1. Lecture 8 Predicate Logic TF4233 Mathematical Logic Semester 2, 2006/2007 Dr. Rolly Intan & Cherry Ballangan, MAIT Informatics Department, Petra Christian University

2. TF4233 Math. Logic Petra Christian University 2 Content 1. Predicate Logic 2. Quantifier 3. Translation Examples

3. TF4233 Math. Logic Petra Christian University 3 1. Predicate Logic “x is greater than 5” (the universe is real numbers) Is it a proposition ??

4. TF4233 Math. Logic Petra Christian University 4 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

5. TF4233 Math. Logic Petra Christian University 5 Predicate logic is… 1. Predicate Logic P(x) : “x is greater than 5” (the universe is real numbers) Subject: x Predicate: “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”

6. TF4233 Math. Logic Petra Christian University 6 2. Quantifier Universal Quantifier  “For all”, “For any”, “For every” Existential Quantifier  “There exists”, “Some”

7. TF4233 Math. Logic Petra Christian University 7 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

8. TF4233 Math. Logic Petra Christian University 8 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

9. TF4233 Math. Logic Petra Christian University 9 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

10. TF4233 Math. Logic Petra Christian University 10 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(A  B(x))  A  (  x)B(x) 2. Quantifier

11. TF4233 Math. Logic Petra Christian University 11 Laws 2  x(A  B(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)(A  B(x)) A  (  x)B(x)  (  x)(A  B(x))  xP(x)  P(c), for any c element univ. P(c )  xP(x) 2. Quantifier

12. TF4233 Math. Logic Petra Christian University 12 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

13. TF4233 Math. Logic Petra Christian University 13 3. Translation Example Example 1 Answer

14. TF4233 Math. Logic Petra Christian University 14 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.”

15. TF4233 Math. Logic Petra Christian University 15 3. Translation Example Example 2 Answer

16. TF4233 Math. Logic Petra Christian University 16 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)

17. TF4233 Math. Logic Petra Christian University 17 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