1. Discrete Mathematics
Logic

2. Propositions
A proposition is a statement or sentence that can be determined to be either true or false (but no both).
Examples:
The only positive integers that divide 7 are 1 and 7 itself.
Buy two tickets for Friday concert.
Earth is the only planet in the universe that contains life.

3. Example Use variable to represent propositions P: 1+1=3
P: It is raining outside
P: Today is Tuesday

4. Connectives
If p and q are propositions, new compound propositions can be formed by using connectives
Most common connectives:
Conjunction (and) ^
Disjunction (or) 
Negation (not) ~
Exclusive-OR v
Condition (if … then) 
Bi-Condition 

5. Example P: It is raining Q: It is cold
Form a new compound statement by combining these two statements
P  Q : It is raining and it is cold
P  Q : It is raining or it is cold

6. Truth table of conjunction
The truth values of compound propositions can be described by truth tables.
Truth table of conjunction
P Q is true only when both P and Q are true. P Q
P  Q T F

7. Example Let P = “A decade is 10 years”
Let Q = “A millennium is 100 years”
P  Q = “A decade is 10 years” and “A millennium is 100 years”
If P is true and Q is false then conjunction is false

8. Truth table of disjunction
The truth table of disjunction is
p  q is false only when both p and q are false
Example: p = "John is a programmer", q = "Mary is a lawyer"
p v q = "John is a programmer or Mary is a lawyer" P Q
P  Q T F

9. Negation Negation of P: in symbols ~P or ⌐P
~P is false when P is true, ~P is true when P is false
Example, P : "John is a programmer"
~P = "John is not a programmer" P ~P T F

10. E.g
P: Paris is the capital of England
~P: Paris is not capital of England

11. Exclusive disjunction
“Either P or Q” (but not both), in symbols P  Q
P  Q is true only when P is true and Q is false, or P is false and Q is true.
Example: p = "John is programmer, q = "Mary is a lawyer"
p v q = "Either John is a programmer or Mary is a lawyer" P Q
P v Q T F

12. More compound statements
Let p, q, r be simple statements
We can form other compound statements, such as
(pq)^r
p(q^r)
(~p)(~q)
(pq)^(~r)
and many others…

13. Example: truth table of (P Q)R

14. Conditional propositions
A conditional proposition is of the form
“If P then Q”
In symbols: P  Q
Example:
P = " A bottle contains acid"
Q = “A bottle has a label”
P  Q = “If a bottle contains acid then it has a label "

15. P  Q is true when both p and q are true
Truth table of P  Q P Q
P  Q T F
P  Q is true when both p and q are true
or when P is false

16. Example
If the mathematics department gets an additional $40,000 then it will hire one new faculty member.
Let P: The Mathematics Department gets an additional $40,000 and Q: The mathematics Department will hire one new faculty member.

17. Hypothesis and conclusion
In a conditional proposition P  Q,
P is called the hypothesis
Q is called the conclusion

18. Example For all real number x if x > 0 then x2 > 0
For example x=3 , 3 > 0 then 32 > 0 both are true.
x=-2 , -2 > 0 is false but -22 > 0

19. Logical equivalence Two propositions are said to be logically
equivalent if their truth tables are identical.
Example: ~P  Q is logically equivalent to P  Q P Q
~P  Q
P  Q T F

20. Converse The converse of p  q is q  p These two propositions
are not logically equivalent p q
p  q
q  p T F

21. They are logically equivalent.
Contrapositive
The contrapositive of the proposition p  q is ~q  ~p.
They are logically equivalent. p q
p  q
~q  ~p T F

22. p  q is logically equivalent to (p  q)^(q  p)
Bi-Conditional
The double implication “p if and only if q” is defined in symbols as p  q
p  q is logically equivalent to (p  q)^(q  p) p q
p  q
(p  q) ^ (q  p) T F

23. Tautology
A proposition is a tautology if its truth table contains only true values for every case
Example: p  p v q p q
p  p v q T F

24. Contradiction
A proposition is a tautology if its truth table contains only false values for every case
Example: p ^ ~p p
p ^ (~p) T F

25. De Morgan’s laws for logic
The following pairs of propositions are logically equivalent:
~ (p  q) and (~p)^(~q)
~ (p ^ q) and (~p)  (~q)

26. Quantifiers
A propositional function P(x) is a statement involving a variable x
For example:
P(x): 2x is an even integer
x is an element of a set D
For example, x is an element of the set of integers
D is called the domain of P(x)

27. For every and for some
Most statements in mathematics and computer science use terms such as for every and for some.
For example:
For every triangle T, the sum of the angles of T is 180 degrees.
For every integer n, n is less than p, for some prime number p.

28. Universal quantifier One can write P(x) for every x in a domain D
In symbols: x P(x)
 is called the universal quantifier

29. Inference Or Deduction
An argument is a sequence of propositions written as below :
p1, p2 , p3 , p4 , p5 ,,…, , pn, therefore q.
(if p1v p2v,.., pn then q) or (if p1 ^p2 ^,.., pn then q)
the propositions p1, .., pn, are called hypotheses (premises) and proposition q is called conclusion (consequent).
The above argument is valid if the consequent can be proven from the premises.
Validity can be shown using truth table or by using theorems / rules.

30. Rules of Inference Addition p , therefore p v q Simplification
(read as if we know p is true, therefore we know p or q is true)
Simplification
p ^ q , therefore p
Conjunction
p , q , therefore p ^ q

31. Rules of Inference (contd..)
Modus Ponens
p  q , p, therefore q
Modus Tollens
~ q, p  q , therefore ~p
Hypothetical Syllogism
p  q, q  r, therefore p  r
Disjunctive Syllogism
p v q , ~ p, therefore q
Resolution
p v q, ~ p v r , therefore q v r

32. Example
State which rule of inference is the basis of the following argument :
“ It is below freezing now. Therefore it is either below freezing or raining now.”
- addition rule
“It is below freezing and raining now. Therefore, it is below freezing now. “
- simplification rule
“Rules : If it snows today, then we will go skiing”
Fact : it snows today. Conclude: We will go skiing”
- modus ponens

33. Problem Show that the hypotheses :
“ It is not sunny this afternoon and it is colder than yesterday.”
Will conclude :
“We will be home by sunset “
Given the following fact :
“We will go swimming only if it is sunny. If we do not go swimming, then we will take a canoe trip. If we take a canoe trip we will be home by sunset”