1. CS203 Discrete Mathematical Structures
Logic
Proposition logic

2. What is logic?
The branch of philosophy concerned with analysing the patterns of reasoning by which a conclusion is drawn from a set of premises, without reference to meaning or context
(Collins English Dictionary)

3. Digital Logic

4. Propositional logic
Simple types of statements, called propositions, are treated as atomic building blocks for more complex statements
1) Alexandria is a port or a holiday resort.
2) Alexandria is not a port.
Therefore, Alexandria is a holiday resort

5. Basic connectives and truth tables
statements (propositions): declarative sentences that are
either true or false--but not both.
Eg. Ahmed Hassan wrote Gone with the Wind.
2+3=5.
not statements:
What a beautiful morning!
Get up and do your exercises.

6. Statement (Proposition)
A Statement is a sentence that is either True or False
2 + 2 = 4
True
Examples:
3 x 3 = 8
False
is a prime
Today is Tuesday.
Non-examples:
x+y>0
x2+y2=z2
They are true for some values of x and y
but are false for some other values of x and y.

7. Propositional connectives
Propositional logic has four connectives
Name
Read as
Symbol
negation
‘not’ 
conjunction
‘and’ 
disjunction
‘or’ 
implication
‘if…then…’ 

8. Logical Operators About a dozen logical operators
Similar to algebraic operators + * - /
In the following examples,
p = “Today is Friday”
q = “Today is my birthday”

9. Logical operators: Not
A “not” operation switches (negates) the truth value
Symbol:  or ~
p = “Today is not Friday” p p T F

10. Logical operators: And
An “and” operation is true if both operands are true
Symbol: 
It’s like the ‘A’ in And
pq = “Today is Friday and today is my birthday” p q
pq T F

11. Logical operators: Or
An “or” operation is true if either operands are true
Symbol: 
pq = “Today is Friday or today is my birthday (or possibly both)” p q
pq T F

12. Logical operators: Conditional
A conditional means “if p then q”
Symbol: 
pq = “If today is Friday, then today is my birthday”
p→q=¬pq p q
pq T F

13. Logical operators: Bi-conditional
A bi-conditional means “p if and only if q”
Symbol: 
Alternatively, it means “(if p then q) and (if q then p)”
Note that a bi-conditional has the opposite truth values of the exclusive or p q
pq T F

14. Boolean operators summary
not
and or
xor
conditional
Bi-conditional p q p q
pq
pq
pq
pq
pq T F
Learn what they mean, don’t just memorize the table!

15. Precedence of operators
Just as in algebra, operators have precedence
4+3*2 = 4+(3*2), not (4+3)*2
Precedence order (from highest to lowest):
¬   → ↔
The first three are the most important
This means that p  q  ¬r → s ↔ t yields: (p  (q  (¬r)) → s) ↔ (t)
Not is always performed before any other operation

16. Example s: Aya goes out for a walk. t: The moon is out.
u: It is snowing.
: If the moon is out and it is not snowing, then
Aya goes out for a walk.
If it is snowing and the moon is not out, then Aya
will not go out for a walk.

17. Translating English Sentences
p = “It is below freezing”
q = “It is snowing”
It is below freezing and it is snowing
It is below freezing but not snowing
It is not below freezing and it is not snowing
It is either snowing or below freezing (or both)
If it is below freezing, it is also snowing
It is either below freezing or it is snowing, but it is not snowing if it is below freezing
That it is below freezing is necessary and
sufficient for it to be snowing
pq
p¬q
¬p¬q
pq
p→q
((pq)¬(pq))(p→¬q)
p↔q

18. Logical Equivalence p q 1 1 1 1 1

19. Tables of Logical Equivalences
Identity laws
Like adding 0
Domination laws
Like multiplying by 0
Idempotent laws
Delete redundancies
Double negation
“I don’t like you, not”
Commutativity
Like “x+y = y+x”
Associativity
Like “(x+y)+z = y+(x+z)”
Distributivity
Like “(x+y)z = xz+yz”
De Morgan L3

20. Tables of Logical Equivalences
Excluded middle
Negating creates opposite
Definition of implication in terms of Not and Or

21. Fundamentals of Logic
A compound statement is called a tautology(T0) if it is
true for all truth value assignments for its component statements.
If a compound statement is false for all such assignments, then it is called a contradiction(F0).
: tautology
: contradiction

22. Propositional Logic - 2 more defn…
A tautology is a proposition that’s always TRUE.
A contradiction is a proposition that’s always FALSE. p p
p  p
p  p T F T F

23. Tautology example Demonstrate that [¬p (p q )]q
is a tautology in two ways:
Using a truth table – show that [¬p (p q )]q is always true
Using a proof (will get to this later).

24. Tautology by truth tablep q ¬p
p q
¬p (p q )
[¬p (p q )]q T F

25. Tautology by truth tablep q ¬p
p q
¬p (p q )
[¬p (p q )]q T F

26. Tautology by truth tablep q ¬p
p q
¬p (p q )
[¬p (p q )]q T F

27. Tautology by truth tablep q ¬p
p q
¬p (p q )
[¬p (p q )]q T F

28. Tautology by truth tablep q ¬p
p q
¬p (p q )
[¬p (p q )]q T F

29. Tautologies, contradictions and programming
Tautologies and contradictions in your code usually correspond to poor programming design. EG:
while(x <= 3 || x > 3)
x++;
if(x > y)
if(x == y)
return “never got here”;

30. dual Let s be a statement. If s contains no logical connectives
other than and , then the dual of s, denoted sd, is the
statement obtained from s by replacing each occurrence of
and by and , respectively, and each occurrence of T
and F by F and T, respectively.
Eg.
The dual of is

31. The Principle of Duality
Theorem () Let s and t be statements.
If , then
First Substitution Rule (replace each p by another statement q)
Ex.
is a tautology. Replace
each occurrence of p by
is also a tautology.

32. Converse
The converse of a logical implication is the reversal of the implication. I.e. the converse of p q is q p.
EG: The converse of “If Donald is a duck then Donald is a bird.”
is “If Donald is a bird then Donald is a duck.”
p q and q p are not logically equivalent.

33. Converse p q
p q
q p
(p q)  (q p)

34. Converse p q
p q
q p
(p q)  (q p) T F

35. Logical Non-Equivalence of Conditional and Conversep q
p q
q p
(p q)  (q p) T F

36. Logical Non-Equivalence of Conditional and Conversep q
p q
q p
(p q)  (q p) T F

37. Logical Non-Equivalence of Conditional and Conversep q
p q
q p
(p q)  (q p) T F

38. Fundamentals of Logic Compare the efficiency of two program segments.
z:=4;
for i:=1 to 10 do
begin
x:=z-1;
y:=z+3*i;
if ((x>0) and (y>0)) then
writeln(‘The value of the sum x+y is’, x+y)
end .
if x>0 then
if y>0 then …
Number of comparisons?
20 vs. 10+3=13
logically equivalent

39. Derivational Proof Techniques
When compound propositions involve more and more atomic components, the size of the truth table for the compound propositions increases
Q1: How many rows are required to construct the truth-table of: ( (q(pr ))  ((sr)t) )  (qr )
Q2: How many rows are required to construct the truth-table of a proposition involving n atomic components?

40. Derivational Proof Techniques
A1: 32 rows, each additional variable doubles the number of rows
A2: In general, 2n rows
Therefore, as compound propositions grow in complexity, truth tables become more and more unwieldy. Checking for tautologies/logical equivalences of complex propositions can become a chore, especially if the problem is obvious.

41. Derivational Proof Techniques
EG: consider the compound proposition
(p p )  ((sr)t) )  (qr )
Q: Why is this a tautology? L3

42. Derivational Proof Techniques
A: Part of it is a tautology (p p ) and the disjunction of True with any other compound proposition is still True:
(p p )  ((sr)t ))  (qr )
T  ((sr)t ))  (qr ) T
Derivational techniques formalize the intuition of this example. L3

43. Propositional Logic - an unfamous 
if NOT (blue AND NOT red) OR red then…
(p  q)  q  p  q
(p  q)  q 
(p  q)  q
DeMorgan’s
(p  q)  q
Double negation
p  (q  q)
Associativity
p  q
Idempotent

44. Tautology by proof
[¬p (p q )]q L3

45. Tautology by proof [¬p (p q )]q  [(¬p p)(¬p q)]q Distributive

46. Tautology by proof [¬p (p q )]q  [(¬p p)(¬p q)]q Distributive
 [ F  (¬p q)]q ULE L3

47. Tautology by proof [¬p (p q )]q  [(¬p p)(¬p q)]q Distributive
 [ F  (¬p q)]q ULE
 [¬p q ]q Identity L3

48. Tautology by proof [¬p (p q )]q  [(¬p p)(¬p q)]q Distributive
 [ F  (¬p q)]q ULE
 [¬p q ]q Identity
 ¬ [¬p q ]  q ULE L3

49. Tautology by proof [¬p (p q )]q  [(¬p p)(¬p q)]q Distributive
 [ F  (¬p q)]q ULE
 [¬p q ]q Identity
 ¬ [¬p q ]  q ULE
 [¬(¬p) ¬q ]  q DeMorgan L3

50. Tautology by proof [¬p (p q )]q  [(¬p p)(¬p q)]q Distributive
 [ F  (¬p q)]q ULE
 [¬p q ]q Identity
 ¬ [¬p q ]  q ULE
 [¬(¬p) ¬q ]  q DeMorgan
 [p  ¬q ]  q Double Negation L3

51. Tautology by proof [¬p (p q )]q  [(¬p p)(¬p q)]q Distributive
 [ F  (¬p q)]q ULE
 [¬p q ]q Identity
 ¬ [¬p q ]  q ULE
 [¬(¬p) ¬q ]  q DeMorgan
 [p  ¬q ]  q Double Negation
 p  [¬q q ] Associative L3

52. Tautology by proof [¬p (p q )]q  [(¬p p)(¬p q)]q Distributive
 [ F  (¬p q)]q ULE
 [¬p q ]q Identity
 ¬ [¬p q ]  q ULE
 [¬(¬p) ¬q ]  q DeMorgan
 [p  ¬q ]  q Double Negation
 p  [¬q q ] Associative
 p  [q ¬q ] Commutative L3

53. Tautology by proof [¬p (p q )]q  [(¬p p)(¬p q)]q Distributive
 [ F  (¬p q)]q ULE
 [¬p q ]q Identity
 ¬ [¬p q ]  q ULE
 [¬(¬p) ¬q ]  q DeMorgan
 [p  ¬q ]  q Double Negation
 p  [¬q q ] Associative
 p  [q ¬q ] Commutative
 p  T ULE L3

54. Tautology by proof [¬p (p q )]q  [(¬p p)(¬p q)]q Distributive
 [ F  (¬p q)]q ULE
 [¬p q ]q Identity
 ¬ [¬p q ]  q ULE
 [¬(¬p) ¬q ]  q DeMorgan
 [p  ¬q ]  q Double Negation
 p  [¬q q ] Associative
 p  [q ¬q ] Commutative
 p  T ULE
 T Domination L3

55. Writing Logical Formula for a Truth Table
Digital logic:
Given a digital circuit, we can construct the truth table.
Now, suppose we are given only the truth table (i.e. the specification,
e.g. the specification of the majority function), how can we construct a digital circuit (i.e. formula) using only simple gates (such as AND, OR, NOT)
that has the same function?

56. Examples
“I don’t study well and fail” is logically equivalent to “If I study well, then I don’t fail”
Write a C program that represents the compound proposition (pq)r

57. Decide whether the following statements are tautologies or contradictions or neither
1. (p  q)  (q  p). 2. (p  q)  (q  ¬p). 3. p  ¬p. 4. (p  ¬q)  (q  ¬p).

58. Use truth table to find - P  Q  P  R P  -Q  R  P  R  - Q
A  B  -C  D  E  F
(P ↔ Q)  (P˅ Q)
[P ˄ (P  Q)]  Q

59. Which of the following equivalent to -(P˄Q)