Quantifiers and Statements

Transcription

1 Daniel Aguilar, Jessica Mean, and Marcus Hughes Math 170 Project (Part 1) Quantifiers and Statements There is basic knowledge that you will need to know before you read the project. Such as: A statement is simply a sentence that can either be answered as true or as false. I play basketball is a statement because it can be answered as true or false. Do your homework is not a statement, because it can t be answered as true or false. A predicate is a statement that contains one or more variables ( x) Existential quantifiers indicates that for some value, there is a statement ( x) Universal quantifiers indicates that for all values, there is a statement Other needed knowledge will be provided at the beginning of each problem* Contributing Mathematicians: Charles Sanders Pierce was a philosopher, logician, and engineer who contributed the concept of qualifiers into symbolic logic. In addition, Gottlob Frege, a logician, contributed and introduced the concept of qualifiers as well. Word problems (state the question and other info that breaks down each question part by part) Section ) Let D be the set of all students at your house, and let M(s) be s is a math major, let C(s) be s is a computer science student, and let E(s) be s is an engineering student. Express each of the following statements using quantifiers, variables, and predicates M(s), C(s), and E(s). NOTE: To do these problems you must first read the statement, and try to point out which predicate are being used and which quantifier is being expressed. The domain is being referred to all the students in my house. A.) There is an engineering student who is a math major. There is meaning at least one, and this means that it is an Existential quantifier ( s). an engineering student predicate E(s). who is meaning also, and. a math major predicate M(s). ( s) D, E(s) M(s)

2 B.) Every computer science student is an engineering student. Every meaning all, and this means that it is a Universal quantifier ( s). is meaning therefore, and. an engineering student predicate E(s). ( s) D, C(s) E(s) C.) No computer science students are engineering students. No meaning none (not all), and this means that it is a Universal quantifier ( s). No negation ~ are meaning therefore, and. an engineering student predicate E(s). ( s) D, ~C(s) E(s) D.) Some computer science students are also math majors. Some meaning at least one, and this means that it is an Existential quantifier ( s). are also meaning therefore, and. math majors predicate M(s). ( s) D, C(s) M(s) E.) Some computer science students are engineering students and some are not. Some meaning at least one, and this means that it is an Existential quantifier ( s).

3 negation ~ are meaning therefore, and. engineering students predicate E(s). and some are not a new statement that is Existential s, it has a AND, and a [( s) D, C(s) E(s)] [( s) D, C(s) ~E(s)] 32.) Let R be the domain of the predicate variable x. Which of the following are true and which are false? Give counter examples for the statements that are false. NOTE: R, the domain, means that x can be all real numbers A.) x > 2 x > 1 is the sign used for implications and it means then or therefore If x is greater than 2, then x is greater than 1 True. Any number greater than 2 is also greater than 1 B.) x > 2 x 2 > 4 is the sign used for implications and it means then or therefore If x is greater than 2, then x squared is greater than 4 True. Any number that is greater than 2 and is squared is then greater than 4 C.) x 2 > 4 x > 2 is the sign used for implications and it means then or therefore If x squared is greater than 4, then x is greater than 2 This statement is False. Counterexample: Let x = 3; 3 2 is greater than 4, but 3 is not greater than 2 D.) x 2 > 4 x > 2 means if and only if

4 x is the distance that x is from 0 on a number line ex) 3 = 3 ; 3 = 3 x squared is greater than 4 if and only if the absolute value of x is greater than 2 True. Any number greater than 2 that is squared or any number less than 2 that is squared is greater than ) Let R be the domain of the predicate variables a, b, c and d. Which of the following are true and which are false? Give counterexamples for statements that are false. NOTE: To do these problems, you must first read the statements and decide whether the predicate satisfy the statement. Also, R, the domain, means that x can be all real numbers. A.) a > 0 and b > 0 ab > 0 a is greater than 0 and b is greater than 0. Then, a times b is greater than 0. True. a is a positive integer and b is a positive integer. So a positive integer multiplied by another positive integer is positive which is greater than 0. B.) a < 0 and b < 0 ab < 0 < means less than a is less than 0 and b is less than 0. Then, a times b is less than 0. False. Since a and b are less than 0, they are negative integers. Two negative integers multiplied together is positive. Counterexample: Let a= 2 and b= 5. So ( 2) ( 5) < 0 is 10 < 0 which is a false statement. C.) ab=0 a=0 or b=0

5 = means equals a times b equals 0. Then, a equals 0 or b equals 0. True. Any number times by 0 is 0. Therefore, if a is 0 then the statement is 0. If b is 0 then the statement is 0. D.) a<b and c<d ac < bd < means less than a is less than b and c is less than d. Then a times c is less than b times d. False. Counterexample: Let a= 3, b= 3, c= 5, and d= 5. So ac= ( 3)( 5)=15. And bd=(3)(50=15. However, 15 is not less than 15. Section ) Let S be the set of students at your school, let M be the set of movies that have ever been released, and let V(s,m) be student s has seen movie m. Rewrite each of the following statements without using the symbol, the symbol, or variables. NOTE: To begin these problems you must identify key symbols and understand the limits of the domain(s). Domain S refers to all the students at CSUMB and the domain M refers to all movies that has ever been released A.) s S such that V(s, Casablanca). s S: There is at least one student at CSUMB V(s, Casablanca): some student has seen movie Casablanca. There is a student at CSUMB who have seen the movie Casablanca. B.) s S, V(s, Star Wars).

6 s S: All students at CSUMB V(s, Star Wars): some student has seen movie Star Wars. All students at CSUMB have seen the movie Star Wars. C.) s S, m M such that V(s, m). s S: All students at CSUMB m M: At least one movie All students at CSUMB have seen at least one movie. D.) m M such that s S, V(s, m). m M: At least one movie s S: All students at CSUMB There is some movie that all students at CSUMB have seen. E.) s S, t S m M such that s =/ t and V(s, m) V(t, m). s S: Some student at CSUMB t S: a different student at CSUMB m s =/ M: At least one movie t: the two students are not the same person : and V(t, m): a different student has seen some movie There are two different students at CSUMB that have seen the same movie.

7 F.) s S and t S such that s =/ t and m M, V(s, m) V(t, m). s S: Some student at CSUMB t S: a different student at CSUMB s =/ m t: the two students are not the same person M: At least one movie : therefore V(t, m): a different student has seen some movie There is a student at CSUMB which every movie that person seen was seen by a different student at CSUMB. **Disclaimer: All examples used in this projects were used from the textbook Discrete Mathematics with Applications (Fourth Edition) Works Cited: S. S. Epp. Discrete Mathematics with Applications (Fourth Edition). Brooks Cole, Boston, 2004.