Logic and Propositional Calculus

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Conditional and Biconditional statementLogical Equivalence

Algebra of PropositionArguments

Propositional function and Quantifier

Logic and Propositional Calculus

Math 301

Dr. Nahid Sultana

October 19, 2012

Math 301 Logic and Propositional Calculus

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IntroductionPropositionTruth TableCompound PropositionTautology and Contradictions

Conditional and Biconditional statement

Logical Equivalence

Algebra of Proposition

Arguments

Propositional function and QuantifierPropositional functionNegation of quantified statement


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PropositionTruth TableCompound PropositionTautology and Contradictions

I Definition: A statement (or proposition) is a declarativesentence that is either true or false but not both.

I Example:

1. The earth is round.2. 2 + 3 = 5.3. Do you speak English?4. 3− x = 5.5. Take two aspirins.

I 1. The earth is round. (statement)2. 2 + 3 = 5. (statement)3. Do you speak English? (not statement, question)4. 3− x = 5. (declarative sentence but not statement)5. Take two aspirins. (not statement, command)


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I Example:




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I Example:




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I Every statement has a truth value; true (denoted by T ) orfalse (denoted by F ).

I The possible truth values of a statement can be representedby a table, called truth table.

I

p

TF

q

TF

p q

T TT FF TF F

I In truth table involving three statements p, q, r , there are8 = 23 possible combinations of truth values.

I In general, a truth table involving n statements contains 2n

possible combinations of truth values.


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I

p

TF

q

TF

p q

T TT FF TF F





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I

p

TF

q

TF

p q

T TT FF TF F





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I

p

TF

q

TF

p q

T TT FF TF F





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I

p

TF

q

TF

p q

T TT FF TF F





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I In Mathematics, x , y , z , ... denotes variables, can be replacedby real numbers, can be combined with operations +,−,×,÷.

I In logic, p, q, r , ... denote propositional variables, can bereplaced by propositions, and can be combined by logicalconnectives (operations) to obtain compound proposition.

I Example:p: The sun is shining todayq: It is coldp and q: The sun is shining and it is cold.

I The truth value of a compound proposition depends on thetruth values of the propositions and the types of theconnective being used.


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I There are three basic logical operations:

1. Conjunction (and)2. Disjunction (or)3. Negation (not)

I Conjunction: If p and q are propositions thenI The conjunction of p and q is the compound proposition ”p

and q”.I Denoted by p ∧ q.I p ∧ q is true when both p and q are true otherwise false.I Truth table:

p q p ∧ qT T TT F FF T FF F F


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I There are three basic logical operations:

1. Conjunction (and)2. Disjunction (or)3. Negation (not)

I Conjunction: If p and q are propositions thenI The conjunction of p and q is the compound proposition ”p

and q”.I Denoted by p ∧ q.I p ∧ q is true when both p and q are true otherwise false.I Truth table:

p q p ∧ qT T TT F FF T FF F F


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I Disjunction: If p and q are two propositions thenI The disjunction of p and q is the compound proposition ”p or

q”.I Denoted by p ∨ q.I p ∨ q is true if at least one of p or q is true; it is false when

both p and q are false.I Truth table:

p q p ∨ qT T TT F TF T TF F F


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I Negation: If p is a proposition then

1. The negation of p is the proposition ”not p”.2. Denoted by ¬p or ∼ p.3. If p is true then ∼ p is false; if p is false then ∼ p is true.4. Truth table:

p ∼ pT FF T

I Example:p: 2 + 2 = 5q: 2 + 2 6= 5 (negation of p)r : It is not the case that 2 + 2 = 5 (negation of p)

I Here p is false, so q and r are true.


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p ∼ pT FF T




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p ∼ pT FF T




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I Truth table of a proposition with several connectives.

I Example:(p ∧ q) ∨ (∼ p)

I Order of priority or the logical connectives:

First ∼ then ∧ and then ∨


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I Truth table of a proposition with several connectives.

I Example:(p ∧ q) ∨ (∼ p)

I Order of priority or the logical connectives:

First ∼ then ∧ and then ∨


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I Consider two examples:

p ∨ (∼ p) and p ∧ (∼ p)

I Truth tables:

I Definition: A compound proposition that is always true for allpossible combination of truth values of the propositionalvariables is called a tautology.

I Definition: A compound proposition that is always false for allpossible combination of truth values of the propositionalvariables is called a contradiction.


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p ∨ (∼ p) and p ∧ (∼ p)

I Truth tables:




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p ∨ (∼ p) and p ∧ (∼ p)

I Truth tables:




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p ∨ (∼ p) and p ∧ (∼ p)

I Truth tables:




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I Implication (Conditional): For two statements p and qI The compound statement ”If p, then q” is called the

implication.I Denoted by p → q; read as ”p implies q”.I p is the hypothesis and q is the conclusion.I p → q is false when p is true and q is false, and is true

otherwise.I Truth table:

p q p → qT T TT F FF T TF F T


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I Biconditional: For two statements p and qI The compound statement ”p iff q” is called a biconditional

statement.I Denoted by p ↔ q.I p ↔ q is true only when p and q have the same truth values.I Truth table:

p q p ↔ qT T TT F FF T FF F T


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I Biconditional: For two statements p and qI The compound statement ”p iff q” is called a biconditional

statement.I Denoted by p ↔ q.I p ↔ q is true only when p and q have the same truth values.I Truth table:

p q p ↔ qT T TT F FF T FF F T


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I Two propositions (compound) P(p, q, ..) and Q(p, q, ..) aresaid to be logically equivalent if they haveidentical truth tables, denoted by

P(p, q, ..) ≡ Q(p, q, ..) .

I Example: ∼ (p ∧ q) and (∼ p) ∨ (∼ q).


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I Two propositions (compound) P(p, q, ..) and Q(p, q, ..) aresaid to be logically equivalent if they haveidentical truth tables, denoted by

P(p, q, ..) ≡ Q(p, q, ..) .

I Example: ∼ (p ∧ q) and (∼ p) ∨ (∼ q).


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I Commutative Properties:

p ∨ q ≡ q ∨ p ; p ∧ q ≡ q ∧ p

I Associative Properties:

p ∨ (q ∨ r) ≡ (p ∨ q) ∨ r ; p ∧ (q ∧ r) ≡ (p ∧ q) ∧ r

I Distributive properties:

p∨ (q∧ r) ≡ (p∨ q)∧ (p∨ r) ; p∧ (q∨ r) ≡ (p∧ q)∨ (p∧ r)

I Idempotent properties:

p ∨ p ≡ p ; p ∧ p ≡ p

I Properties of Negation:

∼ (∼ p) ≡ p ; ∼ (p∨q) ≡ (∼ p)∧(∼ q) ; ∼ (p∧q) ≡ (∼ p)∨(∼ q)


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p ∨ q ≡ q ∨ p ; p ∧ q ≡ q ∧ p


p ∨ (q ∨ r) ≡ (p ∨ q) ∨ r ; p ∧ (q ∧ r) ≡ (p ∧ q) ∧ r


p∨ (q∧ r) ≡ (p∨ q)∧ (p∨ r) ; p∧ (q∨ r) ≡ (p∧ q)∨ (p∧ r)


p ∨ p ≡ p ; p ∧ p ≡ p


∼ (∼ p) ≡ p ; ∼ (p∨q) ≡ (∼ p)∧(∼ q) ; ∼ (p∧q) ≡ (∼ p)∨(∼ q)


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p ∨ q ≡ q ∨ p ; p ∧ q ≡ q ∧ p


p ∨ (q ∨ r) ≡ (p ∨ q) ∨ r ; p ∧ (q ∧ r) ≡ (p ∧ q) ∧ r


p∨ (q∧ r) ≡ (p∨ q)∧ (p∨ r) ; p∧ (q∨ r) ≡ (p∧ q)∨ (p∧ r)


p ∨ p ≡ p ; p ∧ p ≡ p


∼ (∼ p) ≡ p ; ∼ (p∨q) ≡ (∼ p)∧(∼ q) ; ∼ (p∧q) ≡ (∼ p)∨(∼ q)


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p ∨ q ≡ q ∨ p ; p ∧ q ≡ q ∧ p


p ∨ (q ∨ r) ≡ (p ∨ q) ∨ r ; p ∧ (q ∧ r) ≡ (p ∧ q) ∧ r


p∨ (q∧ r) ≡ (p∨ q)∧ (p∨ r) ; p∧ (q∨ r) ≡ (p∧ q)∨ (p∧ r)


p ∨ p ≡ p ; p ∧ p ≡ p


∼ (∼ p) ≡ p ; ∼ (p∨q) ≡ (∼ p)∧(∼ q) ; ∼ (p∧q) ≡ (∼ p)∨(∼ q)


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p ∨ q ≡ q ∨ p ; p ∧ q ≡ q ∧ p


p ∨ (q ∨ r) ≡ (p ∨ q) ∨ r ; p ∧ (q ∧ r) ≡ (p ∧ q) ∧ r


p∨ (q∧ r) ≡ (p∨ q)∧ (p∨ r) ; p∧ (q∨ r) ≡ (p∧ q)∨ (p∧ r)


p ∨ p ≡ p ; p ∧ p ≡ p


∼ (∼ p) ≡ p ; ∼ (p∨q) ≡ (∼ p)∧(∼ q) ; ∼ (p∧q) ≡ (∼ p)∨(∼ q)


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I Definition: A logical argument is a finite set of propositions(premises or hypothesis) followed by a proposition(conclusion).

I Denoted byP1,P2, ....,Pn︸︷︷︸

premises

` Q︸︷︷︸conclusion

I Suppose the premises are all true, then conclusion may beeither true or false.When the conclusion is true then the argument is said to bevalid.When the conclusion is false then the argument is said to beinvalid or fallacy.


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premises




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premises




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I Show that the argument

p, p → q ` q

is valid.


p → q, q ` p

is invalid.


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I Theorem: The argument P1,P2, ....,Pn ` Q is valid iff theproposition (P∧P2 ∧ .... ∧ Pn)→ Q is a tautology.


p → q, q → r ` p → r

is valid.


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I Theorem: The argument P1,P2, ....,Pn ` Q is valid iff theproposition (P∧P2 ∧ .... ∧ Pn)→ Q is a tautology.


p → q, q → r ` p → r

is valid.


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Propositional functionNegation of quantified statement

I Suppose p(x) is an open statement over a domain D. Thenp(x) is a statement for each x ∈ D.

I Generate statement from open statement by the method ofquantification.

I Two types of quantification:1. Universal quantification2. Existential quantification

I The universal quantification of p(x) is the proposition

p(x) is true for all values of x ∈ D ,

it is denoted by ∀x ∈ D, p(x).I Here ∀ is called the universal quantifier.I Let p(x) : −(−x) = x . What is the truth value of∀x ∈ R, p(x)?

I Let q(x) : x + 1 < 4. What is the truth value of ∀x ∈ R, q(x)?


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it is denoted by ∀x ∈ D, p(x).

I Here ∀ is called the universal quantifier.I Let p(x) : −(−x) = x . What is the truth value of∀x ∈ R, p(x)?



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it is denoted by ∀x ∈ D, p(x).I Here ∀ is called the universal quantifier.

I Let p(x) : −(−x) = x . What is the truth value of∀x ∈ R, p(x)?



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I The existential quantification of p(x) is the statement

There exists an x ∈ D for which p(x) is true ,

it is denoted by ∃x ∈ D, p(x).

I Here ∃ is called the existential quantifier.

I Let q(x) : x + 1 < 4. What is the truth value of ∃x ∈ R, q(x)?

I Let q(x) : x + 2 = x . What is the truth value of ∃x ∈ R, q(x)?


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Statement When true? When false?∀x ∈ D, p(x) p(x) is true for every x There is an x for which p(x) is false (counter example)∃x ∈ D, p(x) There is an x for which p(x) is true p(x) is false for every x

Quantification of two variable:

Statement When true? When false?∀x∀y ∈ D, p(x, y) p(x, y) is true for every (x, y) There is a (x, y) for which p(x, y) is false∀y∀x ∈ D, p(x, y)∀x∃y ∈ D, p(x, y) For every x there is a y for which There is an x for which

p(x, y) is true p(x, y) is false for every y∃x∀y ∈ D, p(x, y) There is an x for which For every x there is a y

p(x, y) is true for every y for which p(x, y) is false∃x∃y ∈ D, p(x, y) There is a (x, y) for which p(x, y) is true p(x, y) is false for every (x, y).∃y∃x ∈ D, p(x, y)


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I “Every student in this class has taken calculus course”

∀x ∈ D, p(x) , where p(x) : x has taken calculus course

I Negation: “It is not the case that every student in this classhas taken calculus course”Equivalent statement,“There is a student in this class who has not taken calculuscourse”—This is a existential quantification of the negation of p(x).i.e.

∃x ∈ D,∼ p(x) .

I Therefore,

∼ (∀x ∈ D, p(x)) ≡ ∃x ∈ D,∼ p(x)


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∃x ∈ D,∼ p(x) .

I Therefore,

∼ (∀x ∈ D, p(x)) ≡ ∃x ∈ D,∼ p(x)


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∃x ∈ D,∼ p(x) .

I Therefore,

∼ (∀x ∈ D, p(x)) ≡ ∃x ∈ D,∼ p(x)


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I “There is a student in this class who has taken calculuscourse”

∃x ∈ D, p(x) , where p(x) : x has taken calculus course

I Negation: “It is not the case that There is a student in thisclass who has taken calculus course”Equivalent statement,“Every student in this class has not taken calculus course”—This is the universal quantification of the negation of p(x).i.e.

∀x ∈ D,∼ p(x) .

Therefore,

∼ (∃x ∈ D, p(x)) ≡ ∀x ∈ D,∼ p(x)


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I “There is a student in this class who has taken calculuscourse”

∃x ∈ D, p(x) , where p(x) : x has taken calculus course

I Negation: “It is not the case that There is a student in thisclass who has taken calculus course”Equivalent statement,“Every student in this class has not taken calculus course”—This is the universal quantification of the negation of p(x).i.e.

∀x ∈ D,∼ p(x) .

Therefore,

∼ (∃x ∈ D, p(x)) ≡ ∀x ∈ D,∼ p(x)


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Negation Equivalent statement When negation is true? When negation false?∼ (∃x ∈ D, p(x)) ∀x ∈ D,∼ p(x) p(x) is false for every x There is an x for which

p(x) is true∼ (∀x ∈ D, p(x)) ∃x ∈ D,∼ p(x) There is an x for which p(x) is true for every x

p(x) is false


Logic and Propositional Calculus

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