Module - 1 Discrete Mathematics and Graph Theory

MAT-1014 Discrete Mathematics and Graph Theory
Faculty: Dr.D.Ezhilmaran
Teaching Research Associate: M.Adhiyaman
Department of Mathematics, School of Advanced Sciences, VIT-University,
Tamil Nadu, India
ezhilmaran.d@vit.ac.in
January 31, 2017
Faculty: Dr.D.Ezhilmaran Teaching Research Associate: M.Adhiyaman (VIT)Discrete Mathematics January 31, 2017 1 / 43

Overview
1 Module -1 Mathematical logic and Statement Calculus
Introduction
Statement and Notations
Connectives
Tautologies
Two State Devices and Statement Logic
Equivalance
Implications
Normal forms
The Theory of Inference for the Statement Calculus

Basics of proof techniques
Deﬁnition 1. (Proposition)
A statement or proposition is a declarative sentence that is either true or
false (but not both).
For instance, the following are propositions:
1. 3 > 1 (true).
2. 2 < 4 (true).
3. 4 = 7 (false)
However the following are a not propositions:
1. x is an even number.
2. what is your name?.
3. How old are you?.
4. Close the door.
5. Wow! Wonderful statue.

Deﬁnition 2. (Atomic statements)
Declarative sentences which cannot be further split into simple sentences
are called atomic statements (also called primary statements or primitive
statements).
Example: p is a prime number
Deﬁnition 3. (Compound statements)
New statements can be formed from atomic statements through the use of
connectives such as ”and, but, or etc...” The resulting statement are called
molecular or compound (composite) statements.
Example: If p is a prime number then, the divisors are p and 1 itself

Deﬁnition 4. (truth value)
The truth or falsehood of a proposition is called its truth value.
Deﬁnition 5. (Truth Table)
A table, giving the truth values of a compound statement interms of its
component parts, is called a Truth Table.

Deﬁnition 6. (Connectives)
Connectives are used for making compound propositions. The main ones
are the following (p and q represent given two propositions):
Table 1. Logic Connectives
Name Represented Meaning
Negation ¬p not in p
Conjunction p ∧ q p and q
Disjunction p ∨ q p or q (or both)
Implication p → q if p then q
Biconditional p ↔ q p if and only if q

The truth value of a compound proposition depends only on the value of
its components. Writing F for false and T for true, we can summarize the
meaning of the connectives in the following way:
p q ¬p p ∧ q p ∨ q p → q p ↔ q
T T F T T T T
T F F F T F F
F T T F T T F
F F T F F T T

Definition 7. (Tautology)
A proposition is said to be a tautology if its truth value is T for any
assignment of truth values to its components.
Example: The proposition p ∨ ¬p is a tautology.
Definition 8. (Contradiction)
A proposition is said to be a contradiction if its truth value is F for any
assignment of truth values to its components.
Example: The proposition p ∧ ¬p is a contradiction.
Definition 9.(Contingency)
A proposition that is neither a tautology nor a contradiction is called a
contingency.

p ¬p p ∧ ¬p p ∨ ¬p
T F F T
T F F T
p q p ∨ q ¬p (p ∨ q) ∨ (¬p)
T T T F T
T F T F T
F T T T T
F F F T T

I. Construct the truth table for the following statements
(i) (p → q) ←→ (¬p ∨ q)
(ii) p ∧ (p ∨ q)
(iii) (p → q) → p
(iv) ¬ (p ∧ q) ←→ (¬ p ∨ ¬ q)
(v) (p ∨ ¬q) → q

Solutions
(i) Let S = (p → q) ←→ (¬p ∨ q)
p q ¬p p → q ¬p ∨ q S
T T F T T T
T F F F F T
F T T T T T
F F T T T T

(ii) Let S = p ∧ (p ∨ q)
p q p ∨ q S
T T T T
T F T T
F T T F
F F F F

(iii) Let S = (p → q) → p
p q p → q S
T T T T
T F F T
F T T F
F F T F

(iv) Let S = ¬ (p ∧ q) ←→ ¬ p ∨ ¬ q
p q p ∧ q ¬ (p ∧ q) ¬p ¬q ¬ p ∨ ¬ q S
T T T F F F F T
T F F T F T T T
F T F T T F T T
F F F T T T T T

(v) Let S = (p ∨ ¬q) → q
p q ¬q p ∨ ¬q S
T T F T T
T F T T F
F T F F T
F F T T F

Implications
S.No Formula Name
1 p ∧ q ⇒ p simpliﬁcation
p ∧ q ⇒ q
2 p ⇒ p ∨ q addition
q ⇒ p ∨ q
3 p, q ⇒ p ∧ q
4 p, p → q ⇒ q modus ponens
5 ¬p, p ∨ q ⇒ q disjunctive syllogism
6 ¬q, p → q ⇒ ¬p modus tollens
7 p → q , q → r ⇒ p → r hypothetical syllogism

Logical Equivalence
The compound propositions p → q and ¬p ∨ q have the same truth
values:
p q ¬p p → q ¬p ∨ q
T T F T T
T F F F F
F T T T T
F F T T T

When two compound propositions have the same truth value they are
called logically equivalent.
For instance p → q and ¬p ∨ q are logically equivalent, and it is denoted
by
p → q ⇔ ¬p ∨ q

Deﬁnition 10. (Logically Equivalent)
Two propositions A and B are logically equivalent precisely when A ⇔ B
is a tautology.
Example: The following propositions are logically equivalent:
p ↔ q ⇔ (p → q) ∧ (q → p)
p q p ↔ q p → q q → p (p → q) ∧ (q → p) S
T T T T T T T
T F F F T F T
F T F T F F T
F F T T T T T

Table 2. Logic equivalences
Equivalences Name
p ∧ T ⇔ p Identity law
p ∨ F ⇔ p
p ∨ T ⇔ T Dominent law
p ∧ F ⇔ F
p ∨ p ⇔ p Idempotent law
p ∧ p ⇔ p
p ∨ q ⇔ q ∨ p Commutative law
p ∧ q ⇔ q ∧ p
(p ∨ q) ∨ r ⇔ p ∨ (q ∨ r) Associative law
(p ∧ q) ∧ r ⇔ p ∧ (q ∧ r)

Table 2. Logic equivalences (Continued...)
Equivalences Name
(p ∨ q) ∧ r ⇔ (p ∧ r) ∨ (q ∧ r) Distributive law
(p ∧ q) ∨ r ⇔ (p ∨ r) ∧ (q ∨ r)
(p ∨ q) ∧ p ⇔ p Absorbtion law
(p ∧ q) ∨ p ⇔ p
¬ (p ∧ q) ⇔ ¬ p ∨ ¬ q De morgan’s law
¬ (p ∨ q) ⇔ ¬ p ∧ ¬ q
¬ p ∧ p ⇔ F Negation law
¬ p ∨ p ⇔ T
¬ (¬ p ) ⇔ p

Table 3. Logic equivalences involving implications
Implications
p → q ⇔ ¬p ∨ q
p → q ⇔ ¬q → ¬p
¬(p → q) ⇔ p ∧ ¬q
p ∨ q ⇔ ¬p → q
p ∧ q ⇔ ¬(p → ¬q)
(p → q) ∧ (p → r) ⇔ p → (q ∧ r)
(p → q) ∨ (p → r) ⇔ p → (q ∨ r)
(p → r) ∧ (q → r) ⇔ (p ∨ q) → r)
(p → r) ∨ (q → r) ⇔ (p ∧ q) → r)

Table 4. Logic equivalences involving Biconditions
Biconditions
p ↔ q ⇔ (p → q) ∧ (q → p)
p ↔ q ⇔ ¬p ↔ ¬q
p ↔ q ⇔ (p ∧ q) ∨ (¬p ∧ ¬q)
¬(p ↔ q) ⇔ p ↔ ¬q

Converse, Inverse, Contrapositive
Definition 11. (Converse)
The converse of a conditional proposition p → q is the proposition q → p
Definition 12. (Inverse)
The inverse of a conditional proposition p → q is the proposition
¬ p → ¬ q
Definition 13. (Contrapositive)
The contrapositive of a conditional proposition p → q is the proposition
¬ q → ¬ p.

Example 1
Let us consider the statement,
”The crops will be destroyed, if there is a flood.”
Let F : there is a flood & C : The crops will be destroyed
The symbolic form is, F → C.
Converse (C→F)
i.e., ”if the crops will be destroyed then there is flood.”
Inverse (¬F→¬C)
i.e.,”if there is no flood then the crops won’t be destroyed, .”
Contrapositive (¬C→¬F)
i.e., ”if the crops won’t be destroyed then there is no flood.”

Example 2
If two angles are congruent, then they have the same measure.
Converse If two angles have the same measure, then they are congruent.
Inverse If two angles are not congruent, then they do not have the same
measure.
Contrapositive If two angles do not have the same measure, then they are
not congruent.
Example 3
It rains,They cancel school
Converse If they cancel school, then it rains.
Inverse If it does not rain, then they do not cancel school
Contrapositive If they do not cancel school, then it does not rain.

Inference Theory for Statement Calculus
An argument is a sequence of propositions H1, H2, . . . , Hn called premises
(or hypotheses) followed by a proposition C called conclusion. An
argument is usually written:
H1
H2
...
Hn
implies C
or
H1∧H2∧ . . . ∧Hn ⇒ C
The argument is valid if C is true whenever H1, H2, ..., Hn are true;
otherwise it is invalid.

Example: H1 : p and H2 : p → q then C : q (Modus Ponens)
p q p → q p ∧ (p → q) (p ∧ (p → q)) → q
T T T T T
T F F F T
F T T F T
F F T F T
Notice that (p ∧ (p → q)) → q is a tautology. Therefore it is valid.
Thus p, p → q ⇒ q.

Example: H1 : q and H2 : p → q then C : p
p q p → q q ∧ (p → q) (q ∧ (p → q)) → p
T T T T T
T F F F T
F T T T F
F F T F T
Notice that q ∧ (p → q) → p is a condigency. Therefore it is invalid.

Implication Table:
S.No Formula Name
1 p ∧ q ⇒ p simpliﬁcation
p ∧ q ⇒ q
2 p ⇒ p ∨ q addition
q ⇒ p ∨ q
3 p, q ⇒ p ∧ q
4 p, p → q ⇒ q modus ponens
5 ¬p, p ∨ q ⇒ q disjunctive syllogism
6 ¬q, p → q ⇒ ¬p modus tollens
7 p → q , q → r ⇒ p → r

Rules of inference:
Rule P: A premises can be introduced at any step of derivation.
Rule T: A formula can be introduced provided it is Tautologically implied
by previously introduced formulas in the derivation.
Rule CP: If the conclusion is of the form r → s then we include r as an
additional premises and derive s.
Indirect method: We use negation of the conclusion as an additional
premise and try to arrive a contradiction.
Inconsistent: A set of premises are inconsistent provided their conjunction
implies a contradiction.

Example: Show that ¬q and p → q implies ¬p.
Solution: (A formal proof is as follows):
Step 1. p → q Rule P
Step 2. ¬q → ¬p Rule T
Step 3. ¬q Rule P
Step 4. ¬p Combined {2, 3} and apply Modus Ponens

Example: Show that r is a valid inference from the premises p → q, q → r
and p.
Solution:
Step Derivation Rule
1 p P
2 p → q P
3 q {1, 2}, I4
4 q → r P
5 r {3, 4}, I4

Example: Show that s ∨ r is tautologically implied by p ∨ q, p → r and
q → s.
Solution:
1 p ∨ q P
2 ¬p → q T
3 q → s P
4 ¬p → s {2, 3}, I7
5 ¬s → p T
6 p → r P
7 ¬s → r {5, 6}, I7
8 s ∨ r T

Example: Prove by indirect method that p → q, p ∨ r, ¬q implies r.
Solution: The desired result is r. Include ¬r as a new premise.
1 p ∨ r P
2 ¬r → p T
3 ¬r P(additional premise)
4 p {2, 3}, I4
5 p → q P
6 q {4, 5}, I4
7 ¬q P
8 q ∧ ¬q {6, 7}, I3
The new premise together with the given premises, leads to a
contradiction. Thus p → q, p ∨ r, ¬q implies r.

Example: Prove that p → q, p → r, q → ¬r and p are inconsistent.
Solution: The desired result is false.
1 p P
2 p → q P
3 q {1, 2}, I4
4 q → ¬r P
5 ¬r {3, 4}, I4
6 p → r P
7 ¬p {5, 6}, I6
8 F {1, 7},I3

III. Problems:
(i) Show that r ∨ s is tautologically implied by
c ∨ d, (c ∨ d) → ¬h, ¬h → (a ∧ ¬b) and (a ∧ ¬b) → (r ∨ s).
(ii) Show that r ∧ (p ∨ q) is tautologically implied by p ∨ q, q → r, p → m
and ¬m.
(iii) Show that r → s is tautologically implied by ¬r ∨ p, p → (q → s) and
q.
(iv) Show that p → s is tautologically implied by ¬p ∨ q, ¬q ∨ r and
r → s.

(v) Show that p → (q → s) is tautologically implied by p → (q → r) and
q → (r → s) using CP rule.
(vi) Show that the following premises are inconsistent.
v → l, l → b, m → ¬b and v ∧ m.
(vii) Show that p → ¬s logically follows from the premises
p → (q ∨ r), q → ¬p, s → ¬r and p by indirect method.
(viii) Show that r logically follows from the premises p → q, ¬q and p ∨ r
by indirect method.

Example: Consider the following statements: ’I take the bus or I walk. If I
walk I get tired. I do not get tired. Therefore I take the bus.’ We can
formalize this by calling B= I take the bus, W = I walk and J= I get tired.
The premises are B ∨ W , W → J and ¬J, and the conclusion is B. The
argument can be described in the following steps:
step statement reason
1 W → J P
2 ¬J P
3 ¬W {1, 2}, Modus Tollens
4 B ∨ W P
5 B {3, 4}, Disjunctive Syllogism

(i) Show that the following set of premises is 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.

Let us consider,
E : Jack misses many classes through illness
S : Jack fails high school
A : Jack reads a lot of books
H : Jack is uneducated.
The premises are,

Let us consider,
The premises are,
E → S

Let us consider,
The premises are,
E → S
S → H

Let us consider,
The premises are,
E → S
S → H
A → ¬H and

Let us consider,
The premises are,
E → S
S → H
A → ¬H and
E ∧ A

1 E ∧ A P
2 E T
3 A T
4 E → S P
5 S {2, 4}, I4
6 S → H P
7 H {5, 6}, I4
8 A → ¬H P
9 ¬H {3, 8}, I4
10 H ∧ ¬H {7, 9}, I3

For further queries
Prof.Dr.D.Ezhilmaran, Ph.D.,
Assistant Professor (Senior),
Department of Mathematics,
Vellore Institute of Technology, Tamilnadu, India.
E-mail.ID : ezhilmaran.d@vit.ac.in

Module - 1 Discrete Mathematics and Graph Theory

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