- The document discusses the mathematical foundations of computer science, including topics like mathematical logic, set theory, algebraic structures, and graph theory.
- It specifically focuses on mathematical logic, defining statements, atomic and compound statements, and various logical connectives like negation, conjunction, disjunction, implication, biconditional, and their truth tables.
- It also discusses logical concepts like tautologies, contradictions, contingencies, logical equivalence, and tautological implication through the use of truth tables and logical formulas.
NP completeness. Classes P and NP are two frequently studied classes of problems in computer science. Class P is the set of all problems that can be solved by a deterministic Turing machine in polynomial time.
Content:
1- Mathematical proof (what and why)
2- Logic, basic operators
3- Using simple operators to construct any operator
4- Logical equivalence, DeMorgan’s law
5- Conditional statement (if, if and only if)
6- Arguments
NP completeness. Classes P and NP are two frequently studied classes of problems in computer science. Class P is the set of all problems that can be solved by a deterministic Turing machine in polynomial time.
Content:
1- Mathematical proof (what and why)
2- Logic, basic operators
3- Using simple operators to construct any operator
4- Logical equivalence, DeMorgan’s law
5- Conditional statement (if, if and only if)
6- Arguments
In computer science, divide and conquer (D&C) is an algorithm design paradigm based on multi-branched recursion. A divide and conquer algorithm works by recursively breaking down a problem into two or more sub-problems of the same (or related) type, until these become simple enough to be solved directly. The solutions to the sub-problems are then combined to give a solution to the original problem.
In computer science, merge sort (also commonly spelled mergesort) is an O(n log n) comparison-based sorting algorithm. Most implementations produce a stable sort, which means that the implementation preserves the input order of equal elements in the sorted output. Mergesort is a divide and conquer algorithm that was invented by John von Neumann in 1945. A detailed description and analysis of bottom-up mergesort appeared in a report by Goldstine and Neumann as early as 1948.
In computer science, divide and conquer (D&C) is an algorithm design paradigm based on multi-branched recursion. A divide and conquer algorithm works by recursively breaking down a problem into two or more sub-problems of the same (or related) type, until these become simple enough to be solved directly. The solutions to the sub-problems are then combined to give a solution to the original problem.
In computer science, merge sort (also commonly spelled mergesort) is an O(n log n) comparison-based sorting algorithm. Most implementations produce a stable sort, which means that the implementation preserves the input order of equal elements in the sorted output. Mergesort is a divide and conquer algorithm that was invented by John von Neumann in 1945. A detailed description and analysis of bottom-up mergesort appeared in a report by Goldstine and Neumann as early as 1948.
Discrete Mathematics is a branch of mathematics involving discrete elements that uses algebra and arithmetic. It is increasingly being applied in the practical fields of mathematics and computer science. It is a very good tool for improving reasoning and problem-solving capabilities.
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
Biological screening of herbal drugs: Introduction and Need for
Phyto-Pharmacological Screening, New Strategies for evaluating
Natural Products, In vitro evaluation techniques for Antioxidants, Antimicrobial and Anticancer drugs. In vivo evaluation techniques
for Anti-inflammatory, Antiulcer, Anticancer, Wound healing, Antidiabetic, Hepatoprotective, Cardio protective, Diuretics and
Antifertility, Toxicity studies as per OECD guidelines
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
Embracing GenAI - A Strategic ImperativePeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Acetabularia Information For Class 9 .docxvaibhavrinwa19
Acetabularia acetabulum is a single-celled green alga that in its vegetative state is morphologically differentiated into a basal rhizoid and an axially elongated stalk, which bears whorls of branching hairs. The single diploid nucleus resides in the rhizoid.
1. Mathematical Foundations of
Computer Science
Mathematical logic
Set theory
Algebraic Structures
Elementary Combinatorics
Recurrence relations
Graph theory
2. Mathematical logic
Statement (Proposition) : A declarative sentence to which it is
meaningful to assign one and only one of the truth values “true”
or “false”. We call such sentences Propositions (Statements).
Ex. London is a city. Ex. 2+3 = 4
The following sentences are not statements.
What is your name?
Close the door
For definiteness let us list our assumptions about propositions.
The law of excluded middle: For every proposition p,
either p is true or p is false.
The law of contradiction: For every proposition p, it is not the
case that p is both true and false.
3. Atomic and Compound statements
Atomic statement : A statement which can not be divided
further, is called atomic statement (Simple statement or
primary statement).
These statements are denoted by p,q,r,s,……
Ex. Milk is white Ex. 2+3 = 5
Compound Statement : Two or more simple statements can be
combined to form a new statement. These new statements are
called Compound statements or Molecular Statements or
Propositional function or Statement formulas.
Ex. It is raining today and there are 20 tables in this room.
Compound statements can be formed from atomic statements
through the use of following sentential connectives.
not, and , or , if …then and if and only if .
4. Connectives
Negation: If p is a statement, then the negation of p, written as
~p and read as “ not p ” is a statement.
Ex. p : London is a city.
~p : London is not a city.
The truth table for not p is given below.
p ~p
T
F
5. Conjunction (and) pq
If p and q are two propositions, then the conjunction of p and q
is the statement p q which is read as “ p and q ”.
The statement p q has the truth value T whenever both p and
q have truth value T; otherwise it has the truth value false.
Conjunctive syllogism: If p q is false and p is true, then q is
false.
p q p q
F
F
T
T
F
T
F
T
6. Disjunction ( Or ) pq
If p and q are two propositions, then the disjunction of p and q
is the statement p q which is read as “ p or q”.
The statement pq has the truth value F only when both p and
q have truth value F; otherwise it has the truth value T.
Disjunctive syllogism: If p q is true and p is false, then q is
true.
p q p q
F
F
T
T
F
T
F
T
7. Implication (Conditional) pq
If p and q are two propositions, then the statement pq
which is read as “ if p, then q ” or “ p implies q “.
The statement pq has truth value F only when p is true
and q is false; otherwise it has a truth value T.
A false antecedent p implies any proposition q.
A true consequent q is implied by any proposition q
p q pq
F
F
T
T
F
T
F
T
8. Biconditional (if and only if) pq
Biconditional : If p and q are two propositions, then the
statement pq, which is read as “p if and only if q” is called a
biconditional statement.
The statement pq has the truth value T whenever both p and
q have identical truth values.
p q pq
F
F
T
T
F
T
F
T
9. More on Implication
The opposite of pq is p q
The converse of pq is q p
The contra positive of pq is q p
Note : pq is logically equivalent to q p
i.e., pq q p
or pq q p
* Ex. p: Today is Sunday
q: Today is Holiday
p q : If today is Sunday, then today is
Holiday
q p : If today is not Holiday, then today is not
Sunday
10. Well formed formulas
A well formed formula can be generated by the following rules.
1. A statement variable standing alone is a well formed formula.
2. If P is a well formed formula, then ~P is a well formed
formula.
3. If P and Q are well formed formulas, then (PQ) , (PQ) ,
(PQ) and (PQ) are well formed formulas.
4. A string of symbols containing the statement
variables,connectives and parenthesis is a well formed formula,
iff it can be obtained by finitely many applications of the rules
1,2 and 3.
Ex. (PQ) , (PQ) , (P (PQ) ) , (P (Q R)) and
(PQ) (PQ) are well formed formulas.
Ex. PQ , (PQ )Q ) and (P Q ) (Q) are not well
formed formulas.
11. Truth tables
Our basic concern is to determine the truth value of a statement
formula for each possible combination of the truth values of the
component statements.
A table showing all such truth values is called the truth table of
the formula.
Ex.1 Construct truth table for the statement formula P Q
P Q Q P Q
F
F
T
T
F
T
F
T
T
F
T
F
12. Truth tables - Examples
Ex : 2 Construct the truth table for (PQ) P
P Q PQ P (PQ) P
F
F
T
T
F
T
F
T
F
T
T
T
T
T
F
F
T
T
T
T
13. Truth tables - Examples
Ex.3 Construct the truth table for (PQ) (QP)
P Q PQ QP (PQ) (QP)
F
F
T
T
F
T
F
T
T
T
F
T
T
F
T
T
T
F
F
T
Note:
(PQ) {(PQ) (QP)}
14. Tautology and Contradiction
Tautology : A propositional function (Statement formula) whose
value is true all all possible values of the propositional variables
is called a Tautology ( A Universally valid formula or a logical
truth).
Ex: P P is a tautology.
Ex. ( P P ) Q is a tautology.
Contradiction (Absurdity): A propositional function whose truth
value is always false is called a Contradiction
Ex. P P is a Contradiction .
Ex. ( P P ) Q is a Contradiction
Contingency: A propositional function that is neither a tautology
nor a contradiction is called a Contingency.
Ex. P Q , P Q , P Q, ….
15. Logical Equivalence & Tautological Implication
Two propositional functions P and Q are logically equivalent, if
they have same truth tables. Then we write
P Q or P Q
Ex: (P ) P
Ex: ( P Q ) ( P Q ).
Note : The symbol is not a connective
A Statement P is said to tautologically imply a Statement Q if
and only if PQ is a tautology.We shall denote this as P Q.
Here, P and Q are related to the extent that, Whenever P has
the truth value T then so does Q.
Every logical implication is an implication, but all implications
are not logical implications.
16. More on Implications
If P Q and Q P , then PQ.
If PQ then PQ is a tautology.
Ex: Show that ( P Q ) ( P Q )
Since columns 3 and 5 are identical, The result follows
P Q PQ P PQ
F
F
T
T
F
T
F
T
T
T
F
T
T
T
F
F
T
T
F
T
17. Ex.Construct truth table for [(pq) (r)] p]
The truth table is given below
p q r pq r (pq) (r) [(pq) (r)] p]
F F F
F F T
F T F
F T T
T F F
T F T
T T F
T T T
F
F
F
F
F
F
T
T
T
F
T
F
T
F
T
F
T
F
T
F
T
F
T
T
F
T
F
T
T
F
T
T
18. Ex. Show that (PQ) (Q P)
Let us prove the result using truth table.
P Q PQ Q P (Q P)
F
F
T
T
F
T
F
T
T
T
F
T
T
F
T
F
T
T
F
F
T
T
F
T
19. Ex. Using truth tables, show that ( P Q ) (Q)
is a tautology
The truth table is given below.
P Q P Q ( P Q ) Q ( P Q ) (Q)
F
F
T
T
F
T
F
T
T
T
F
T
F
F
T
F
T
F
T
F
T
T
T
T
20. Equivalences
Commutative laws:
P Q Q P
P Q Q P
Asociative laws:
( P Q ) R P ( Q R )
( P Q ) R P ( Q R )
Distributive laws:
P ( Q R ) ( P Q ) ( P R )
P ( Q R ) ( P Q ) ( P R )
Demorgan’s laws:
( P Q) P Q
( P Q) P Q
21. More Equivalences
( P ) P (Double negation)
P P P
P P P
P P T
P P F
R ( P P ) R
R ( P P ) R
R ( P P ) T
R ( P P ) F
P Q ( P Q)
( P Q ) (P Q)
P Q ( Q P )
22. More Equivalences
• P F P
• P T T
• P F F
• P T P
• P ( Q R) ( P Q ) R
( P Q ) (P Q)
• (P Q ) [( P Q) ( Q P )]
• ( P Q ) [( P Q) ( P Q )]
• Absorption laws
• P ( P Q ) P
• P ( P Q ) P
23. Ex. Without using truth tables, Show that
P ( Q R) ( P Q ) R
Proof:
L.H.S = P (Q R)
P (Q R) (Since A B ( A B))
P (Q R)
(P Q) R (By associative property)
( P Q ) R (By demorgan’s law)
( P Q ) R
= R.H.S
24. Ex. Without using truth tables, Show that
( P Q ) P is a tautology.
Proof:
Consider, ( P Q ) P
( Q P ) P ( By commutative law )
Q (P P ) ( By associative property)
Q T
T
( P Q ) P is a tautology.
25. Ex. Show that the Statement formula
( P Q ) (PQ) P is a tautology.
Proof : Consider,
{( P Q ) (PQ)} P (Associative law)
{(P Q ) (PQ)} P ( Demorgan’s law)
{P (Q Q)} P (Distributive law)
{P T } P
{P } P
T
( P Q ) (PQ) P is a tautology
26. Ex. Show that [{( P Q ) ( P Q )} R ] R
Proof: L.H.S = {( P Q ) ( P Q )} R
{ T } R (Since P Q ( P Q))
R
= R.H.S
Ex. Show that {( P Q ) ( P Q )} is a Contradiction.
Proof : Let P Q = R
Consider, {( P Q ) ( P Q )}
{ R R }
F
{ ( P Q ) ( P Q )} is a contradiction.
27. Ex. Show that (P (Q R)) ( Q R ) (P R) R
Proof : Consider,
{P (Q R)} ( Q R ) (P R)
{(P Q) R} {( Q R ) (P R)}, By associative law
{ (P Q) R} {(Q P ) R} , By distributive law
{(P Q) R} {(Q P ) R} , By Demorgan’s law
{(P Q) (Q P ) } R, By distributive law
{T } R (Since, A A T)
R
29. Normal forms
Elementary product:A product of the variables and their
negations in a formula is called an Elementary product.
Ex: P, PQ, PQ, PQ R
Elementary Sum: A Sum of the variables and their negations in
a formula is called an Elementary Sum.
Ex: P, P Q, P Q, P Q R
Disjunctive normal form: A formula which is equivalent to a
given formula and which consists of a sum of elementary
products is called a disjunctive normal form.
Ex: (P ) ( PQ ) (PQ).
Ex: ( PQ ) (PQ) (PQ R ).
30. Normal forms (contd.,)
Conjunctive normal form: A formula which is equivalent to a
given formula and which consists of a product of elementary
sums is called a conjunctive normal form.
Ex: (P ) ( P Q ) (P Q).
Ex: ( P Q ) (P Q) (P Q R ).
Min terms: Let P and Q are two statement variables. Let us
construct all possible formulas which consist of conjunctions of
P or its negation and conjunctions of Q or its negation.
For two variables P and Q, there are 22 such formulas given by
PQ, PQ, PQ, PQ
These formulas are called ‘min terms’.
31. Normal forms (contd.,)
For three variables P,Q and R, there are 23 such formulas given
by
PQ R, PQ R, PQ R, PQ R,
PQ R, PQ R, PQ R, PQ R
These min terms are denoted by m0, m1 , …, m7 respectively.
In general, there are 2n min terms for n variables.
Principal Disjunctive normal form (Sum of products canonical
form) : For a given formula, an equivalent formula consisting of
disjunctions of min terms only is known as Principal Disjunctive
normal form .
32. Ex. Obtain the Principal Disjunctive normal forms of the following
PQ , P Q, (PQ)
Solution:
PQ (PQ) (PQ) (PQ)
P Q (PQ) (PQ) (PQ)
(PQ) (PQ) (PQ) (PQ)
P Q PQ P Q PQ (PQ)
F
F
T
T
F
T
F
T
T
T
F
T
F
T
T
T
F
F
F
T
T
T
T
F
33. Ex. Obtain the Principal Disjunctive normal form of the following
P {(PQ) (P Q)}
Given formula is, [ P {(PQ) (P Q)} ] = A (say)
The truth table for A is given below.
A (PQ) (PQ) (PQ)
Which is the PDNF for A .
P Q PQ P Q {(PQ)
(P Q)}
A
F
F
T
T
F
T
F
T
T
T
F
T
F
F
F
T
F
F
F
T
T
T
F
T
34. Ex. Obtain the Principal Disjunctive normal form of the following
(P Q) (Q R) (P R )
Solution: Consider, (P Q) (Q R) (P R )
{(P Q) (R R)}
{(P P) (Q R) }
{(P R ) (Q Q)}
(PQ R) (PQ R) (PQ R) (PQ R)
Which is the PDNF for the given formula.
35. Ex. Obtain the Principal Disjunctive normal form of the following
(P Q) (P R )
= A (say)
A (PQ R) (PQ R) (PQ R) = (m1, m6, m7)
P Q R P Q P (P R) A
F
F
F
F
T
T
T
T
F
F
T
T
F
F
T
T
F
T
F
T
F
T
F
T
T
T
F
F
F
F
T
T
T
T
T
T
F
F
F
F
F
T
F
T
T
T
T
T
F
T
F
F
F
F
T
T
36. Principal Conjunctive normal forms
(Product of Sums canonical forms)
Max terms: For a given number of variables, the max term
consists of disjunctions in which each variable or its negation,
but not both, appears only once.
For two variables P and Q, there are 22 such formulas given by
(P Q), (P Q), (P Q), (P Q).
These formulas are called ‘max terms’.
For three variables P,Q and R, there are 23 such formulas given
by
P Q R , P Q R, P Q R, P Q R,
P Q R, P Q R, P Q R, P Q R
These max terms are denoted by M0, M1 , …, M7 respectively.
In general, there are 2n Max terms for n variables.
37. PCNF (Contd.,)
Mi = mi
M0 = m0
= (PQ R) = (P Q R)
M1 = m1
= (PQ R) = (P Q R)
M2 = m2
= (PQ R) = (P Q R)
Principal Conjunctive normal form (Product of Sums canonical
form) : For a given formula, an equivalent formula
consisting of conjunctions of max terms only is known as
Principal Conjunctive normal form.
38. Ex. Obtain the Principal Conjunctive normal forms of the following
PQ , P Q, (PQ)
The PCNF’s are
PQ (P Q)
P Q (P Q) (P Q) (P Q)
(PQ) (P Q) (P Q)
P Q PQ P Q PQ (PQ)
F
F
T
T
F
T
F
T
T
T
F
T
F
F
F
T
T
F
F
T
F
T
T
F
39. EX. Obtain the Principal Conjunctive normal form of the formula
given by (P R) (Q P)
Solution: (P R) (Q P)
(P R) {(PQ) (QP)}
(P R) (P Q) (Q P)
{ (P R) (Q Q) }
{ (P Q) (R R) }
{ (Q P) (R R) }
(P Q R) (P Q R) (P Q R)
( P Q R) (P Q R)
= (0,2,3,4,5)
Which is the required PCNF.
40. Max terms and Min terms
*
P Q R Min terms mi Max terms Mi
F
F
F
F
T
T
T
T
F
F
T
T
F
F
T
T
F
T
F
T
F
T
F
T
m0 : PQ R
m1 : PQ R
m2 : PQ R
m3 : PQ R
m4 : PQ R
m5 : PQ R
m6 : PQ R
m7 : PQ R
M0 : P Q R
M1 : P Q R
M2 : P Q R
M3 : P Q R
M4 : P Q R
M5 : P Q R
M6 : P Q R
M7 : P Q R
41. Ex. Obtain the Principal Conjunctive normal form and Principal
disjunctive normal form of A, where A = (P Q) (P R )
The PCNF of A = (0,2,4,5)
A (P Q R) (P Q R) (P Q R) (P Q R)
P Q R P Q P P R A
F
F
F
F
T
T
T
T
F
F
T
T
F
F
T
T
F
T
F
T
F
T
F
T
F
F
F
F
F
F
T
T
T
T
T
T
F
F
F
F
F
T
F
T
F
F
F
F
F
T
F
T
F
F
T
T
42. Contd.,
The PDNF of A = (1,3,6,7)
A (PQ R) (PQ R) (PQ R) (PQ R)
43. Implications ,Arguments,Inferences
Inference (Argument): From a set of premises (called
Hypotheses) {H1, H2, …., Hn } a conclusion C follows logically
iff H1 H2 …. Hn C.
• The rules of inference are criteria for determining the validity of
an argument.
• Any conclusion which is arrived at by following these rules is
called a valid conclusion, and the argument is called a valid
argument.
• The following statements are equivalent.
• 1. {H1 , H2 , …. , Hn } C is a logical implication.
• 2. ( H1 H2 …. Hn) C is a tautology.
• 3. {H1 , H2 ,…. , Hn } C is a valid argument.
44. Rules of Inference
There are two rules of Inference
1) Rule P: A premise may be introduced at any point in the
derivation.
2) Rule T: A formula S may be introduced in a derivation if S is
tautologically implied by and/or equivalent to any one or more
of the preceding formulas in thederivation.
48. Rules of Inference (contd.,)
Simplification rules:
(P Q) P
(P Q) P is a tautology.
P logically follows from (P Q)
(P Q) Q
(P Q) Q is a tautology.
Q logically follows from (P Q)
Addition rules:
• P (P Q)
P (P Q) is a tautology
(P Q) logically follows from P
49. Rules of Inference (contd.,)
Q ( P Q )
Q (P Q) is a tautology
(P Q) logically follows from Q
P (P Q)
P (P Q) is a tautology
(P Q) logically follows from P
Q ( P Q)
Q (P Q) is a tautology
(P Q) logically follows from Q
(P Q) P
(P Q) P is a tautology (or) P follows from (P Q)
50. Rules of Inference (Contd.,)
(P Q ) (Q)
(P Q ) (Q) is a tautology
Q logically follows from (P Q)
Disjunctive syllogism
{P, P Q} Q
{P ( P Q)} Q is a tautology.
The inference P Q
P
----------------
Q is valid
51. Modus ponens (Rule of detachment)
{P, PQ} Q
{ P (PQ) } Q is a Tautology
The argument
PQ
P
------------
Q is valid
Ex: The following argument is valid.
A) If today is a Sunday then today is a Holiday
B) Today is Sunday
C : Hence, Today is Holiday
52. Modus tollens
{ PQ, Q } P
{ (PQ) Q} (P) is a Tautology
The argument
PQ
Q
------------
P is valid
Ex: The following argument is valid.
A) If today is a Sunday then today is a Holiday
B) Today is not Holiday
C : Hence, Today is not Sunday
53. Rule of Transitivity (Hypothetical Syllogism)
{ PQ, QR } (PR)
{ (PQ) (QR} (PR) is a Tautology
The argument
PQ
QR
------------
PR is valid
Ex: The following argument is valid.
A) If I Study well, then I will get distinction.
B) If I get distinction, then I will get a Good Job.
C: If I Study well, then I will get a good job
54. Dilemma
The Inference
P Q
P R
Q R
------------
R is a valid Inference.
{P Q, PR, QR } R is a logical implication.
{(PQ) (PR} (QR) } R is a Tautology
55. Constructive dilemma
The Inference
P Q
P R
Q S
------------
R S is a valid Inference.
{P Q, PR, QS } ( R S ) is a logical implication.
{(PQ) (PR} (QS) } (R S) is a Tautology
56. Destructive Dilemma
The Inference
P R
Q S
R S
----------------
P Q is a valid Inference.
{ PR, QS, R S } (P Q ) is a logical implication.
{(PR) (QS) (R S )} (P Q) is a Tautology
57. Conjunction and Conjunctive Syllogism
Conjunction
P, Q
----------
(PQ)
Conjunctive Syllogism:
{(PQ), P } Q
{(PQ) P } Q is a tautology.
(PQ)
P
--------
Q
58. Fallacies
1. The fallacy of affirming the Consequent (or affirming the
converse):
PQ
Q
_________
P Fallacy
Ex: Consider, the following argument
If today is Mahatma Gandhi’s Birth day, then today is October 2nd.
Today is October 2nd.
Today is Mahatma Gandhi’s Birth day.
The argument is not valid
59. 2. Fallacy of denying the antecedent
( Or Assuming the opposite)
Consider the following
PQ
P
_________
Q Fallacy
Ex: Consider the following argument:
H1 : If today is Sunday, then today is Holiday
H2 : Today is not Sunday
C : Today is not Holiday
The argument is not Valid.This is the fallacy of assuming the
opposite.
60. The non sequitur fallacy
P , Q
---------
R is a fallacy.
Ex: Consider the following argument:
1. India’s Capital is New Delhi
2. Milk is White
C: Sun rises in the East.
The conclusion does not follow from the premises.
Hence, the argument is invalid.
61. Ex: Show that R follows logically from the premises
PQ, QR, P
Proof: Consider the premises,
PQ -----(1)
QR -----(2)
P ------(3)
{1} (1) P Q Rule P
{2} (2) P Rule P
{1, 2} (3) Q Rule T, (1), (2), and I11.
{4} (4) Q R Rule P
{1, 2, 4} (5) R Rule T, (3), (4) and I11.
62. Ex: Show that R follows logically from the premises
PQ, QR, P
Proof: Consider the premises,
PQ -----(1)
QR -----(2)
P ------(3)
From (1) and (2), By the rule of transitivity,we have
PR --------(4)
From (3) and (4), By the rule of Modus ponens,
R follows.
R logically follows from the given premises
63. Ex: Show that P follows logically from the premises
PQ, QR, R
Proof: Consider the premises,
PQ -----(1)
QR -----(2)
R ------(3)
From (1) and (2), By the rule of transitivity,we have
PR --------(4)
From (3) and (4), By the rule of Modus tollens,
P follows.
P logically follows from the given premises
64. Ex: Show that R follows logically from the premises
PQ, QR, PM, M
Proof: Consider the premises,
P Q -----(1)
Q R -----(2)
P M -----(3)
M ------(4)
From (3) and (4), By the rule of Modus tollens, we have
P --------(5)
From (1) and (5), By the rule of Disjunctive Syllogism,we have
Q --------(4)
From (2) and (4), By the rule of Modus ponens,
R follows.
65. Ex: Show that (R S) follows logically from the premises
C D, (C D) H, H (A B), (A B) (R S )
Proof: Consider the premises,
(C D) -----(1)
(C D) H -----(2)
H (A B) -----(3)
(A B) (R S ) ------(4)
From (2),(3) and (4), By the rule of Transitivity, we have
(C D) (R S ) --------(5)
From (1) and (5), By the rule of Modus ponens,
(R S) follows.
66. Ex: Show that S follows logically from the premises
P (R S), RP, P
Proof: Consider the premises,
P (R S) -----(1)
R P -----(2)
P -----(3)
From (1) and (3), By the rule of Modus ponens, we have
(R S) ------(4)
From (2), By Contra positive equivalence, we have
P R -------(5)
(3) and (5), By the rule of Modus ponens, we have
R --------(6)
From (4) and (6), By the rule of Modus ponens, S follows.
67. Ex: Show that W follows logically from the premises
TR, S, T W, R S.
Proof: Consider the premises,
T R ------(1)
S -----(2)
T W -----(3)
R S -----(4)
From (1), By Contra positive equivalence, we have
R T -------(5)
From, (5) and (3), By the rule of Transitivity, we have
R W --------(6)
From (4) and (2), By the rule of Disjunctive syllogism,we have
R ---------- (7)
From(6)and (7), By the rule of Modus ponens, W follows
68. Ex: Show that TP follows logically from the premises
R(ST), R W, P S, W
Proof: Consider the premises,
R(ST) ------(1)
R W -----(2)
P S -----(3)
W -----(4)
From (2) and (4), By the rule of Disjunctive syllogism,we have
R ---------(5)
From(1)and (5), By the rule of Modus ponens, we have
S T ---------(6)
From, (3) and (6), By the rule of Transitivity, we have
P T ---------(7)
( T P ) (By Contra positive equivalence)
69. Conditional Proof (CP rule)
Theorem: If {H1, H2, …., Hn } and P imply Q, then
{H1, H2, …., Hn } imply (PQ).
Proof: From our assumption we have,
(H1 H2 …. Hn P) Q
This assumption means (H1 H2 …. Hn P) Q is a tautology.
Using the equivalence P (Q R) (P Q) R
We can say that (H1 H2 …. Hn) ( PQ ) is a tautology.
Hence the theorem.
Rule CP : If we can derive Q from P and a set of premises,then
we can derive PQ from the set of premises alone
70. Ex:Show that RS can be derived from the premises
p (Q S), RP, Q
Solution: Instead of deriving RS, we shall include R as an
additional premise and show S first.
p (Q S) …..(1)
RP …..(2)
Q ……(3)
R …….(4)
From (2) and (4), By the rule of Disjunctive syllogism,we have
P ---------(5)
From(1)and (5), By the rule of Modus ponens, we have
Q S ………….(6)
From(3)and (6), By the rule of Modus ponens, S follows
By CP rule, RS follows from the given premises.
71. Consistency, Inconsistency and Proof by Contradiction
A set of formulas {H1, H2, …., Hn} is said to be consistent, if
their conjunction has truth value T for some assignment of the
truth values to the atomic variables appearing in H1, H2, …., Hn .
A set of formulas {H1, H2, …., Hn} is said to be inconsistent, if
their conjunction implies a contradiction. that is
(H1 H2 …. Hn ) (R R) where R is any formula.
Proof by Contradiction :
In order to show that,a conclusion C logically follows from the
premises H1, H2, …., Hn ,We assume that C is false and Consider
C as additional premise.
If the new set of premises is inconsistent, then our assumption
is wrong. Hence C follows.
72. Ex: Show that (PQ) follows from (P Q)
Solution: Let us introduce (PQ) as an additional premise and
show that this leads to contradiction.
(PQ) ….(1)
Which is equivalent to
(PQ) ….(2)
From (2), P follows
Given that, (P Q) …..(3)
From (3), P follows
But, P and P cannot be simultaneously true (Contradiction).
Our assumption is false.
Hence (PQ) follows from (P Q)
73. Ex: Show that P follows from the premises PQ, (P Q)
Solution: Let us introduce P as an additional premise and show
that this leads to contradiction.
P ….(1)
PQ …..(2)
(PQ) ….(3)
From (1) and (2), By the rule of Moden ponens, we have
Q …….(4)
From (1) and (4), We have
( PQ) …….(5)
But, (3) and (5) cannot be simultaneously true (Contradiction).
Our assumption is false.
Hence, P follows from the premises PQ, (P Q)
74. Ex: Show that the following set of premises are inconsistent.
P Q, P R, Q R, P
Proof: Consider the premises,
P Q ------(1)
P R -----(2)
Q R -----(3)
P ----(4)
From (1) and (3), By the rule of transitivity, we have
P R …….(5)
From(2)and (4), By the rule of Modus ponens, R follows
From(4)and (5), By the rule of Modus ponens, R follows
But, R and R cannot be simultaneously true (Contradiction).
Hence, the given premises are inconsistent.
75. Ex: Show that the following set of premises are inconsistent.
R M, R S, M, S
Proof: Consider the premises,
R M ------(1)
R S ----(2)
M -----(3)
S ----------(4)
From (1) and (3), By the rule of Disjunctive Syllogism,we have
R ……….(5)
From(2)and (4), By the rule of Disjunctive Syllogism, We have
R ………..(6)
But, R and R cannot be simultaneously true (Contradiction).
Hence, the given premises are inconsistent.
76. Ex: Verify that the following argument is valid by using the rules of
inference (Here, H1 , H2 , …. are premises and C is conclusion) :
H1 : If Joe is a Mathematician, then he is ambitious.
H2 : If Joe is an early riser, then he does not like oat meal.
H3 : If Joe is ambitious, then he is an early riser
C : Hence, if Joe is a Mathematician, then he does not like oat
meal.
Solution: Let us make the following representations
p : Joe is a Mathematician.
q : Joe is ambitious
r : Joe is an early riser
s : Joe likes oat meal
The symbolic form of the given argument is
77. Contd.,
H1 : pq ….(1)
H2 : rs ….(2)
H3 : qr …..(3)
From (1) and (3), By the rule of transitivity, we have
pr ……(4)
From (4) and (2), By the rule of transitivity, we have
p s ……(5)
i.e., if Joe is a Mathematician, then he does not like oat meal.
The conclusion logically follows from the premises.
Hence, the argument is valid
78. Ex: Verify that the following argument is valid by using the rules of
inference (Here, H1 , H2 , …. are premises and C is conclusion) :
H1 : If Cliffton does not live in France, then he does not
speak French.
H2 : Cliffton does not drive a Datsun.
H3 : If Cliffton lives in France, then he rides a Bicycle.
H4 : Either Cliffton speaks French,or he drives a Datsun.
C : Hence, Cliffton drives a bicycle.
Solution: Let us make the following representations
p : Cliffton lives in France.
q : Cliffton speaks French.
r : Cliffton drives a Datsun.
s : Cliffton drives a Bicycle.
The symbolic form of the given argument is
79. Contd.,
H1 : p q …..(1)
H2 : r …..(2)
H3 : p s …..(3)
H4 : q r …….(4)
From (2) and (4), By the rule of Disjunctive Syllogism,we have
q ……..(5)
(1) q p …….(6)
From (5) and (6), By the rule of Modus ponens, we have
P ……(7)
From (3) and (7), By the rule of Modus ponens, s follows
The conclusion logically follows from the premises.
Hence, the argument is valid
80. Ex:Using Symbolic logic, Show that the following
premises are inconsistent
1. If Jack misses many classes through illness,then he fails high
school.
2. If Jack fails high school, then he is uneducated.
3. If Jack reads a lot of books, then he is not uneducated.
4. Jack misses many classes through illness and reads a lot of
books.
Solution: Let us make the following representations
p : Jack misses many classes through illness
q : Jack fails high school
r : Jack is uneducated
s : Jack reads a lot of books
Now, the given premises can be represented as
81. Contd.,
p q …..(1) q r …..(2)
s r …..(3) p s …..(4)
From (1) and (2), By transitivity, p r …..(5)
From(3), By Contra positive equivalence, r s ….. (6)
From (5) and (6), By transitivity, we have
p s …..(7)
From(4), we have
p …..(8)
From (7) and (8), By the rule of modus ponens, s follows
From (4), s follows
But, s and s cannot be simultaneously true (Contradiction).
Hence, the given premises are inconsistent
82. Ex:Using Symbolic logic,prove the following argument
If A works hard, then either B or C will enjoy themselves.
If B enjoys himself, then A will not work hard.
If D enjoys himself, then C will not enjoy himself.
Therefore, If A works hard, then D will not enjoy himself .
Solution: Let us use the following representations.
A : A works hard.
B : B will enjoy himself.
C : C will enjoy himself.
D : D will enjoy himself.
Now, we have to show that, A D follows from
A (B C) , B A and D C
83. Contd.,
A (B C) ….(1) B A ….(2)
D C ….(3) A …. (4) ( Additional premise)
From, (1) and (4), By modus ponens, We have
(B C) ……(5)
(2) A B …. (6)
From, (4) and (6), By modus ponens, B ….(7) follows.
From (5) and (7), By the rule of Disjunctive Syllogism, we have
C ….(8)
(3) C D …. (9)
From (8) and (9), By modus ponens, D follows
Hence, By CP rule, A D follows