Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
Direct Proof and Proof by Contrapositive
Math 401
Dr. Nahid Sultana
October 20, 2012
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
Introduction
Proposition
Truth Table
Compound Proposition
Tautology and Contradictions
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
Quantifier
Negation of quantified statement
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
I
Proposition
Truth Table
Compound Proposition
Tautology and Contradictions
Definition: A statement (or proposition) is a declarative
sentence that is either true or false but not both.
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
I
I
Proposition
Truth Table
Compound Proposition
Tautology and Contradictions
Definition: A statement (or proposition) is a declarative
sentence that is either true or false but not both.
Example:
1.
2.
3.
4.
5.
The earth is round.
2 + 3 = 5.
Do you speak English?
3 − x = 5.
Take two aspirins.
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
I
I
I
Proposition
Truth Table
Compound Proposition
Tautology and Contradictions
Definition: A statement (or proposition) is a declarative
sentence that is either true or false but not both.
Example:
1.
2.
3.
4.
5.
The earth is round.
2 + 3 = 5.
Do you speak English?
3 − x = 5.
Take two aspirins.
1.
2.
3.
4.
5.
The earth is round. (statement)
2 + 3 = 5. (statement)
Do you speak English? (not statement, question)
3 − x = 5. (declarative sentence but not statement)
Take two aspirins. (not statement, command)
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
I
Proposition
Truth Table
Compound Proposition
Tautology and Contradictions
Definition: An open sentence is a declarative sentence that
contains one or more variables, and becomes a statement
when values from their respective domains are substituted for
these variables.
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
I
I
Proposition
Truth Table
Compound Proposition
Tautology and Contradictions
Definition: An open sentence is a declarative sentence that
contains one or more variables, and becomes a statement
when values from their respective domains are substituted for
these variables.
Example: p(x) : x(x − 1) = 6 is an open sentence over the
domain R.
1. For what values of x is p(x) a true statement?
2. For what values of x is p(x) a false statement?
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
I
Proposition
Truth Table
Compound Proposition
Tautology and Contradictions
Every statement has a truth value; true (denoted by T ) or
false (denoted by F ).
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
Proposition
Truth Table
Compound Proposition
Tautology and Contradictions
I
Every statement has a truth value; true (denoted by T ) or
false (denoted by F ).
I
The possible truth values of a statement can be represented
by a table, called truth table.
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
Proposition
Truth Table
Compound Proposition
Tautology and Contradictions
I
Every statement has a truth value; true (denoted by T ) or
false (denoted by F ).
I
The possible truth values of a statement can be represented
by a table, called truth table.
p q
p
q
T T
T
T
T F
F
F
F T
F F
I
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
Proposition
Truth Table
Compound Proposition
Tautology and Contradictions
I
Every statement has a truth value; true (denoted by T ) or
false (denoted by F ).
I
The possible truth values of a statement can be represented
by a table, called truth table.
p q
p
q
T T
T
T
T F
F
F
F T
F F
In truth table involving three statements p, q, r , there are
8 = 23 possible combinations of truth values.
I
I
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
Proposition
Truth Table
Compound Proposition
Tautology and Contradictions
I
Every statement has a truth value; true (denoted by T ) or
false (denoted by F ).
I
The possible truth values of a statement can be represented
by a table, called truth table.
p q
p
q
T T
T
T
T F
F
F
F T
F F
In truth table involving three statements p, q, r , there are
8 = 23 possible combinations of truth values.
I
I
I
In general, a truth table involving n statements contains 2n
possible combinations of truth values.
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
I
Proposition
Truth Table
Compound Proposition
Tautology and Contradictions
In Mathematics, x, y , z, ... denotes variables, can be replaced
by real numbers, can be combined with operations +, −, ×, ÷.
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
Proposition
Truth Table
Compound Proposition
Tautology and Contradictions
I
In Mathematics, x, y , z, ... denotes variables, can be replaced
by real numbers, can be combined with operations +, −, ×, ÷.
I
In logic, p, q, r , ... denote propositional variables, can be
replaced by propositions, and can be combined by logical
connectives (operations) to obtain compound proposition.
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
Proposition
Truth Table
Compound Proposition
Tautology and Contradictions
I
In Mathematics, x, y , z, ... denotes variables, can be replaced
by real numbers, can be combined with operations +, −, ×, ÷.
I
In logic, p, q, r , ... denote propositional variables, can be
replaced by propositions, and can be combined by logical
connectives (operations) to obtain compound proposition.
I
Example:
p: The sun is shining today
q: It is cold
p and q: The sun is shining and it is cold.
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
Proposition
Truth Table
Compound Proposition
Tautology and Contradictions
I
In Mathematics, x, y , z, ... denotes variables, can be replaced
by real numbers, can be combined with operations +, −, ×, ÷.
I
In logic, p, q, r , ... denote propositional variables, can be
replaced by propositions, and can be combined by logical
connectives (operations) to obtain compound proposition.
I
Example:
p: The sun is shining today
q: It is cold
p and q: The sun is shining and it is cold.
I
The truth value of a compound proposition depends on the
truth values of the propositions and the types of the
connective being used.
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
I
Proposition
Truth Table
Compound Proposition
Tautology and Contradictions
There are three basic logical operations:
1. Conjunction (and)
2. Disjunction (or)
3. Negation (not)
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
I
Proposition
Truth Table
Compound Proposition
Tautology and Contradictions
There are three basic logical operations:
1. Conjunction (and)
2. Disjunction (or)
3. Negation (not)
I
Conjunction: If p and q are propositions then
I
I
I
I
The conjunction of p and q is the compound proposition ”p
and q”.
Denoted by p ∧ q.
p ∧ q is true when both p and q are true otherwise false.
Truth table:
p q p∧q
T T
T
T F
F
F T
F
F F
F
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
I
Proposition
Truth Table
Compound Proposition
Tautology and Contradictions
Disjunction: If p and q are two propositions then
I
I
I
I
The disjunction of p and q
q”.
Denoted by p ∨ q.
p ∨ q is true if at least one
both p and q are false.
Truth table:
p
T
T
F
F
Math 401
is the compound proposition ”p or
of p or q is true; it is false when
q
T
F
T
F
p∨q
T
T
T
F
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
I
Proposition
Truth Table
Compound Proposition
Tautology and Contradictions
Negation: If p is a proposition then
1.
2.
3.
4.
The negation of p is the proposition ”not p”.
Denoted by ¬p or ∼ p.
If p is true then ∼ p is false; if p is false then ∼ p is true.
Truth table:
p ∼p
T
F
F
T
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
I
Negation: If p is a proposition then
1.
2.
3.
4.
I
Proposition
Truth Table
Compound Proposition
Tautology and Contradictions
The negation of p is the proposition ”not p”.
Denoted by ¬p or ∼ p.
If p is true then ∼ p is false; if p is false then ∼ p is true.
Truth table:
p ∼p
T
F
F
T
Example:
p: 2 + 2 = 5
q: 2 + 2 6= 5 (negation of p)
r : It is not the case that 2 + 2 = 5 (negation of p)
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
I
Proposition
Truth Table
Compound Proposition
Tautology and Contradictions
Negation: If p is a proposition then
1.
2.
3.
4.
The negation of p is the proposition ”not p”.
Denoted by ¬p or ∼ p.
If p is true then ∼ p is false; if p is false then ∼ p is true.
Truth table:
p ∼p
T
F
F
T
I
Example:
p: 2 + 2 = 5
q: 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.
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
Proposition
Truth Table
Compound Proposition
Tautology and Contradictions
I
Truth table of a proposition with several connectives.
I
Example:
(p ∧ q) ∨ (∼ p)
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
Proposition
Truth Table
Compound Proposition
Tautology and Contradictions
I
Truth table of a proposition with several connectives.
I
Example:
(p ∧ q) ∨ (∼ p)
I
Order of priority or the logical connectives:
First ∼ then ∧
Math 401
and then ∨
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
I
Proposition
Truth Table
Compound Proposition
Tautology and Contradictions
Consider two examples:
p ∨ (∼ p) and p ∧ (∼ p)
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
I
Proposition
Truth Table
Compound Proposition
Tautology and Contradictions
Consider two examples:
p ∨ (∼ p) and p ∧ (∼ p)
I
Truth tables:
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
I
Proposition
Truth Table
Compound Proposition
Tautology and Contradictions
Consider two examples:
p ∨ (∼ p) and p ∧ (∼ p)
I
Truth tables:
I
Definition: A compound proposition that is always true for all
possible combination of truth values of the propositional
variables is called a tautology.
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
I
Proposition
Truth Table
Compound Proposition
Tautology and Contradictions
Consider two examples:
p ∨ (∼ p) and p ∧ (∼ p)
I
Truth tables:
I
Definition: A compound proposition that is always true for all
possible combination of truth values of the propositional
variables is called a tautology.
I
Definition: A compound proposition that is always false for all
possible combination of truth values of the propositional
variables is called a contradiction.
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
I
Implication (Conditional): For two statements p and q
I
I
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I
I
The compound statement ”If p, then q” is called the
implication.
Denoted by p ⇒ q; read as ”p implies q”.
p is the hypothesis and q is the conclusion.
p ⇒ q is false when p is true and q is false, and is true
otherwise.
Truth table:
p q p⇒q
T T
T
T F
F
F T
T
F F
T
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
I
Let
p(n) : n2 − n + 1 is prime ; q(n) : n3 − n + 1 is prime
be open sentences over the domain S = {2, 3, 5}. Determine
the truth or falseness of p(n) ⇒ q(n) for each n ∈ S.
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
I
Let
p(n) : n2 − n + 1 is prime ; q(n) : n3 − n + 1 is prime
be open sentences over the domain S = {2, 3, 5}. Determine
the truth or falseness of p(n) ⇒ q(n) for each n ∈ S.
I
The converse of the statement p ⇒ q is q ⇒ p .
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
I
Let
p(n) : n2 − n + 1 is prime ; q(n) : n3 − n + 1 is prime
be open sentences over the domain S = {2, 3, 5}. Determine
the truth or falseness of p(n) ⇒ q(n) for each n ∈ S.
I
The converse of the statement p ⇒ q is q ⇒ p .
I
The contrapositive of p ⇒ q is (∼ q) ⇒ (∼ p).
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
I
Biconditional: For two statements p and q
I
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I
I
The compound statement ”p iff q” is called a biconditional
statement.
Denoted by p ⇔ q.
p ⇔ q is true only when p and q have the same truth values.
Truth table:
p q p⇔q
T T
T
T F
F
F T
F
F F
T
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
I
Biconditional: For two statements p and q
I
I
I
I
I
The compound statement ”p iff q” is called a biconditional
statement.
Denoted by p ⇔ q.
p ⇔ q is true only when p and q have the same truth values.
Truth table:
p q p⇔q
T T
T
T F
F
F T
F
F F
T
p: 18 is odd
q: 25 is even
Is p ⇔ q true or false.
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
I
Biconditional: For two statements p and q
I
I
I
I
I
I
The compound statement ”p iff q” is called a biconditional
statement.
Denoted by p ⇔ q.
p ⇔ q is true only when p and q have the same truth values.
Truth table:
p q p⇔q
T T
T
T F
F
F T
F
F F
T
p: 18 is odd
q: 25 is even
Is p ⇔ q true or false.
p ⇔ q is also stated as
”p is a necessary and sufficient condition for q ”
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
I
Example: p: 18 is odd
and q: 25 is even
Is p ⇔ q true or false.
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
I
Two propositions (compound) P(p, q, ..) and Q(p, q, ..) are
said to be logically equivalent if they have
identical truth tables, denoted by
P(p, q, ..) ≡ Q(p, q, ..) .
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
I
Two propositions (compound) P(p, q, ..) and Q(p, q, ..) are
said to be logically equivalent if they have
identical truth tables, denoted by
P(p, q, ..) ≡ Q(p, q, ..) .
I
Example: ∼ (p ∧ q) and (∼ p) ∨ (∼ q).
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
I
Commutative Properties:
p∨q ≡q∨p ; p∧q ≡q∧p
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
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
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
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 )
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
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
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
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)
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
I
Definition: A logical argument is a finite set of propositions
(premises or hypothesis) followed by a proposition
(conclusion).
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
I
Definition: A logical argument is a finite set of propositions
(premises or hypothesis) followed by a proposition
(conclusion).
I
Denoted by
Q
P1 , P2 , ...., Pn `
|{z}
{z
}
|
conclusion
premises
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
I
Definition: A logical argument is a finite set of propositions
(premises or hypothesis) followed by a proposition
(conclusion).
I
Denoted by
Q
P1 , P2 , ...., Pn `
|{z}
{z
}
|
conclusion
premises
I
Suppose the premises are all true, then conclusion may be
either true or false.
When the conclusion is true then the argument is said to be
valid.
When the conclusion is false then the argument is said to be
invalid or fallacy.
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
I
Show that the argument
p, p → q ` q
is valid.
I
Show that the argument
p → q, q ` p
is invalid.
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
I
Theorem: The argument P1 , P2 , ...., Pn ` Q is valid iff the
proposition (P∧ P2 ∧ .... ∧ Pn ) → Q is a tautology.
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
I
Theorem: The argument P1 , P2 , ...., Pn ` Q is valid iff the
proposition (P∧ P2 ∧ .... ∧ Pn ) → Q is a tautology.
I
Show that the argument
p → q, q → r ` p → r
is valid.
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
I
Quantifier
Negation of quantified statement
Suppose p(x) is an open statement over a domain D. Then
p(x) is a statement for each x ∈ D.
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
I
I
Quantifier
Negation of quantified statement
Suppose p(x) is an open statement over a domain D. Then
p(x) is a statement for each x ∈ D.
Generate statement from open statement by the method of
quantification.
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
I
I
I
Quantifier
Negation of quantified statement
Suppose p(x) is an open statement over a domain D. Then
p(x) is a statement for each x ∈ D.
Generate statement from open statement by the method of
quantification.
Two types of quantification:
1. Universal quantification
2. Existential quantification
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
I
I
I
Quantifier
Negation of quantified statement
Suppose p(x) is an open statement over a domain D. Then
p(x) is a statement for each x ∈ D.
Generate statement from open statement by the method of
quantification.
Two types of quantification:
1. Universal quantification
2. 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).
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
I
I
I
Quantifier
Negation of quantified statement
Suppose p(x) is an open statement over a domain D. Then
p(x) is a statement for each x ∈ D.
Generate statement from open statement by the method of
quantification.
Two types of quantification:
1. Universal quantification
2. Existential quantification
I
The universal quantification of p(x) is the proposition
p(x) is true for all values of x ∈ D ,
I
it is denoted by ∀x ∈ D, p(x).
Here ∀ is called the universal quantifier.
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
I
I
I
Quantifier
Negation of quantified statement
Suppose p(x) is an open statement over a domain D. Then
p(x) is a statement for each x ∈ D.
Generate statement from open statement by the method of
quantification.
Two types of quantification:
1. Universal quantification
2. Existential quantification
I
The universal quantification of p(x) is the proposition
p(x) is true for all values of x ∈ D ,
I
I
it is denoted by ∀x ∈ D, p(x).
Here ∀ is called the universal quantifier.
Let p(x) : −(−x) = x. What is the truth value of
∀x ∈ R, p(x)?
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
I
I
I
Quantifier
Negation of quantified statement
Suppose p(x) is an open statement over a domain D. Then
p(x) is a statement for each x ∈ D.
Generate statement from open statement by the method of
quantification.
Two types of quantification:
1. Universal quantification
2. Existential quantification
I
The universal quantification of p(x) is the proposition
p(x) is true for all values of x ∈ D ,
I
I
I
it is denoted by ∀x ∈ D, p(x).
Here ∀ is called the universal quantifier.
Let p(x) : −(−x) = x. What is the truth value of
∀x ∈ R, p(x)?
Let q(x) : x + 1 < 4. What is the truth value of ∀x ∈ R, q(x)?
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
I
Quantifier
Negation of quantified statement
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).
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
I
Quantifier
Negation of quantified statement
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.
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
I
Quantifier
Negation of quantified statement
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)?
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
I
Quantifier
Negation of quantified statement
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)?
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
Statement
∀x ∈ D, p(x)
∃x ∈ D, p(x)
When true?
p(x) is true for every x
There is an x for which p(x) is true
Quantifier
Negation of quantified statement
When false?
There is an x for which p(x) is false (counter example)
p(x) is false for every x
Quantification of two variable:
Statement
∀x∀y ∈ D, p(x, y )
∀y ∀x ∈ D, p(x, y )
∀x∃y ∈ D, p(x, y )
∃x∀y ∈ D, p(x, y )
∃x∃y ∈ D, p(x, y )
∃y ∃x ∈ D, p(x, y )
When true?
p(x, y ) is true for every (x, y )
When false?
There is a (x, y ) for which p(x, y ) is false
For every x there is a y for which
p(x, y ) is true
There is an x for which
p(x, y ) is true for every y
There is a (x, y ) for which p(x, y ) is true
There is an x for which
p(x, y ) is false for every y
For every x there is a y
for which p(x, y ) is false
p(x, y ) is false for every (x, y ).
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
I
Quantifier
Negation of quantified statement
“Every student in this class has taken calculus course”
∀x ∈ D, p(x) , where p(x) : x has taken calculus course
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
I
Quantifier
Negation of quantified statement
“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 class
has taken calculus course”
Equivalent statement,
“There is a student in this class who has not taken calculus
course”
—This is a existential quantification of the negation of p(x).
i.e.
∃x ∈ D, ∼ p(x) .
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
I
Quantifier
Negation of quantified statement
“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 class
has taken calculus course”
Equivalent statement,
“There is a student in this class who has not taken calculus
course”
—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)
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
I
Quantifier
Negation of quantified statement
“There is a student in this class who has taken calculus
course”
∃x ∈ D, p(x) , where p(x) : x has taken calculus course
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
I
Quantifier
Negation of quantified statement
“There is a student in this class who has taken calculus
course”
∃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 this
class 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)
Math 401
Direct Proof and Proof by Contrapositive
Outline
Introduction
Conditional and Biconditional statement
Logical Equivalence
Algebra of Proposition
Arguments
Propositional function and Quantifier
Quantifier
Negation of quantified statement
Negation
∼ (∃x ∈ D, p(x))
Equivalent statement
∀x ∈ D, ∼ p(x)
When negation is true?
p(x) is false for every x
∼ (∀x ∈ D, p(x))
∃x ∈ D, ∼ p(x)
There is an x for which
p(x) is false
Math 401
When negation false?
There is an x for which
p(x) is true
p(x) is true for every x
Direct Proof and Proof by Contrapositive