Download The Foundations: Logic and Proof

The Foundations: Logic and Proof
CS104
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1.3 Propositional Equivalences
•Introduction
•Logical Equivalences
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Introduction
Definition1
 A compound proposition that is always true, no matter what
the truth values of the propositions that occurs in it, is
called a tautology.
 A compound proposition that is always false is called a
contradiction.
 A compound proposition that is neither a tautology or a
contradiction is called a contingency.
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p
p
p  p
p  p
F
T
T
F
T
F
T
F
Logic Equivalence
Definition2:
The compound propositions p and q are called logically
equivalent if p ↔ q is a tautology. The notation p ≡ q denotes
that p and q are logically equivalent.
•Compound propositions that have the same truth values in all
possible cases are called logically equivalent.
De Morgan’s
Laws
Example:
Show that ¬(p ν q) and p  q are logically equivalent.
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p
q
F
T
F
T
F
F
T
T
p  q (p  q)
F
T
T
T
T
F
F
F
p
q
p  q
T
F
T
F
T
T
F
F
T
F
F
F
Logic Equivalence
• Example3:
Show that ¬p ν q and p → q are logically equivalent.
Solution on the book
Show that p  (q  r)  (p  q)  (p  r)
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Logic Equivalence
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Equivalence
Name
p Λ T  p and p ν F  p
Identity laws
p ν T  T and p Λ F  F
Domination laws
p ν p  p and p Λ p  p
Idempotent laws
¬( ¬ p)  p
Double negation law
p ν q  q ν p and p Λ q  q Λ p
Commutative laws
(p ν q) ν r  p ν (q ν r) and (p Λ q) Λ r  p Λ (q Λ
r)
Associative laws
p  (q  r)  (p  q)  (p  r)
p  (q  r)  (p  q)  (p  r)
Distributive laws
¬(p ν q)  ¬p Λ ¬q and ¬(p Λ q)  ¬p ν ¬q
De Morgan’s laws
p  (p  q)  p and p  (p  q)  p
Absorption laws
p ν ¬p  T and p Λ ¬p  F
Negation laws
Logic Equivalence
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Logic Equivalence
Example6: Show that ¬(p → q ) and p Λ ¬q are logically equivalent.
Solution:
¬(p → q ) ≡ ¬(¬p ν q)
≡ ¬(¬p) Λ ¬q by the second De Morgan law
≡ p Λ ¬q by the double negation law
Example8: Show that (p Λ q) → (p ν q) is a tautology.
Solution: To show that this statement is a tautology, we will use logical
equivalences to demonstrate that it is logically equivalent to T.
(p Λ q) → (p ν q) ≡ ¬(p Λ q) ν (p ν q)
≡ (¬ p ν ¬q) ν (p ν q) by the first De Morgan law
≡ (¬ p ν p) ν (¬ q ν q) by the associative and
communicative law for disjunction
≡TνT
≡T
Note: The above examples can also be done using truth tables.
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1.4 Predicates and Quantifiers
•Introduction
•Predicates
•Quantifiers
•Universal quantifier
•Quantifier with restricted domains
•Logical equivalences involving quantifiers
•Negating quantified expressions
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•Translating from English into logical expressions
Predicates and Quantifiers
• Propositional logic cannot adequately express the meaning of all
statements in mathematics and natural language !
e.g. we know that : “ every computer connected to the network is
functioning properly “ ,
• Predicate Logic :
is a powerful type of logic that could be used to express the
meaning of wide range of statements in mathematics and CS.
• Two main concepts are introduced:
 Predicate
 Quantifiers.
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Predicates and Quantifiers
• Predicate:
• Statements involving variables such as : x > 3, x = y + 3 , x + y = z are
neither True nor false when the value of variables are not specified.
• This section discusses how proposition can be produced from such
statements
• The statement x>3 has two parts :
the first part: subject of statement ,variable x
the second part: predicate , a property that the subject of a statement
can have , “ is greater than 3”
•
The statement could be denoted by : P(x) , propositional function ,where p
is a predicate(is greater than 3), x is a variable.
• Once we assign a value to variable x, P(x) becomes a proposition that could
be True or False.
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Predicates
• Example:
Let P(x) : x > 3 , what is the truth value of P(4) ?
setting x = 4 , 4 > 3 , true
Let P (x,y) : x = y + 3, what is the truth value of P(1,2) , “ involve more than
one variable ! “
1 = 2 + 3 , false
• A statement of the form P(x1, x2, …., xn) is the value of the propositional
function P at the n-tuple (x1, x2, …., xn) and P is called n-ary predicate
• Propositional function occurs in computer programs as in the following
example.
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Predicates
 Example:
 Consider the statement
if x > 0 then x := x + 1.
Here P(x) = “x > 0”.
If P(x) is true for the value of x, then the assignment statement
x := x + 1 is executed, x is increased by 1.
If P(x) is false for the value of x, then the assignment statement
is not executed, x remains same
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Predicates
Usage of Predicate:
• Predicate are used to establish the correctness of
computer program, i.e. to show that computer programs
always produce the desired output when given valid
input.
• Precondition: statements that describe valid input
• Postcondition: condition that the output should satisfy
when the program has run
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Predicates
Example:
The following code is designed to interchange the values of two
variables x and y:
temp := x
x := y
y := temp
The predicate for precondition:
P (x ,y) where P (x , y) is x = a , y = b
The predicate for postcondition:
Q ( x , y) where Q ( x , y) is x = b , y = a
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Quantifiers
Quantification :
• It is a way to create a proposition from a propositional function.
• It expresses the extent to which a predicate is true over a range of
elements.
• In English, the words all, some , many, none and few are used in
quantification.
Types of Quantification:
1. Universal quantification: Which tells that a predicate is true for
every element under consideration.
2. Existential quantification: Which tells that there is one or more
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element under consideration for which the predicate is true.
Universal Quantifiers
Universal Quantifiers:
• Many mathematical statements assert that a property is true for all
values of a variable in a particular domain.
• The universal quantification of P(x) for particular domain is the
statement “P(x) is true for all values of x in the domain”.
• The notation x P(x) denotes the universal quantification of P(x).
Here  is called universal quantifier.
• The meaning of the universal quantification of P(x) changes when
the domain is changed.
• An element for which P(x) is false is called a counterexample of x
P(x).
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Universal Quantifiers
Example:
Let P(x) = “x+1 > x”. What is the truth value of the quantification x P(x),
where the domain consists of all real numbers?
Because P(x) is true for all real numbers x, the quantification x P(x) is true.
Example:
Let Q(x) = “x < 2”. What is the truth value of the quantification x Q(x),
where the domain consists of all real numbers?
Because Q(x) is not true for every real number x, because, for instance, Q(3) is
false. That is, x=3 is a counterexample for the statement x Q(x).
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Universal Quantifiers
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Existential Quantifiers
Existential Quantifier
• Many mathematical statements assert that there is an element with
a certain property.
• The existential quantification of P(x) is the statement “There exists
an element x in the domain such that P(x)”.
• The notation  x P(x) denotes the existential quantification of P(x).
Here  is called existential quantifier.
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Existential Quantifiers
Example:
Let P(x) = “x > 3”. What is the truth value of the quantification x
P(x), where the domain consists of all real numbers?
Because P(x) is sometimes true - for instance, when x=4 – the
existential quantification of P(x), which is x P(x), is true.
Example:
Let P(x) = “x = x+1”. What is the truth value of the quantification  x
P(x), where the domain consists of all real numbers?
Because P(x) is false for every real number x, the quantification  x
P(x) is false.
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Predicates and Quantifiers
Statement
x P(x)
When True?
P(x) is true for every x
When False?
There is an x for
which P(x) is false
 x P(x)
There is an x for which
P(x) is false for every
P(x) is true
x
Predicates and Quantifiers
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Translating English Sentences
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Translating English into Logical Expression
• Translating from English to logical expression becomes even more
complex when quantifiers are needed ! , and it could be done in
four main steps:
1. Determine the quantifier used in the sentence
2. Determine the variable in the sentence
3. Determine the domain of this variable
4. Determine the predicate
5. Write complete sentence
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Translating English into Logical Expression
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Translating English into Logical
Expression
Quantifiers with Restricted Domains
Quantifiers with Restricted Domains:
• An abbreviated notation is used to restrict the domain of a quantifier. In this
notation, a condition, a variable must satisfy, is included after the quantifier.
• The restriction of a universal quantification is the same as the universal
quantification of a conditional statement.
For instance, x < 0(x2 > 0) is another way of expressing x (x < 0 → x2 > 0).
• The restriction of an existential quantification is the same as the existential
quantification of a conjunction.
For instance,  z>0 (z2=2) can be expressed as z (z > 0 ˄ z2 = 2).
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Quantifiers with Restricted Domains
Example:
What do the statement x < 0(x2 > 0), and z > 0(z2 = 2) mean?
The statement x < 0(x2 > 0) states that for every number x with x < 0, x2 > 0.
That is, it states “The square of a negative real number is positive”.
The statement is the same as x (x < 0 → x2 > 0).
The statement z > 0(z2 = 2) states that there exist a real number z with z > 0
such
that z2=2.
That is, it states “There is positive square root of 2”.
The statement is the same as z (z > 0 ˄z2 = 2).
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Logical Equivalences Involving
Quantifiers
Precedence of Quantifiers:
•
The quantifiers  and  have higher precedence then all logical operators from
propositional calculus.
•
For example, xP(x) v Q(x) is the disjunction of xP(x) and Q(x). In other words, it
means (xP(x)) v Q(x) rather than x (P(x) v Q(x))
Logical Equivalences Involving Quantifiers:
Statements involving predicate and quantifiers are logically equivalent if and only
if they have the same truth value no matter which predicates are substituted into
these statements and which domain of discourse is used for the variables in these
propositional functions.
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Logical Equivalences Involving
Quantifiers
Example:
Show that x (P(x)  Q(x)) and xP(x)  x Q(x) are logically equivalent.
Let x (P(x)  Q(x)) is true
=> if a is in the domain, then P(a)  Q(a) is true
=> P(a) and Q(a) are true, for all a in the domain
=> xP(x) and xQ(x) are true
=> xP(x)  xQ(x) is true
So, they are logically equivalent.
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Negation Quantified Expression
 Example: “Every student in your class has taken a course
in calculus”
Domain consists of the students in your class.
P(x) is the statement “x has taken a course in calculus”
x P(x).
 Negation of the statement: “It is not the case that every
student in your class has taken a course in calculus”. This
equivalent to “There is a student in your class who has
not taken a course in calculus”.
 This is simply the existential quantification x  P(x) .
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Negation Quantified Expression
De Morgan’s laws for quantifiers
When is Negation
True?
When False?
 x P(x)
Equivalent
Statement
x P(x)
For every x,
P(x) is false
There is an x for
which P(x) is true
x P(x)
 x P(x)
There is an x for which
P(x) is false
P(x) is true for
every x
Negation
Example:
Show that x(P(x)Q(x)) and x(P(x)  Q(x)) are logically equivalent.
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1.5 Nested Quantifiers
•Introduction
•Order of quantifiers
•Translating mathematical statements into
statements involving nested quantifiers
•Translating from nested quantifiers into English
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Nested Quantifiers
• Two quantifiers are said to be nested if one is within the scope of the
other, such as x y (x + y = 0)
o Note that xy (x + y = 0) is same as x Q(x), where Q(x) is yP(x,
y), where P(x, y) is x + y = 0.
o It says that for every real number x there is a real number y such
that x + y = 0. This states that every real number has an additive
inverse.
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Nested Quantifiers
Example:
Assume that the domain for the variables x and y consists of all real
numbers.
• The statement : xy (x + y = y + x)
says that x + y = y + x for all real numbers x and y. This is commutative
law for addition of real numbers.
• The statement : xyz (x + (y + z) = (x + y) + z) is associative law for
addition of real numbers.
Nested Quantifiers
• The order of the quantifiers is important unless all the quantifiers are universal
quantifiers or all are existential quantifiers.
Example:
Let P(x, y) be the statement “x + y = y + x”. What are the truth values of the
quantifications xyP(x, y) and yxP(x, y), where the domain for all variables
consists of all real numbers?
The quantification xyP(x, y) is the proposition “For all real numbers x, for all real
numbers y, x + y = y + x”. Since P(x, y) is true for all real numbers x and y, the
proposition xyP(x, y) is also true.
Also, yxP(x, y) is the proposition “For all real numbers y, for all real numbers x, x
+ y = y + x” = xyP(x, y), so it is true.
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Nested Quantifiers
Note: Order of nested universal quantifiers in a statement without other
quantifiers can be changed without changing meaning of quantified statement.
Example:
Let Q(x, y) be the statement “x + y = 0”. What are the truth values of the
quantifications yxQ(x, y) and xyQ(x, y), where the domain for all variables
consists of all real numbers?
• yxQ(x, y) = “There is a real number y such that for every real number x, x +
y = 0”. Since there is no real number y such that x + y = 0, for all real numbers
x, the proposition yxQ(x, y) is false.
• xyQ(x, y) = “For every real number x, there is a real number y such that x +
y = 0”. Given a real number x, there is a real number y, namely y = -x, such that
x + y =0. So, xyQ(x, y) is true.
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Nested Quantifiers
The following table summarizes the meanings of the different possible
quantifications involving two variables:
Statement
When True?
When False?
xyP(x, y)
yxP(x, y)
P(x, y) is true for every pair x, y.
There is a pair x, y for which P(x, y)
is false
xyP(x, y),
For every x there is a y for which
P(x, y) is true
There is an x such that P(x, y) is
false for every y
xyP(x, y)
There is an x such that P(x, y) is
true for every y
For every x there is a y for which
P(x, y) is false
xyP(x, y)
yxP(x, y)
There is a pair x, y for which P(x, y) P(x, y) is false for every pair x, y.
is true
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Nested Quantifiers
Example:
Let Q(x, y, z) be the statement “x + y = z”. What are the truth values of the
quantifications xyzQ(x, y, z) and zxyQ(x, y, z), where the domain
for all variables consists of all real numbers?
xyzQ(x, y, z) = “For all real numbers x and for all real numbers y there
is a real number z such that x + y = z” is true.
The order of quantification is important here. Since zxyQ(x, y, z) =
“There is a real number z such that for all real numbers x and for all real
numbers y, x + y = z”, is false, because there is no z that satisfies th
equation x + y = z for all for all real numbers x and y
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Nested Quantifiers
Example: Translate the statement “The sum of two positive integers is
always positive” into a logical expression.
Let x and y be two positive integers, then we can write the statement as
“For all positive integers x and y, x + y is positive”
1. Determine the quantifier used in the sentence : universal 
2. Determine the variable in the sentence: x and y
3. Determine the domain of this variable : positive integers
4. Determine the predicate: p(x,y) is “ x + y >0 “
5. Write complete sentence : xy (x + y > 0)
Also, it can be expressed as xy((x > 0) ⌃ (y > 0) → (x + y > 0)) where
the domain for both variables consists of integers.
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Nested Quantifiers
Example: Translate the statement “Every real number except zero has a
multiplicative inverse” into a logical expression.
Example: Translate the statement x(C(x) v y(C(y) ⌃ F(x, y)))
into English, where C(x)=“x has a computer”, F(x, y) = “x and y are friends”,
and the domain for both x and y consists of all students in your class.
The statement says that for every student x in your class, x has a computer or
there is a student y such that y has computer and x and y are friends.
In other words, every student in your class has a computer or has a friend who
has a computer.
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Nested Quantifiers
Example:
Translate the statement xyz((F(x, y) ⌃ F(x, z) ⌃ (y ≠ z)) → F(y, z))
into English, where F(a, b) means a and b are friends and the domain for x, y and
consists of all students in your class.