Discrete mathematical structures-mca vtu,1st semester cbcs scheme-module system-module 1

1. MODULE 1- RULES
OF INFERENCE

2. INTRODUCTION
• An argument in propositional logic is a sequence of
propositions.
• All but the final proposition in the argument are called
premises and the final proposition is called the conclusion.
• An argument is valid if the truth of all its premises implies that
the conclusion is true.

3. • An argument form in propositional logic is a sequence of
compound propositions involving propositional variables.
• An argument form is valid no matter which particular
propositions are substituted for the propositional variables in
its premises, the conclusion is true if the premises are all true.

4. VALID ARGUMENTS IN
PROPOSITIONAL LOGIC
• Example:
• “If you have a current password, then you can log onto the
network.”
• “You have a current password.”
Therefore,
• “You can log onto the network.”

5. Use p to represent “You have a current password” and q to
represent “You can log onto the network.” Then, the argument
has the form
p → q
p
∴ q

6. RULES OF INFERENCE FOR
PROPOSITIONAL LOGIC

8. 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.”

9. • State which rule of inference is the basis of the following
argument: “It is below freezing and raining now. Therefore, it
is below freezing now.”

10. • State which rule of inference is used in the argument:
• If it rains today, then we will not have a barbecue today. If we
do not have a barbecue today, then we will have a barbecue
tomorrow. Therefore, if it rains today, then we will have a
barbecue tomorrow

11. USING RULES OF INFERENCE TO BUILD
ARGUMENTS
• When there are many premises , several rules of inference are
often needed to show that given argument is valid.
• Example is as follows:
• Show that the premises “It is not sunny this afternoon and it is
colder than yesterday,” “We will go swimming only if it is sunny,”
“If we do not go swimming, then we will take a canoe trip,” and “If
we take a canoe trip, then we will be home by sunset” lead to the
conclusion “We will be home by sunset.”

12. • Show that the premises “If you send me an e-mail message,
then I will finish writing the program,” “If you do not send me
an e-mail message, then I will go to sleep early,” and “If I go
to sleep early, then I will wake up feeling refreshed” lead to
the conclusion “If I do not finish writing the program, then I
will wake up feeling refreshed.”

13. • Show that the following argument is valid. If today is Tuesday,
I have test on mathematics or economics. If my economics
professor is sick, I will not have test in economics. Today is
Tuesday and my economics professor is sick. Therefore I have
a test in mathematics

14. RESOLUTION PRINCIPLE
• A variable or a negation of a variable is called a literal
• A disjunction of literal is called a sum and conjunction of
literals is called product
• For any two clauses C1 and C2, if there is a literal L1 in C1 i.e
complementary to a literal L2 in C2, delete L1 and L2 from C1
and C2 respectively and construct the disjunction of the
remaining clauses. This constructed clause is a resolvent of C1
and C2

15. RESOLUTION PRINCIPLE (FORMAL
DEFINITION)
• Given S set of clauses , a resolution or deduction of C from S
is finite sequence C1, C2, … Ck of such clauses such that each
C1 is either is a clause in S or a resolvent of the clause
preceding C and Ck=C. A deduction of empty [] is called
refutation or proof of S

16. EXAMPLE
• Show that the following argument is valid using resolution
principle. If today is Tuesday, I have test on mathematics or
economics. If my economics professor is sick, I will not have
test in economics. Today is Tuesday and my economics
professor is sick. Therefore I have a test in mathematics

17. FALLACIES
• Type of incorrect reasoning
• Is the following argument valid?
• If you do every problem in this book, then you will learn
discrete mathematics. You learned discrete mathematics.
Therefore, you did every problem in this book.

18. RULES OF INTERFERENCE FOR
QUANTIFIED STATEMENTS

19. UNIVERSAL INSTANTIATION
• P ( C ) is true, when c is the member of domain
• Eg: all women are wise, that “lisa is wise” where lisa is the
member of domain women.

20. UNIVERSAL GENERALIZATION
• We show that the quantification is true by taking a arbitrary
value C from the domain and showing that P (C ) is true.
• Lisa is wise , where lisa is from domain women . Then
• All women are wise

21. EXISTENTIAL INSTANTIATION
• Concludes that there is an element c in the domain which P ©
is true , if we know the quantification is true

22. EXISTENTIAL GENERALIZATION
• Conclude that quantifiction is true when particular element c
with P (C ) is true is known

23. EXAMPLE
• Show that the premises “Everyone in this discrete
mathematics class has taken a course in computer science”
and “Marla is a student in this class” imply the conclusion
“Marla has taken a course in computer science.”

24. • Show that the premises “A student in this class has not read
the book,” and “Everyone in this class passed the first exam”
imply the conclusion “Someone who passed the first exam has
not read the book.”