Download Predicates and Quantified Statements

The Logic of
Quantified Statements
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Definition of Predicate
 Predicate is a sentence that
 contains finite number of variables;
 becomes a statement when
specific values are substituted
for the variables.
 Ex:  let predicate P(x,y) be “x>2 and x+y=8”
 when x=5 and y=3,
P(5,3) is “5>2 and 5+3=8”
 Domain of a predicate variable is
the set of all possible values of the variable.
 Ex (cont.): D(x)= ; D(y)=R
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Truth Set of a Predicate
• If P(x) is a predicate and
x has domain D,
then the truth set of P(x) is
all xD such that P(x) is true.
(denoted {xD | P(x)} )
• Ex: P(x) is “5<x<9” and D(x)=Z.
Then {xD | P(x)} ={6, 7, 8}
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Universal Statement and
Quantifier
Let P(x) be “x should take Math306”;
D={Math majors} be the domain of x.
Then “all Math majors take Math306”
is denoted xD, P(x)
and is called universal statement.
 is called universal quantifier;
expressions for : “for all”, “for arbitrary”,
“for any”, “for each”.
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Truth and Falsity of Universal
Statements
 Universal statement “xD, P(x)”
 is true iff P(x) is true for every x in D;
 is false iff P(x) is false for at least one x.
(that x is called counterexample)
 Ex: 1) Let D be the set of even integers.
“xD yD, x+y is even” is true.
2) Let D be the set of all NBA players.
“xD, x has a college degree” is false.
Counterexample: Kobe Bryant.
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Existential Statement and
Quantifier
 Let P(x) be “x(x+2)=24”;
D =Z be the domain of x.
Then
”there is an integer x such that x(x+2)=24”
is denoted “xD, P(x)”
and is called existential statement.
  is called existential quantifier;
expressions for  : “there exists”, “there is a”,
“there is at least one”,
“we can find a”.
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Truth and Falsity of Existential
Statements
 Existential statement “ xD, P(x)”
 is true iff P(x) is true for at least one x in D;
 is false iff P(x) is false for all x in D.
 Ex: 1) Let D be the set of rational numbers.
“ xD, x 2  2 x  1  0 ” is true.
2) Let D = Z.
“ xD, x(x-1)(x-2)(x-3)<0” is false.
Why? Hint: Use proof by division into cases.
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Negations of Quantified
Statements
The negation of universal statement
“xD, P(x)” is
the existential statement “xD, ~P(x)”
Example: The negation of
“All NBA players have college degree”
is “There is a NBA player
who doesn’t have college degree”.
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Negations of Quantified
Statements
The negation of existential statement
“ xD, P(x)” is
the universal statement “ xD, ~P(x)”
 Example: The negation of
“ x Z such that x(x+1)<0”
is “ x Z, x(x+1) ≥ 0”.
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Statements containing multiple
quantifiers
Ex: 1) xR, yZ such that |x-y|<1.
2) For any building x in the city
there is a fire-station y such that
the distance between x and y
is at most 2 miles.
3) xZ such that y[3,5], x<y.
4) There is a student who solved all
the problems of the exam correctly.
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Truth values of multiply
quantified statements
Ex:  Students = {Joe, Ann, Bob, Dave}
 2 groups of languages:
Asian languages={Japanese,Chinese,Korean};
European languages={French, German,
Italian, Spanish}.
 Joe speaks Italian and French;
Ann speaks German, French and Japanese;
Bob speaks Spanish, Italian and Chinese;
Dave speaks Japanese and Korean.
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Truth values of multiply
quantified statements
Ex(cont.): Determine truth values of the
following statements:
1)  a student S s.t.  language L,
S speaks L.
2)  a student S s.t. for  language group G
 L in G s.t. S speaks L.
3)  a language group G s.t. for  student S
 L in G s.t. S speaks L.
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Negating multiply quantified
statements
Example:
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The negation of “for xR, yR s.t. y  x “
is logically equivalent to
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“xR s.t. for yR, y  x “.
Generally,
the negation of x, y s.t. P(x,y)
is logically equivalent to
x s.t. y, ~P(x,y)
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Negating multiply quantified
statements
Example:
The negation of “ xR s.t. yZ, x>y“
is logically equivalent to
“xR yZ s.t. x≤y“.
Generally,
the negation of x s.t. y, P(x,y)
is logically equivalent to
x y s.t. ~P(x,y)
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The Relation among , , Λ, ν
Let Q(x) be a predicate;
D={x_1, x_2, …, x_n} be the domain of x.
Then
 xD, Q(x) is logically equivalent to
Q(x_1) Λ Q(x_2) Λ … Λ Q(x_n) ;
 xD, Q(x) is logically equivalent to
Q(x_1) ν Q(x_2) ν …
ν Q(x_n) .
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Universal Conditional Statement
 Definition:  x, if P(x) then Q(x) .
 Example:  undergrad S,
if S takes CS300,
then S has taken CS240.
Negation of universal conditional statement:
 x such that P(x) and ~Q(x)
 Ex(cont.):  undergrad who takes CS300
but hasn’t taken CS240.
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Variations of universal conditional
statements
Variations of xD, if P(x) then Q(x):
• Contrapositive: xD, if ~Q(x) then ~P(x);
• Converse: xD, if Q(x) then P(x);
• Inverse: xD, if ~P(x) then ~Q(x).
 The original statement is logically equivalent to
its contrapositive.
 Converse is logically equivalent to inverse.
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Necessary and Sufficient
Conditions
• “x, P(x) is a sufficient condition for
Q(x)”
means “x, if P(x) then Q(x)”
• “x, P(x) is a necessary condition
for Q(x)”
means “x, if Q(x) then P(x)”
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Validity of Arguments with
Quantified Statements
Argument form is valid means that
for any substitution of the predicates,
if the premises are true,
then the conclusion is also true.
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Valid Argument Forms:
Universal Instantiation
• x D, P(x);
aD;
P(a).
• If some property is true
for everything in a domain,
then it is true
for any particular thing in the domain.
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Valid Argument Forms:
Universal Instantiation
Ex: 1) All Italians are good cooks;
Tony is an Italian;
 Tony is a good cook.
2) For x,y R, x  y  ( x  y)( x  y)
74.5, 73.5 R
 74.52  73.52  (74.5  73.5)(74.5  73.5)
2
2
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Testing validity by diagrams
• Ex: All integers are rational numbers;
5 is an integer;
 5 is a rational number.
Integers
5
Rational numbers
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Testing validity by diagrams
• Ex: All logicians are mathematicians;
John is not a mathematician;
 John is not a logician.
Logicians
John
Mathematicians
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Testing validity by diagrams:
Converse Error
• Ex: All Math majors are taking Math306;
Bill is taking Math306;
 Bill is a Math major.
Math majors
Bill
Math306 class
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