Logic and proof
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This document introduces basic concepts in propositional logic, including: 1. Propositions are declarative statements that are either true or false. Compound propositions consist of simple propositions connected by logical operators like AND and OR. 2. Truth tables define logical connectives like conjunction, disjunction, conditional, biconditional, and negation. Equivalences between statements can be shown through truth tables. 3. Logical implications can be proven without truth tables by showing that if the antecedent is true, the consequent must also be true. Dual statements and De Morgan's laws are also introduced.
![Unit I
LOGIC AND PROOFS
1.1 INTRODUCTION
PROPOSITION (OR) STATEMENT:
Proposition is a declarative statement that is either true or false but not both. The truth value of
proposition is true or false.
Truth table
It displays the relationship between the truth values of proposition.
Negation of a proposition
If P is a proposition, then its negation is denoted by ¬P or ~p and is defined by the following truth
table.
EXAMPLE
P - Ram is intelligent
¬P -Ram is not intelligent
proposition is a declarative sentence which is either true or false but not both.
COMPOUND PROPOSITION
It is a proposition consisting of two or more simple proposition using logical operators.
1.2 LOGICAL CONNECTIVES
(1) DISJUNCTION (OR)
The disjunction of two proposition P and Q is the proposition P˅Q [read as P or Q ] and is
defined by the following truth table.
P Q P˅Q
P ¬P
T F
F T](https://image.slidesharecdn.com/logicandproof-200726210310/85/Logic-and-proof-1-320.jpg)
![p: you will be work hard.
q: you will be rewarded.
¬p: You will not be work hard.
¬ q: You will no tbe rewarded.
Converse: q→ p, If you will be rewarded then you will be work hard
Contrapositive: ¬ q→ p,if You will not be rewarded then You will not be work hard.
Inverse: ¬ p→ ¬ q, if You will not be work hard then You will no tbe rewarded.
Example 4.Prove that 𝑷 → 𝑸 ˄ 𝑸 → 𝑹 → 𝑷 → 𝑹
Proof:
Let S: 𝑃 → 𝑄 ˄ 𝑄 → 𝑅 → 𝑃 → 𝑅
To prove: S is a tautology
The last column shows that S is a tautology
1.3 PROPOSITIONAL EQUIVALENCE:
Logical Equivalence:
Let p and q be two statements formulas, p is said to be logically equivalent to q if p & q have the
same set of truth values or equivalently p & q are logically equivalent if 𝑝 ↔ 𝑞 is tautology.
Hence, 𝑝 ⇔ 𝑞 if and only if 𝑝 ↔ 𝑞 is a tautology.
Logical Implication or Tautological Implication
A statement formula A logically implies another, statement formula B if and only if 𝐴 → 𝐵 is a
tautology.
∴ 𝐴 ⇒ 𝐵 [A logically iff 𝐴 → 𝐵 is tautology, implies B]
If 𝐴 ⇒ 𝐵 , then
A is called antecedent
P Q R 𝑃 → 𝑄 𝑄 → 𝑅 𝑃 → 𝑅 𝑃 → 𝑄 ˄ 𝑄 → 𝑅 S
T T T T T T T T
T T F T F F F T
T F T F T T F T
T F F F T T T T
F T T T T T T T
F T F T F T F T
F F T T T T T T
F F F T T T T T](https://image.slidesharecdn.com/logicandproof-200726210310/85/Logic-and-proof-7-320.jpg)
![≡￢(￢p∧￢q) ∨(q∧￢q ) ∨ ((￢p∧p) ∨(q∧p)
≡￢(￢p∨q) ∨F∨F∨(q∧p)
≡￢(￢p∨q) ∨(q∧p)
≡(p∨q) ∧ (q∧p).
Consider ( P Q ) ( P R ) ( P Q ) ( P R ) (( P Q ) ( P R )) ( 2 )
Using (1) and (2)
(( P Q ) ( P Q ) ( P R )) (( P Q ) ( P R ))
[( P Q ) ( P R )] [( P Q ) ( P R )] T
Prove the following equivalences by proving the equivalences of the dual
(( P Q ) ( P Q )) ( P Q ) P
Solution: It‟s dual is
(( P Q ) ( P Q )) ( P Q ) P
Consider,
(( P Q ) ( P Q )) ( P Q ) P Reasons
(( P Q ) ( P Q )) ( P Q )
((Q P ) ( Q P )) ( P Q )
((Q Q ) P ) ( P Q )
(T P ) ( P Q )
P ( P Q )
P
(Demorgan‟s law)
(Commutative law)
(Distributive law)
( P P T )
( P T P )
(Absorption law)
Obtain DNF of Q ( P R ) (( P R ) Q ) .
Solution:
Q ( P R ) (( P R ) Q )
(Q ( P R )) ( (( P R ) Q ) (D e m o r g a n la w )](https://image.slidesharecdn.com/logicandproof-200726210310/85/Logic-and-proof-9-320.jpg)
Logic and proof
- 1. Unit I LOGIC AND PROOFS 1.1 INTRODUCTION PROPOSITION (OR) STATEMENT: Proposition is a declarative statement that is either true or false but not both. The truth value of proposition is true or false. Truth table It displays the relationship between the truth values of proposition. Negation of a proposition If P is a proposition, then its negation is denoted by ¬P or ~p and is defined by the following truth table. EXAMPLE P - Ram is intelligent ¬P -Ram is not intelligent proposition is a declarative sentence which is either true or false but not both. COMPOUND PROPOSITION It is a proposition consisting of two or more simple proposition using logical operators. 1.2 LOGICAL CONNECTIVES (1) DISJUNCTION (OR) The disjunction of two proposition P and Q is the proposition P˅Q [read as P or Q ] and is defined by the following truth table. P Q P˅Q P ¬P T F F T
- 2. T T T T F T F T T F F F (1) CONJUNCTION (AND) If P and Q are two propositions , then the conjunction of P and Q is denoted by P˄Q ( read as P and Q ) and is defined by following truth table. P Q P˄Q T T T T F F F T F F F F CONDITIONAL AND BI- CONDITIONAL PROPOSITION (1) Conditional proposition If p and q are propositions, then the implication “If p then q “ denoted by p→q , called the conditional statement of p and q , is defined by following truth table. p q p→q T T T T F F F T T F F T NOTE p→q is false when p is true and q is false. Otherwise it is true. The different situations where the conditional statements applied are listed below. (1) If p then q (2) p only if q (3) q whenever p (4) q is necessary for p (5) q follows from p (6) q when p (7) p is sufficient for q (8) p implies q Converse, contrapositive and Inverse statement If p→q is a conditional statement, then (1) q→p is called converse of p→q (2) ¬q→¬p is called contrapositive of p→q (3) ¬p→¬q is called inverse of p→q EXAMPLE p : Ram is a computer science student q : Ram study DBMS
- 3. p→q: If Ram is a computer science student, then the will study DBMS. (2) Bi-conditional proposition If p and q are proposition, then the proposition p if and only if q, denoted by 𝑝 ↔ 𝑞is called the bi-conditional statement and is defined by the following truth table. p q T T T T F F F T F F F T NOTE 𝑃 ↔ 𝑄is true if both p and q have same truth values. Otherwise 𝑃 ↔ 𝑄 is false. EXAMPLE P: You can take the flight q: You buy a ticket p↔q: You can take the flight if and only if buy a ticket. Symbolize the statements using Logical Connectives Example: 1 The automated reply can be sent when the file system is full. P: The automated reply can be sent Q: The file system is full Solution: Symbolic form :q→¬ p EXAMPLE: 2 Write the symbolized form of the statement. If either Ram takes C++ or Kumar takes pascal, then Latha will take Lotus. R:Ram takes C++ K:Kumar takes Pascal L:Latha takes Lotus Solution:
- 4. Symbolic form: (R˅K)→L Example 3 Let p,q,r represent the following propositions, P:It is raining q: The sun is shining r: There are clouds in the sky Symbolize the following statements. (1) If it is raining, then there are clouds in the sky (2) If it is not raining, then the sun is not shining and there are clouds in the sky. (3) The sun is shining if and only if it is not raining. Solution: Symbolic form: (1) p → r (2) ¬p → (¬q˄r) (3) q ↔ ¬r Example: 4 Symbolize the following statements: (1) If the moon is out and it is not snowing, then Ram goes out for a walk. (2) If the moon is out, then if it is not snowing,Ram goes out for a walk. (3) It is not the case that Ram goes out for a walk if and only if it is not snowing or the moon is out. Solution: Let the propositions be, P: The moon is out Q: It is snowing R: Ram goes out for a walk. Symbolic form: (1) (p˄¬q) → r (2) p → (¬q → r) (3) ¬(r ↔ (¬q˅p)) Example: 5 Symbolize the following using the propositions.
- 5. P:I finish writing my computer program before lunch q: I shall play Tennis in afternoon. r: The sun is shining s: The boundary is low. (1) If the sun is shining, I shall play tennis in the afternoon. (2) Finishing the writing of my computer program before lunch is necessary for playing tennis in this afternoon. (3) Low boundary and sunshine are sufficient to play Tennis in this afternoon. Solution: Symbolic form: (1) r → q (2) q → p (3) (s˄r) → q Construction of Truth Tables EXAMPLE: 1 Show that the truth values of the formula 𝑃˄(𝑃 → 𝑄) → 𝑄 are independent of their components. Solution: The truth table for the formula is, 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 The truth values of the given formula are all true for every possible truth values of P and Q. Therefore, the truth value of the given formula is independent of their components. Example 1. Without constructing the truth table show that p→ (q → p) ≡ ￢p(p→ q) Solution p→ (q → p) ≡ p→ (￢q ∨p) ≡ ￢p∨ (￢q ∨p) ≡ ￢p∨ (p∨￢q)
- 6. ≡ (￢p∨p)∨￢q ≡ T∨￢q ≡ T. Example 2. Prove that p→ q is logically prove that (￢p∨q) Solution: p q 𝑝 → 𝑞 ¬𝑝 ˅ ∨ 𝑞 T T T T T F F F F T T T F F T T EXAMPLE: 2 Write the symbolized form of the statement. If either Ram takes C++ or Kumar takes pascal, then Latha will take Lotus. R:Ram takes C++ K:Kumar takes Pascal L:Latha takes Lotus Solution: Symbolic form: (R˅K)→L Tautology. A statement that is true for all possible values of its propositional variables is called a tautology universely valid formula or a logical truth. Example:1. Write the converse, inverse, contra positive of ‘If you work hard then you will be rewarded’ Solution: p: you will be work hard. q: you will be rewarded. ¬p: You will not be work hard. ¬ q: You will no tbe rewarded. Converse: q→ p, If you will be rewarded then you will be work hard Contrapositive: ¬ q→ p,if You will not be rewarded then You will not be work hard. Inverse: ¬ p→ ¬ q, if You will not be work hard then You will no tbe rewarded. Example:2. Write the converse, inverse, contra positive of ‘If you work hard then you will be rewarded’ Solution:
- 7. p: you will be work hard. q: you will be rewarded. ¬p: You will not be work hard. ¬ q: You will no tbe rewarded. Converse: q→ p, If you will be rewarded then you will be work hard Contrapositive: ¬ q→ p,if You will not be rewarded then You will not be work hard. Inverse: ¬ p→ ¬ q, if You will not be work hard then You will no tbe rewarded. Example 4.Prove that 𝑷 → 𝑸 ˄ 𝑸 → 𝑹 → 𝑷 → 𝑹 Proof: Let S: 𝑃 → 𝑄 ˄ 𝑄 → 𝑅 → 𝑃 → 𝑅 To prove: S is a tautology The last column shows that S is a tautology 1.3 PROPOSITIONAL EQUIVALENCE: Logical Equivalence: Let p and q be two statements formulas, p is said to be logically equivalent to q if p & q have the same set of truth values or equivalently p & q are logically equivalent if 𝑝 ↔ 𝑞 is tautology. Hence, 𝑝 ⇔ 𝑞 if and only if 𝑝 ↔ 𝑞 is a tautology. Logical Implication or Tautological Implication A statement formula A logically implies another, statement formula B if and only if 𝐴 → 𝐵 is a tautology. ∴ 𝐴 ⇒ 𝐵 [A logically iff 𝐴 → 𝐵 is tautology, implies B] If 𝐴 ⇒ 𝐵 , then A is called antecedent P Q R 𝑃 → 𝑄 𝑄 → 𝑅 𝑃 → 𝑅 𝑃 → 𝑄 ˄ 𝑄 → 𝑅 S T T T T T T T T T T F T F F F T T F T F T T F T T F F F T T T T F T T T T T T T F T F T F T F T F F T T T T T T F F F T T T T T
- 8. B is called consequent. Further 𝐴 ⇒ 𝐵 guarantees that B has the truth value T whenever A has the truth value T. ∴ In order to show any of the given implications, it is sufficient to show that an assignment of the truth value T to the antecedent of the given conditional leads to the truth value T for the consequent. 1. Prove without using truth table 𝑃 → 𝑄 ˄ ¬𝑄 ⇒ ¬𝑃 Proof: Antecedent: 𝑃 → 𝑄 ˄ ¬𝑄 Consequent: ¬𝑃 Assume that, the antecedent has the truth value T. ∴ ¬𝑄 And 𝑃 → 𝑄 both are true. ⇒ Truth value of Q is F and the truth value of P is also F. ∴ Consequent ¬𝑃 is true. ∴ The truth of the antecedent implies the truth of the consequent. ∴ 𝑃 → 𝑄 ˄ ¬𝑄 ⇒ ¬𝑃 Example:1Without constructing the truth table show that p→ (q → p) ≡ ￢p(p→ q) Solution p→ (q → p) ≡ p→ (￢q ∨p) ≡ ￢p∨ (￢q ∨p) ≡ ￢p∨ (p∨￢q) ≡ (￢p∨p)∨￢q ≡ T∨￢q ≡ T. Example 2:Show that ￢(p↔q) ≡ (p∨q) ∧￢(p∧q) without constructing the truth table Solution : ￢(p↔q) ≡ (p∨q) ∧￢(p∧q) ￢(p↔q) ≡ ￢(p→q) ∧ (q→p) ≡￢(￢p∨q) ∧ (￢q∨p) ≡￢(￢p∨q) ∧￢q) ∨((￢p∨q) ∧p)
- 9. ≡￢(￢p∧￢q) ∨(q∧￢q ) ∨ ((￢p∧p) ∨(q∧p) ≡￢(￢p∨q) ∨F∨F∨(q∧p) ≡￢(￢p∨q) ∨(q∧p) ≡(p∨q) ∧ (q∧p). Consider ( P Q ) ( P R ) ( P Q ) ( P R ) (( P Q ) ( P R )) ( 2 ) Using (1) and (2) (( P Q ) ( P Q ) ( P R )) (( P Q ) ( P R )) [( P Q ) ( P R )] [( P Q ) ( P R )] T Prove the following equivalences by proving the equivalences of the dual (( P Q ) ( P Q )) ( P Q ) P Solution: It‟s dual is (( P Q ) ( P Q )) ( P Q ) P Consider, (( P Q ) ( P Q )) ( P Q ) P Reasons (( P Q ) ( P Q )) ( P Q ) ((Q P ) ( Q P )) ( P Q ) ((Q Q ) P ) ( P Q ) (T P ) ( P Q ) P ( P Q ) P (Demorgan‟s law) (Commutative law) (Distributive law) ( P P T ) ( P T P ) (Absorption law) Obtain DNF of Q ( P R ) (( P R ) Q ) . Solution: Q ( P R ) (( P R ) Q ) (Q ( P R )) ( (( P R ) Q ) (D e m o r g a n la w )
- 10. (Q ( P R )) (( P R ) Q ) (D e m o rg a n la w ) (Q ( P R )) (Q Q ) (( P R ) P R ) (( P R ) Q ) (E x te n d e d d istrib u te d la w ) ( P Q R ) F ( F R R ) ( P Q R ) (N eg a tio n la w ) ( P Q R ) ( P Q R ) (N e g a tio n la w ) Obtain Pcnf and Pdnf of the formula ( P Q ) ( P Q ) Solution: Let S = ( P Q ) ( P Q ) P Q P Q P Q P Q S Minterm Maxterm T T F F F F T P Q T F F T T T T P Q F T T F T T T P Q F F T T T F F P Q PCNF: P Q and PDNF: ( P Q ) ( P Q ) ( P Q ) Obtain PDNF of P P Q P . Solution: P P Q P ~ ~ P ( P (~ Q P )) P ( P ~ Q ) ( P P ) (~ (~ P T ) ( P ~ Q ) ( P P ) P (Q ~ Q ) ( P ~ Q )) ( P (Q ~ Q )) (~ (~