Logic is used to reason about the truth or falsity of statements. Propositional logic deals with Boolean functions while predicate logic deals with quantified Boolean functions. Statements can be combined using logical connectives like AND, OR, IMPLIES. Their truth values are determined using truth tables. Logical statements can be translated between English and symbolic notation. Predicate logic involves functions whose values depend on variables that range over a domain. Quantifiers like "for all" and "for some" are used to make assertions about predicates over a domain.

1. LOGIC

2. 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)

3. 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

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

5. 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

6. 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.

7. Truth Tables - NOT
P 5P
T F
F T

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

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

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

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

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

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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

19. A Simple Tautology (continued)
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

20. A Simple Tautology (continued)
Remember your implication table!
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

21. A Simple Tautology (continued)
Remember your implication table!
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

22. 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

23. 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

24. 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.

25. 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.

26. 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)}

27. 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)

28. 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.

29. 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))