then
5(œ(x ε D), P(x)) ] (›(x ε D), 5P(x))
and
5(›(x ε D), P(x)) ] (œ(x ε D), 5P(x))

Logic

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