Unit-I
Mathematical logic
by
P. Swapna
Sreyas Institute of Engineering and
Technology
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1
Proposition or statement : sentence either it is true or
false. Denoted by A,B,C…P,Q,R,S,…..
ex: India is a country
1+101=110
Close the door
1. Primary or simple or automic statements
2. Compound statements : Using logical connective in
between two sentences.
p
¬p
 Negation(¬or~): not in the sentence.
T
F
Ex : p:I went to college yesterday.
F
T
¬p : I did not go to college yesterday.

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Logical connectives : not, and ,or ,“if….then”, “if
and only if”.
 Conjunction (∧) : using “and”-true only if both are
true.
 Disjunction (v) : using “or”-true if anyone is true.
 Exclusive Disjunction ( ⊻ ) : “exclusive or”-true
only when p is true or q is true but not both.
 Conditional (→) : “if….then” –false only when p
is true and q is false.
 Bi conditional (↔) : “if and only if” or (p → q) ∧
(q → p) – true only if both are true or both are
false.

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EX: p: 2 is an integer q: 9 is multiple of 3
¬p , p ∧ q , p ∨ q , p → q , P ↔ q , P ⊻ q
Truth table:

p
q
¬p
p∧q
p∨q
p→q
P↔q
P⊻q
T
T
F
T
T
T
T
F
T
F
F
F
T
F
F
T
F
T
T
F
T
T
F
T
F
F
T
F
F
T
T
F

Statement formulas: any formula
Ex: ¬p , p ∧ q , p ∨ q , p ∧ q → r ,p ∧ p ∨ q
 Well formed formulas : Cannot be interpreted in
more than one way
(p ∧ q) → r
p∧q→r
Ex:
Mathematical logic
p ∧ (q → r)
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


Tautology (To) : True for all
possible truth values
Contradiction (Fo) : False for all
possible truth values
P
¬p pv¬p
T
F
T
F
T
T
p
¬p
p∧¬p
T
F
F
F
T
F
Contingency : A compound proposition that can be
true or false
p
q
Pvq
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T
T
T
T
F
T
F
T
T
F
F
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F
 Construct
truth table for the following
 ¬(p ∧ q) v ¬ (q ↔ p)
p
q
p∧q
¬(p ∧ q)
q↔p
¬ (q ↔ p)
¬(p ∧ q) v
¬ (q ↔ p)
T
T
T
F
T
F
F
T
F
F
T
F
T
T
F
T
F
T
F
T
T
F
F
F
T
T
F
T
(p v q ) v r
 p→ (q→ ¬r)
 [(p ∧ q) v ¬r] ↔ p
 q ↔(¬ p v ¬q)

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u and v have identical truth values /u↔v is a
tautology denoted by u⇔v
 Example:
 Prove that (p → q) ⇔ ((¬p)vq)
Soln:

p
q
(p → q)
T
T
T
F
T
T
F
F
F
F
F
T
T
T
T
F
F
T
T
T
¬p
(¬p) v q
(p → q) and (¬p) v q have same truth values so
logically equivalent
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Law of Double negation:
¬ ( ¬p ) ⇔ p
 Idempotent Laws :
(p v p) ⇔ p and (p ∧ p) ⇔ p
 Identity Laws :
( p v Fo) ⇔ p and (p ∧ To ) ⇔ p
 Inverse Laws :
( p v ¬p) ⇔ To and ( p ∧ ¬p) ⇔ Fo
 Domination Laws :
( p v To) ⇔ To and (p ∧ Fo ) ⇔ Fo

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Commutative Laws :
(p v q) ⇔ (q v p) and (p ∧ q) ⇔ (q ∧ p)
 Absorption Laws :
[p v (p ∧ q) ] ⇔ p and [p ∧ (p v q) ] ⇔ p
 Demorgan Laws :
¬ (p v q) ⇔ ¬p ∧ ¬q and ¬ (p ∧ q) ⇔ ¬p v ¬q
 Associative Laws:
(p v (q v r)) ⇔ (p v q) v r and
(p ∧ (q ∧ r)) ⇔ (p ∧ q) ∧ r
 Disributive Laws :
(p v (q ∧ r)) ⇔ (p v q) ∧ (p v r)
(p ∧ (q v r)) ⇔ (p ∧ q) v (p ∧ r)

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 Simplify
the following compound propositions
using the laws of logic
(p v q) ∧ ¬ ((¬p) v q)
Soln: (p v q) ∧ ¬ ((¬p) v q)
⇔ (p v q) ∧ ( p ∧ ¬q) (Demorgan law)
⇔ ((p v q) ∧ p ) ∧ (¬q) (Associative law)
⇔ (p ∧ (p v q)) ∧ (¬q) (Commutative law)
⇔ p ∧ (¬q) (Absorption law)

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¬(¬((p v q) ∧ r) v ¬q)
Soln:¬(¬((p v q) ∧ r) v ¬q)
⇔ ¬ (¬ ((p v q) ∧ r) ∧ q) (Demorgan law)
⇔ ((p v q) ∧ r) ∧ q) (law of double negation)
⇔ (p v q) ∧ (r ∧ q) (Associative law)
⇔ (p v q) ∧ (q ∧ r) (commutative law)
⇔((p v q) ∧ q) ∧ r (Associative law)
⇔ (q ∧ r)
(Absorption law)

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 Prove
the following logical equivalences
without using truth tables
 P v [p ∧ (p v q) ] ⇔ p
Soln:p v [p ∧ (p v q) ]
⇔ p v p (Absorption law)
⇔p
(Idempotent law)
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v q v (¬p ∧¬q ∧ r)] ⇔ (p v q v r)
Soln: ¬p ∧¬q ∧ r
⇔ (¬p ∧¬q) ∧ r (Associative law)
⇔ ¬(p v q) ∧ r (Demorgan law)
Therefore [p v q v (¬p ∧¬q ∧ r)]
⇔ (p v q) v [¬(p v q) ∧ r ]
⇔ [(p v q) v ¬(p v q)] ∧ [(p v q) v r]
(Distributive law)
⇔ To ∧ [p v q v r] (inverse law and associative
law)
⇔ [p v q v r] ∧ To (Commutative law)
⇔ (p v q v r)
 [p
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
Tautological implication (⇒) :
A ⇒ B /A→ B is a tautology
Ex: Prove that (p → q) ⇒ (¬ q→ ¬ p)
Soln:
p
q
¬p
¬q
(p → q)
(¬ q→ ¬ p)
(p → q) →
(¬ q→ ¬ p)
T
T
F
F
T
T
T
T
F
F
T
F
F
T
F
T
T
F
T
T
T
F
F
T
T
T
T
T
Therefore (p → q) ⇒ (¬ q→ ¬ p)
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 Prove
that (¬ q ∧(p → q) ) ⇒ ¬ p
 ¬ (p → q) ⇒ p
 (p → ( q →r)) ⇒(( p →q) →( p →r))
 q ⇒ ( p →r)
 (p ∧ q) ⇒ ( p →q)
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

NAND (↑) : ‘not’ and ‘and’
(p ↑ q) = ¬ (p ∧ q) ⇔ ¬p v ¬q
NOR (↓) : ‘not’ and ‘or’
(p ↓q) = ¬ (p v q) ⇔ ¬p ∧ ¬q
p
q
p↑q
p ↓q
T
T
F
F
T
F
T
F
F
T
T
F
F
F
T
T
For any propositions p,q prove the following
 ¬ (p ↓q) ⇔ (¬p ↑ ¬q)
Soln: ¬ (p ↓q) ⇔ ¬(¬ (p v q) )
⇔¬(¬p ∧ ¬q)
⇔ (¬p ↑ ¬q)

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¬ (p ↑ q) ⇔ (¬p ↓ ¬q)
Soln: ¬ (p ↑ q) ⇔ ¬(¬ (p ∧ q) )
⇔ ¬(¬p v ¬q)
⇔ (¬p ↓ ¬q)
 For any propositions p,q,r prove the following
 p ↑( q ↑ r) ⇔ ¬p v (q ∧ r)
Soln: p ↑( q ↑ r) ⇔ ¬(p ∧ ( q ↑ r) )
⇔ ¬(p ∧ ¬ ( q ∧ r) )
⇔ ¬p v (q ∧ r)
 (p ↑ q) ↑ r ⇔ (p ∧ q) v ¬r
Soln: (p ↑ q) ↑ r ⇔ ¬((p ↑ q) ∧ r)
⇔¬(¬ (p ∧ q) ∧ r)
⇔ (p ∧ q) v ¬r
 (p ↓q) ↓ r ⇔ (p v q) ∧ ¬ r
 p ↓(q ↓ r) ⇔ ¬ p Mathematical
∧ (q v logic
r)

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 Elementary
product : product of variables and
their negations
Ex: p , ¬p , ¬ p ∧ q , ¬ q ∧ p ∧ ¬ p , p ∧ ¬ p
 Elementary sum : sum of variables and their
negations
Ex: p , ¬p , ¬ p v q , ¬ q v p v ¬ p , p v ¬ p
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Disjunctive Normal Forms (d.n.f) : sum of
elementary products
2. Conjunctive Nomal Form (c.n.f) : Product of
elementary sums
 Procedure to obtain d.n.f:
 Replace → , ↔ by ∧ , ∨ and ¬
(p → q) ⇔ ((¬p)vq)
p ↔ q ⇔ (p ∧ q) ∨ ¬ (p ∨ q) ⇔ (¬ p ∨ q) ∧ (¬ q
∨ p)
 Use Demorgans law
 Apply disrtibutive law
1.
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( p ∨ q) ∧ ( r ∨ s) ⇔ (( p ∨ q) ∧ r ) ∨(( p ∨ q) ∧ s)
⇔ ( p ∧ r) ∨ ( p ∧ s) ∨( q ∧ r) ∨ ( q ∧ s)

Ex: Obtain d.n.f of p ∧ (p → q)
Soln: p ∧ (p → q) ⇔ p ∧ (¬ p ∨ q)
⇔ (p ∧ ¬ p ) ∨ (p ∧ q)
Note: d.n.f and c.n.f are not unique.

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Minterms : conjunction in which each statement
variable or its negation , but not both , appears
only once.
Ex : p ∧ q , ¬p ∧ q, p ∧ ¬q, ¬p ∧ ¬q
 Maxterms : disjunctions in which each statement
variable ,or its negation , but not both , appears
only once.
 Principal Disjunctive Normal Form (P.D.N.F) :
sum of products canonical form -disjunctions of
minterms
 Principal Conjunctive Normal Form (P.C.N.F):
Product of sums canonical form –Conjunctions of
maxterms

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Procedure to obtain P.D.N.F or P.C.N.F:
1. Replace → , ↔ by ∧ , ∨ and ¬
2. Use Demorgans law followed by distibutive law
3. Elementary product which is contradiction is
dropped. Identical minterms appearing in the
disjunction are deleted.
 Quantifiers: ‘for all’ , ‘for every’ , ‘for each’ ,
‘for any’ , ‘for some’ , ‘there exists’
 Existential Quantifier: ‘for some’ , ‘there exists’
 Universal Quantifier : ‘for all’ , ‘for every’, ‘for
each’ , ‘for any’

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q → p: is converse of p → q
 ¬ p → ¬ q is inverse or opposite of p → q
 ¬ q → ¬ p is contrapositive of p → q
 Ex: p: 2 is an integer q: 9 is multiple of 3
1. p → q:if 2 is an integer then 9 is multiple of 3
2. q → p: if 9 is multiple of 3 then 2 is an integer q
→ p is converse of p → q
3. ¬ p → ¬ q : if 2 is not an integer then 9 is not
multiple of 3
4. ¬ q → ¬ p : if 9 is not multiple of 3 then 2 is not
an integer

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