20130905170921MTK3013-Chapter1.3 Predicates and Quantifiers
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Transcript of 20130905170921MTK3013-Chapter1.3 Predicates and Quantifiers
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1.3 PREDICATES AND
QUANTIFIER
MTK3013
DISCRETE STRUCTURES
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Predicates
A predicate is a statement that contains
variables.
Example:
P(x) :x > 3
Q(x,y) :x =y + 3
R(x,y,z) :x +y =z
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Predicates
A predicate becomes a proposition if thevariable(s) contained is(are)
Assigned specific value(s) Quantified
P(x) :x > 3.What are the truth values of
P(4) andP(2)? Q(x,y) :x =y + 3. What are the truth values
ofQ(1,2) and Q(3,0)?
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Quantifiers
Two types of quantifiers
Universal Existential
Universe of discourse - the particular
domain of the variable in a propositional
function
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Universal Quantification
P(x) is true for all values ofx in the universe
of discourse.
x P(x)
for allx,P(x)
for everyx
,P(x)
The variablex is bound by the universal
quantifier, producing a proposition
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Example
U= {all real numbers},P(x):x+1 >x
What is the truth value of x P(x)
U= {all real numbers}, Q(x):x < 2
What is the truth value of x Q(x)
U= {all students in MTK3013}
R(x) :x has an account on Tabung Haji
What does x R(x) mean?
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For universal quantification
P(x) P(x1) P(x2) P(xn)
If the elements in the universe of discourse can belisted, U= {x1, x2, , xn}x P(x)P(x
1
) P(x2
) P(xn
)
Example
U= {positive integers not exceeding 3} andP(x):
x2 < 10 What is the truth value of x P(x)
P(1) ^ P(2) ^ P (3)
T ^ T ^ T
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Existential Quantification
P(x) is truefor somex in the universe ofdiscourse
x P(x) for somex,P(x)
There exists anx such thatP(x)
There is at least onex
such thatP(x)
The variablex is bound by the existentialquantifier, producing a proposition
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Example
U= {all real numbers}, P(x):x> 3 What is the truth value of x P(x)
U= {all real numbers}, Q(x):x=x+ 1 What is the truth value ofx Q(x)
U= {all students in MTK 3013}
R(x) :xhas an account on Tabung Haji What does x R(x) mean?
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For existential quanti f ication
P(x)P(x1) P(x2) P(xn)
If the elements in the universe of discourse
can be listed, U= {x1, x2, , xn}
x P(x) P(x1) P(x2)
P(xn)
Example
U= {positive integers not exceeding 4} andP(x):x2 > 10 What is the truth value ofx P(x)
P(1) v P(2) v P(3) v P(4)
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Binding Variables
Boundvariable: if a variable is quantified
Free variable: Neither bound nor assigned aspecific value
Example: x P(x) x Q(x,y)
Scope of Quantifiers: Part of a logicalexpression to which a quantifier is applied
Example: x (P(x) Q(x)) x R(x)
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Negation of Quantifiers
Distributing a negation operator across a
quantifier changes a universal to an
existential and vice versa.
~x P(x) x ~P(x)
~x P(x) x ~P(x)
Example:
P(x) :x has taken a course in calculus
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Translating from English
Many ways to translate a given sentence
Goal is to produce a logical expression that is
simple and can be easily used in subsequentreasoning
Steps:
Clearly identify the appropriate quantifier(s)
Introduce variable(s) and predicate(s)
Translate using quantifiers, predicates, andlogical operators
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Example
Every student in this class has studied
Discrete Structures
Solution 1
Assume, U = {all students in MTK 3013}
Solution 2
Assume, U = {all people}
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Example
Some student in this class has visited
Mexico
Solution 1
Assume, U = {all students in MTK 3013}
Solution 2
Assume, U = {all people}
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More Example
C(x):x is a COMPUTING student
M(x):x is an MULTIMEDIA student
S(x):x is a smart student
U= {all students in MTK 3013}
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More Example (Cont..)
Everyone is a COMPUTING student.
x C(x)
Nobody is an MULTIMEDIA student.
x ~M(x) or ~x M(x)
All COMPUTING students are smart students.
x [C(x)S(x)]
Some COMPUTING students are smartstudents.
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Use implication or
conjunction? Universal quantifiers usually take implications
All COMPUTING students are smart students.
x[C(x)S(x)] Correct
x[C(x) S(x)] Incorrect
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Use implication or
conjunction? Existential quantifiers usually take
conjunctions
Some COMPUTING students are smart
students.
x [C(x) S(x)] Correctx [C(x) S(x)] Incorrect
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More Example
No COMPUTING student is an MULTIMEDIAstudent.
Ifx is a COMPUTING student, then that student is notan MULTIMEDIA student.
x [C(x) ~M(x)] There does not exist a COMPUTING student who is
also an MULTIMEDIA student.
~x [C(x) M(x)] If any MULTIMEDIA student is a smart student
then he is also a COMPUTING student.x [(M(x) S(x))C(x)]
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(13b) Determine truth value. U={Z}
n (2n = 3n)
(16b) Determine truth value U={R
}n (x2 = -1)
Exercise 17 (Page 47)
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Nested Quantifiers
Quantifiers that occur within the scope of
other quantifiers
Example:
P(x,y): x + y = 0, U={R}
xy P(x,y)
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Quantifications of Two
Variables
For all pair x,y P(x,y).
xy P(x,y) yx P(x,y)
For every x there is a y such that P(x,y).
xy P(x,y)
There is an x such that P(x,y) for all y.
xy P(x,y) There is a pair x,y such that P(x,y).
xy P(x,y) yx P(x,y)
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Translating statements
with nested quantifiers
U= {all real numbers}x y (x + y = y + x)
x y (x + y = 0)x y ( (x > 0) (y < 0) (xy < 0) )
U= {all students in cs2813}
C(x): x has a computerF(x,y): xandyare friends
x ( C(x) y (C(y) F(x,y)) )
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Translating Sentences
U= {all people}
If a person is female and is a parent, then this
person is someones mother.
U= {all integers}
The sum of two positive integers is positive.
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Is the order of quantifiers
important?
If the quantifiers are of the same type, then
order does not matter If the quantifiers are of different types,
then order is important
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Example U={R}
Q(x,y): x+y=0
What are the truth values for
y x Q(x,y) andx y Q(x,y)
y x Q(x,y): There exist at least one y such that forevery real number x, Q(x,y) is true, i.e. x+y=0.
FALSE (not for every, only when y is x).
Butx y Q(x,y): For every real number x, there is a real
number y such that Q(x,y) is true, i.e x+y =0.TRUE (for every x when y is x)
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