Proposition Logic in Discrete Structure
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This document provides an overview of propositional logic: 1. It defines propositions as statements that can be either true or false, and propositional variables connected by logical connectives like AND and OR. 2. It explains the five main connectives - negation, conjunction, disjunction, implication, and biconditional - and provides their truth tables. 3. It discusses well-formed formulas, tautologies, contradictions, and contingencies. Equivalences between logical statements and duality principles are also covered. 4. Normal forms like CNF and DNF are defined, along with concepts like minterms and maxterms. Rules of inference for building logical arguments are outlined
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2. If P and Q are WFFs then P,Q,P Q,P Q,P → Q and P Q are also WFFs.
3. All WFFs are obtained by the above procedures applied a finite number of times.
For example, the following are WFFs
p, p q, p → q, p (q → r) (p q)→ r, (p → q)→(q → p)
Note: In order to avoid excessive use of parenthesis, we adopt an order of precedence for
logical Operators.
, , , → and
Tautologies
A Tautology is a formula which is always true for every value of its propositional variables.
Example − Prove [(A → B) A] → B is a tautology
The truth table is as follows −
A B A → B (A → B) A [(A → B) A] → B
True True True True True
True False False False True
False True True False True
False False True False True
As we can see every value of [(A → B) A] → B is “True”, it is a tautology.
Contradictions
A Contradiction is a formula which is always false for every value of its propositional
variables.
Example − Prove (A VB) [(¬A) (¬B)] is a contradiction
The truth table is as follows −](https://image.slidesharecdn.com/unit4dstl-230123080850-0f8e901c/85/Proposition-Logic-in-Discrete-Structure-4-320.jpg)
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A B A
B
¬A ¬B (¬A)
(¬B)
(A V B) [(¬A)
(¬B)]
True True True False False False False
True False True False True False False
False True True True False False False
False False False True True True False
As we can see every value of (A B) [(¬A) (¬B)] is “False”, it is a
Contradiction.
Contingency
A Contingency is a formula which has both some true and some false values for
every value of its propositional variables.
Example − Prove (A B) (¬A) a contingency
The truth table is as follows −
A B A B ¬A (A B) (¬A)
True True True False False
True False True False False
False True True True True
False False False True False
As we can see every value of (A B) (¬A) has both “True” and “False”, it
is a contingency.
Propositional Equivalences](https://image.slidesharecdn.com/unit4dstl-230123080850-0f8e901c/85/Proposition-Logic-in-Discrete-Structure-5-320.jpg)
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Two statements X and Y are logically equivalent if any of the following two conditions −
• The truth tables of each statement have the same truth values.
• The bi-conditional statement X Y is a tautology.
Example−Prove¬(A B)and[(¬A) (¬B)]are
equivalent Testing by 1st method (Matching truth
table)
A B A
B
¬ (A B) ¬A ¬B [(¬A) (¬B)]
True True True False False False False
True False True False False True False
False True True False True False False
False False False True True True True
Here, we can see the truth values of ¬ (A B) and [(¬A) (¬B)] are same, hence
the statements are equivalent.
Testing by 2nd method (Bi-conditionality)
A B ¬ (A
B)
[(¬A)
(¬B)]
[¬ (A B)] [(¬A)
(¬B)]
True True False False True
True False False False True
False True False False True
False False True True True
As [¬ (A B)] [(¬A) (¬B)] is a tautology, the statements are equivalent.](https://image.slidesharecdn.com/unit4dstl-230123080850-0f8e901c/85/Proposition-Logic-in-Discrete-Structure-6-320.jpg)
Proposition Logic in Discrete Structure
- 1. ` 1 Definition: Propositional Logic A proposition is a collection of declarative statements that has either a truth value "true” or a truth value "false". A propositional consists of propositional variables and connectives. We denote the propositional variables by capital letters (A, B, C etc). The connectives connect the propositional variables. Some examples of Propositions are given below: • "Man is Mortal", it returns truth value“TRUE” as T. • "12 + 9 = 3 − 2", it returns truth value “FALSE” as F. The following is not a Proposition • "A is less than 2". It is because unless we give a specific value of A, we cannot say whether the statement is true orfalse. Connectives In propositional logic generally we use five connectives which are OR (), AND (), Negation/ NOT (¬), Conditional or Implication / if-then (→), Bi conditional or If and only if (). Negation (¬) − The negation of a proposition A (written as ¬A) is false when A is true and is true when A is false. The truth table is as follows – A ¬A True False False True AND ( ) − The AND operation of two propositions A and B (written as A B) is true if both the propositional variable A and B is true.
- 2. ` 2 The truth table is as follows − A B A B True True False True False False False True False False False True OR ()− The OR operation of two propositions A and B (written as A B) is true if at least any of the propositional variable A or B is true. The truth table is as follows − A B A B T T T T F T F T T F F F Implication / if-then (→) − An implication A→B is False if A is true and B is false. The rest cases are true.
- 3. ` 3 The truth table is as follows − A B A → B True True True True False False False True True False False True If and only if ()− A B is bi-conditional logical connective which is true when p and q are both false or both are true. The truth table is as follows − A B A B True True True True False False False True False False False True Well Formed Formulas (WFFs) The well formed formulas (WFFs) or statement formulas or logic formulas are defined recursively (or inductively) as below. 1. Propositional variables p,q,r,… and propositional constants F,T are well formed formulas. They are known as primitive WFFs.
- 4. ` 4 2. If P and Q are WFFs then P,Q,P Q,P Q,P → Q and P Q are also WFFs. 3. All WFFs are obtained by the above procedures applied a finite number of times. For example, the following are WFFs p, p q, p → q, p (q → r) (p q)→ r, (p → q)→(q → p) Note: In order to avoid excessive use of parenthesis, we adopt an order of precedence for logical Operators. , , , → and Tautologies A Tautology is a formula which is always true for every value of its propositional variables. Example − Prove [(A → B) A] → B is a tautology The truth table is as follows − A B A → B (A → B) A [(A → B) A] → B True True True True True True False False False True False True True False True False False True False True As we can see every value of [(A → B) A] → B is “True”, it is a tautology. Contradictions A Contradiction is a formula which is always false for every value of its propositional variables. Example − Prove (A VB) [(¬A) (¬B)] is a contradiction The truth table is as follows −
- 5. ` 5 A B A B ¬A ¬B (¬A) (¬B) (A V B) [(¬A) (¬B)] True True True False False False False True False True False True False False False True True True False False False False False False True True True False As we can see every value of (A B) [(¬A) (¬B)] is “False”, it is a Contradiction. Contingency A Contingency is a formula which has both some true and some false values for every value of its propositional variables. Example − Prove (A B) (¬A) a contingency The truth table is as follows − A B A B ¬A (A B) (¬A) True True True False False True False True False False False True True True True False False False True False As we can see every value of (A B) (¬A) has both “True” and “False”, it is a contingency. Propositional Equivalences
- 6. ` 6 Two statements X and Y are logically equivalent if any of the following two conditions − • The truth tables of each statement have the same truth values. • The bi-conditional statement X Y is a tautology. Example−Prove¬(A B)and[(¬A) (¬B)]are equivalent Testing by 1st method (Matching truth table) A B A B ¬ (A B) ¬A ¬B [(¬A) (¬B)] True True True False False False False True False True False False True False False True True False True False False False False False True True True True Here, we can see the truth values of ¬ (A B) and [(¬A) (¬B)] are same, hence the statements are equivalent. Testing by 2nd method (Bi-conditionality) A B ¬ (A B) [(¬A) (¬B)] [¬ (A B)] [(¬A) (¬B)] True True False False True True False False False True False True False False True False False True True True As [¬ (A B)] [(¬A) (¬B)] is a tautology, the statements are equivalent.
- 7. ` 8 Laws of Propositional Logic: S.No Name of Laws Primal Form Dual Form 1 Idempotent Law p p p p p p 2 Identity Law p F p p T p 3 Dominant Law p T T p F F 4 Complement Law p p T p p F 5 Commutative Law p q q p p q q p 6 Associative Law p (q r) (p q) r p (q r) (p q) r 7 Distributive Law p(qr) (p q)(p r) p(q r) (p q)(p r) 8 Absorption Law p (p q) p p (p q) p 9 De Morgan’s Law (p q) p q (p q) p q 10 Double Negation Law (p) p - Logical Equivalences involving Conditional Statements Logical Equivalences involving Biconditional Statements Inverse, Converse, and Contra-positive A conditional statement has two parts − Hypothesis and Conclusion. Example of Conditional Statement − “If you do your homework, you will not be punished.” Here, "you do your homework" is the hypothesis and "you will
- 8. ` 9 not be punished" is the conclusion. Inverse –An inverse of the conditional statement is the negation of both the hypothesis and the conclusion. If the statement is “If p, then q”, the inverse will be “If not p, then not q”. The inverse of “If you do your homework, you will not be punished” is “If you do not do your homework, you will be punished.” Converse−The converse of the conditional statement is computed by interchanging the hypothesis and the conclusion. If the statement is “If p, then q”, the inverse will be “If q, Then p”. The converse of "If you do your homework, you will not be punished" is "If you will not be punished, you do not do your homework”. Contra-positive –The contra-positive of the conditional is computed by interchanging the hypothesis and the conclusion of the inverse statement. If the statement is “If p, then q”, the inverse will be “If not q, then not p”. The Contra- positive of "If you do your homework, you will not be punished” is "If you will be punished, you do your home work”. Duality Principle Duality principle set states that for any true statement, the dual statement obtained by interchanging unions into intersections (and vice versa) and interchanging Universal set into Null set (and vice versa) is also true. If dual of any statement is the statement itself, it is said self-dual statement.
- 10. ` 11 Elementary Product: A product of the variables and their negations in a formula is called an elementary product. If p and q are any two atomic variables, then p, p q, q p, p q are some examples of elementary products. Elementary Sum: A sum of the variables and their negations in a formula is called an elementary sum. If P and Q are any two atomic variables, then p, p q, q p, p q are some examples of elementary sums. Normal Forms: We can convert any proposition in two normal forms − 1. Conjunctive Normal Form (CNF) 2.Disjunctive Normal Form (DNF) Conjunctive Normal Form A compound statement is in conjunctive normal form if it is obtained by operating AND among variables (negation of variables included) connected with ORs. Examples: 1. (p q) (q r) 2. (p q r) (s r) Disjunctive Normal Form A compound statement is in disjunctive normal form if it is obtained by operating OR among variables (negation of variables included) connected with ANDs. Example: (p q) (p q) (p q r) Functionally Complete set A set of logical operators is called functionally complete if every compound proposition is logically equivalent to a compound proposition involving only this set of logical operators. ,, form a functionally complete set of operators. Minterms: For two variables p and q there are 4 possible formulas which consist of conjunctions of p,q or it’s negation given by p q, p q, p q, p q
- 11. ` 12 Maxterms: For two variables p and q there are 4 possible formulas which consist of disjunctions of p,q or its negation given by p q, p q, p q, p q Principal Disjunctive Normal Form: For a given formula an equivalent formula consisting of disjunctions of minterms only is known as principal disjunctive normal form (PDNF). Principal Conjunctive Normal Form: For a given formula an equivalent formula consisting of conjunctions of maxterms only is known as principal conjunctive normal form (PCNF). Problems:
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- 13. ` 14 Rule of inference to build arguments: RuleP: A premise may be introduced at point in the derivation Rule T: A formula S may be introduced in a derivation if S is a tautologically implied by any one or more of the preceeding formulas in the derivation. Rule CP: If we can derive S from R and a set of premises, then we can derive R →S from the set of premises alone. Rule CP is also called deduction theoem.
- 14. ` 15 Examples:
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- 16. ` 17 Quantifiers: The variable of predicates is quantified by quantifiers. There are two types of quantifier in Predicate logic − Universal Quantifier and Existential Quantifier. Universal Quantifier: Universal quantifier states that the statements within its scope are true for every value of the specific variable. It is denoted by the symbol ∀. ∀x P(x) is read as for every value of x, P(x) is true. Example: "Man is mortal" can be transformed into the propositional form ∀x P(x) where P(x) is the predicate which denotes x is mortal and the universe of discourse is all men. Existential Quantifier: Existential quantifier states that the statements within its scope are true for some values of the specific variable. It is denoted by the symbol ∃.∃x P(x) is read as for some values of x, P(x) is true.
- 17. ` 18 Example: "Some people are dishonest" can be transformed into the propositional form ∃x P(x) where P(x) is the predicate which denotes x is dishonest and the universe of discourse is some people. Nested Quantifiers: If we use a quantifier that appears within the scope of another quantifier, it is called nested quantifier.
- 18. ` 19 Symbolize the following statements: (a)All men are mortal (b)All the world loves alover (c) X is the father of mother of Y (d)No cats has atail (e) Some people who trust others are rewarded Solution:
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