lecture 8 predicate logic tf4233 mathematical logic semester 2, 2006/2007 dr. rolly intan &...
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Lecture 8 Predicate Logic
TF4233 Mathematical LogicSemester 2, 2006/2007
Dr. Rolly Intan & Cherry Ballangan, MAITInformatics Department, Petra Christian University
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Content
1. Predicate Logic
2. Quantifier
3. Translation Examples
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1. Predicate Logic
“x is greater than 5”
(the universe is real numbers)
Is it a proposition ??
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Are these propositions?
4 is greater than 5. 8 is greater than 5. For all x, x is greater than 5. There is x such that x is greater than 5.
(the universe is real numbers)
1. Predicate Logic
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Predicate logic is…
1. Predicate Logic
P(x) : “x is greater than 5”(the universe is real numbers)
Subject: xPredicate: “is greater than 5”
P(4) : “4 is greater than 5” P(8) : “8 is greater than 5” x P(x) : “For all x, x is greater 5” x P(x) : “There is x such that x is greater than 5”
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2. Quantifier
Universal Quantifier
• “For all”, “For any”, “For every”
Existential Quantifier
• “There exists”, “Some”
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Example
clever(x) : “x is clever”(universe: students)
x clever(x) : “All students are clever” x clever(x) : “Some students are clever” x clever(x) : “Not all students are clever” x clever(x) : “No students is clever”
2. Quantifier
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Negation
x P(x) x P(x) x P(x) x P(x)
Example NOT ALL students are clever
• SOME students are NOT clever
NO students is clever• ALL students are NOT clever
2. Quantifier
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For more than one variable
The order of the quantifiers is important! Example:
• Let P(x, y) : ”x + y = 0” (universal: Real)x y P(x, y)
y x P(x, y)
• Let P(x, y, z): “x + y = z” x y z P(x, y, z)
z x y P(x, y, z)
2. Quantifier
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Laws 1 x ( A(x) B(x) ) (x) A(x) (x) B(x) x (A(x) B(x)) (x)A(x)(x)B(x) (x)A(x) (x)A(x) (x)A(x) (x)A(x) (x)A(x)(x)B(x) x(A(x)B(x)) (x)(A(x)B(x)) (x)A(x)(x)B(x) x(AB(x)) A(x)B(x)
2. Quantifier
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Laws 2
x(AB(x)) A(x)B(x)
(x)(A(x)B) (x)(A(x)B)
(x)A(x)B (x)(A(x)B)
A(x)B(x) (x)(AB(x))
A(x)B(x) (x)(AB(x))
xP(x) P(c), for any c element univ.
P(c ) xP(x)
2. Quantifier
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3. Translation Example
Translate into predicate logic: i) “All lions are fierce.” ii) ”Some lions do not drink coffee.” iii) “Some fierce creature don’t drink
coffee.”
Example 1
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3. Translation Example
Example 1 Answer
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3. Translation Example
Example 2
i). “All Hummingbirds are richly colored.”
ii). “No large birds live on honey.”iii). “Birds that do not live on honey are
dull in color.”iv). “Hummingbirds are small.”
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3. Translation Example
Example 2 Answer
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Exercises
Translate the following statements into predicate logic:
• All students in this class have studied Calculus.
• No students in this class have studied Discrete Mathematics.
If P(x) means “x + 1 > x” (the universe is real number), determine the truth value of the following statements: xP(x)
xP(x)
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Exercises
Negate the following statements: x |x| = x
x x2 = x
• All students live in dormitories
• All math majors are males
• Some students are 25 years or older
• If there is a riot then someone is killed
• It is daylight and all the people have arisen