1. Advanced Algorithms Analysis and DesignBy
Dr. Nazir Ahmad Zafar
Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

2. Lecture No 3 Logic and Proving Techniques
Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

3. Today Covered Tools used for proving algorithms Propositional Logic
Predicate Logic
Proofs using
Truth Tables
Logical Equivalences
Counter Example
Contradiction
Rule of Inference
Probability as Analysis Tool
Series and Summation etc.
Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

4. Propositional and Predicate Logic
Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

5. Logical Connectives
Proposition: Simplest statements also called atomic formula
Propositions may be connected based on atomic formula.
Logical connectives, in descending order of operator precedence
Symbol
Name
Pronunciation 
negation
not 
conjunction
and 
disjunction or 
implication
implies 
equivalence
if and only if
Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

6. Negation, Conjunction and Disjunction
The conjunction p  q is true only if p and q both are true otherwise false
The conjunction follows the commutative property, i.e. p  q = q  p
Disjunction
The disjunction p  q is false if both p and q are false otherwise true
The disjunction follows the commutative property as well, i.e. p  q = q  p
Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

7. Implication The p is antecedent and q is consequent
The antecedent is stronger than consequent.
Commutative property does not hold, i.e.
(p  q)  (q  p) p q
p  q
q  p
 p  q t f
Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

8. Bi-implication The equivalence p  q means p  q & q  p
Commutative property does hold, i.e.
(p  q) = (q  p) p q
p  q
q  p
p  q & q  p t f
Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

9. Predicates and Quantifiers
Predicate: P(x)  x < 5
Example:  x : N | x2 = x  x < 2
For all quantifier
 x, P(x) is true  P(x) is true for all x.
Existential Quantifier
 x, P(x) is true  P(x) is true for some value of x.
Logical Equivalences
 x, P(x) is logically equivalent to  ( x, P(x))
 x, P(x) is logically equivalent to ( x, P(x))
 x, (P(x)  Q(x)) means x, P(x)  Q(x)
Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

10. Proving Techniques
Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

11. Proof using Truth Table: (p  q  r)  (p  (q  r))
t t t t t t f
t f t f f t
f t t t t t
f t t t f t
Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

12. De Morgan’s Laws (p  q) =  p   q p q p  q (p  q) p q p  qt f
Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

13. De Morgan’s Laws 2. (p  q) =  p   q p q p  q (p  q) p qt f
Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

14. Proof using Counter Example, Contraposition
Counter Example To prove  x (A(x)  B(x)) is false, we show some object x for which A(x) is true and B(x) is false.
Proof
 ( x (A(x)  B(x))) 
x, (A(x)  B(x))) 
 x, (A(x)  B(x)) 
 x, A(x)  B(x))
Contraposition To prove A  B, we show ( B)  ( A)
x is divisible by 4  x is divisible by 2 
x is not divisible by 2  x is not divisible by 4 
Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

15. Proof by Contradiction
Contradiction To prove A  B,
Steps in Proof
We assume A and to prove that B
On contrary suppose that  B and
Then prove B, it will be contradiction
Further analysis
A  B  (A  B)  B Contradiction
A  B  (A  B) is false
Assuming (A  B) is true, and discover a contradiction (such as A  A), then conclude (A  B) is false, and so A  B.
Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

16. Problem: Proof by Contradiction
Prove:
[B  (B  C)]  C, by contradiction
Proof:
Suppose [B  (B  C)], to prove C
On contrary, assume C
C  [B  (B  C)] must be true
 C  [B  ( B  C)]
 C  [(B   B)  (B  C)]
 C  [f  (B  C)]
 C  B  C = C  C  B = f  B = f
 False, Contradiction  C
Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

17. Rules of Inference
A rule of inference is a general pattern that allows us to draw some new conclusion from a set of given statements. If we know P then we can conclude Q.
Modus ponens
If {B  (B  C)} then {C}, example in last slide
Proof:
Suppose B  (B  C) then B
B  C
Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

18. Rules of Inference Syllogism If {A  B  B  C} then {A  C} Proof
Suppose A  B  B  C, To prove A  C B C
Rule of cases
If {B  C  B  C} then {C}
B, true, implies C true
 B, true, implies C true
Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

19. Two Valued Boolean Logic
Boolean values = B = {0, 1}, there are two binary operations:
+ = or = 
· = and = 
2. Closure properties:
 x, y  B, x + y  B
 x, y  B, x . y  B
3. Identity element:
x + 0 = 0 + x = x
x · 1 = 1 . x = x
Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

20. Two Valued Boolean Logic
4. Commutative:
x + y = y + x
x · y = y · x
5. Distributive:
x · (y + z) = (x · y) + (x · z)
x + (y · z) = (x + y) · (x + z)
6. Complement:
 x  B,  x’  B such that
x + x’ = 1, x · x’ = 0
Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

21. Tautologies and Truth Table
Tautology:
Any statement which is always true is called a tautology
Example
Show [B  (B  C)]  C is a tautology:
Proof
B C (B  C) (B  (B  C)) (B  (B  C))  C
For every assignment for B and C, the statement is True, hence the above statement is a tautology.
Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

22. Probability as Analysis Tool
Elementary events
Suppose that in a given situation an event, or an experiment, may have any one, and only one, of k outcomes, s1, s2, …, sk. Assume that all these outcomes are mutually exclusive.
Universe The set of all elementary events is called the universe of discourse and is denoted
U = {s1, s2, …, sk}.
Probability of an outcome si
Associate a real number Pr(si), such that
0  Pr(si)  1 for 1  i  k;
Pr(s1) + Pr(s2) + … + Pr(sk) = 1
Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

23. Event Event Let S  U. Then S is called an event, and Sure event
U = {s1, s2, …, sk}, if S = U
Impossible event
S = , Pr() = 0
Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

24. Arithmetic and Geometric Series
Dr Nazir A. Zafar Advanced Algorithms Analysis and Design

25. Conclusion Propositional Logic Predicate Logic
We have discussed various techniques of proving
Truth Tables
Logical Equivalence
Counter Example
Contraposition
Contradiction
Rule of Inference
Probability can be used for average cost analysis
Series and summation are very helpful in simplification
Dr Nazir A. Zafar Advanced Algorithms Analysis and Design