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