Group 5. Jeremy Gutierrez. Jesus Ochoa Perez. Alvaro Gonzalez. MATH 170: Discrete Mathematics. Dr. Lipika Deka. March 14, 2014.

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1 Gutierrez, Perez, & Gonzalez 1 Group 5 Jeremy Gutierrez Jesus Ochoa Perez Alvaro Gonzalez MATH 170: Discrete Mathematics Dr. Lipika Deka March 14, 2014 Project Part 1

2 Gutierrez, Perez, & Gonzalez 2 Hello today we will be working on Negating Quantifier problem(s) in Discrete Mathematics or Math 170 (according to CSUMB), but before we start working on the problem(s) we must know the following: Predicates, and Quantifiers Predicates: In Math 170 Predicates are functions with finitely many variables that becomes a statement when specific values are applied. Domain of Predicate: All possible values that the variable can take Truth of a Predicate: The set of all elements in the domain that make it a true statement both. *Statements in Math 170 are English sentences that are can be either true or false but not Quantifiers: Discrete Math quantifiers are logical symbols that make assertions about the set of values which make the formula true or false. There are two types of quantifiers, The Universal quantifier and the Existential quantifier. Universal Quantifiers: The symbol is used in the universal quantifier so x ε D, P(x): P(x) is True for all values in the domain. *D is domain, P(x) is predicate, x is variable, is for all Existential Quantifier: In Discrete math Existential Quantifier means that not all but some variable exists in the domain that makes the predicate or P(x) true so x ε D, P(x) : Whatever variable makes the statement true. Now that we understand what terms are these we are will know get to know the Negation of Quantifiers. In Discrete Math the negative sign of something will be ~ instead of your average sign. ~ means not it works like a negative sign. Negation of quantifiers: ~ ( ) can be seen as ( ) which be equivalent to. ~ ( ) can be seen as ( ) which will be equivalent to *Note: When distributing ~ to (something) the term will be called De Morgan s Law a law where one distributes the negative to the values that are next to it or in the parenthesis *Note: Negate the quantifier and the predicate expression that is following the quantifier Now we understand the stuff so We can work on Negating quantifiers so here is a problem's from chapter 3.2 and 3.3 from Discrete Mathematics with Applications textbook that was written by Susanna S. Epp (Epp, 2011).

3 Gutierrez, Perez, & Gonzalez 3 Section 3.2 #8 Consider the statement "There are no simple solutions to life's problems." Write an informal negation for the statement, and then write the statement formally using quantifiers and variables. Informal negation: There is at least one simple solution to life's problems Formal Statement: Key Points: simple solutions x, x is not a simple solution to life's problems. Predicate: A predicate is a function with finitely many variables that becomes a statement when specific values of the variables are applied. Domain of Predicate: All possible values that the variables can take. Universal quantifier ( ) is where P(x) is true for all values in the domain. Existential quantifiers (Э) There exist in some x value in the domain for which P(x) is true. Section 3.2 #25 Each of the following statements is true. In each case write the converse of the statement, and give a counterexample showing that the converse is false. a. If n is any prime number that is greater than 2, then n +1 is even. Converse: If n+1 is even, then n is any prime number that is greater than 2. Counterexample: Let n = 21, then n +1 = 22. In this case 22 is even and n is not a prime number that is greater than 2. b. If m is any given odd integer, then 2m is even.

4 Gutierrez, Perez, & Gonzalez 4 Converse: If 2m is even, then m is any given odd integer. Counterexample: Let m = 6, then 2m = 12. In this case 12 is an even number and m in not an odd integer. c. If two circles intersect in exactly two points, then they do not have a common center. Converse: If two circles do not have a common center, then they intersect in exactly two point. Counterexample: If circle #1 has a radius of 1 and it has a center at ( 0,0 ) then a second circle can have a radius of 1 and have a center of ( 3,0 ) and at no point will the two circles intersect. Section 3.2 #34 Write the contrapositive for each of the following statements. a. If n is prime, then n is not divisible by any prime number between 1 and inclusive. (Assume that n is a fixed integer that is greater than 1.) Contrapositive: If n is divisible by any prime number between 1 and inclusive, then n is not prime. b. If A and B do not have any elements in common, then they are disjoint. (Assume that A and B are fixed sets.) Contrapositive: If A and B are not disjoint, then they have elements in common.

5 Gutierrez, Perez, & Gonzalez 5 Section 3.3 #24 Use the laws for negating universal and existential statements to derive the following rules: * We have to find how to get the from the left side to the right side. Let s try this one out. * The 3 lines that are stacked on each other means that it is equivalent to a. ( x D( y E(P(x, y)))) x D( y E( P(x, y))) 1. Distribute the ~ to the Parenthesis ( x D ( y E(P(x, y)))) *Only will be affected by ~ 2:Distribute the ~to the rest of the parenthesis *remember ~( ) (equivalent to) x D (~( y E(P(x, y)))) 3. Result of distribution is x D( y E( P(x, y))) 4. Solution: ( x D( y E(P(x, y)))) x D (~( y E(P(x, y)))) x D( y E( P(x, y))) Now let s work on ch3.3 problem 24 b. b. ( x D( y E(P(x, y)))) x D( y E( P(x, y)))

6 Gutierrez, Perez, & Gonzalez 6 1. Distribute ~to parenthesis ( x D ( y E(P(x, y)))) 2. Distribute ~ to other parenthesis. x D ( ~( y E(P(x, y)))) 3. Result x D ( y E( P(x, y))) 4. Solution: ( x D( y E(P(x, y)))) x D(~( y E(P(x, y)))) x D( y E( P(x, y)))

7 Gutierrez, Perez, & Gonzalez 7 Bibliography We have used De Morgan s law in order to perform the operations to get us to the solutions that in Problem 24 from Chapter 3 of the Discrete Mathematics with Applications textbook. Augustus De Morgan was born on June 27th, 1806 and died on March 18th, 1871.Augustus De Morgan was an English mathematician and logician who received his education at Trinity College, Cambridge, in 1828 De Morgan became a math professor who not only taught but as well published works such as Elements of Arithmetic (1830). Later in his career had created De Morgan s law which later on (and still) contributes to the mathematics to this very day. De Morgan s law is a rule where it is used to distribute to quantifiers, and to logic operations. De Morgan s law can be interpreted as distributing the negative sign to math operations. The laws names are theorems that have the possibility of altering statements and formulas into create something that is new. His laws where that the negation of a statement is equivalent to the disjunction of the original statement and the negation of a statement is equivalent to the disjunction of the negation of the original statement. De Morgan was one of those Mathematicians who helped developed mathematical logic and modern math theories and his work (laws) are still put into great use of the mathematical world.

8 Gutierrez, Perez, & Gonzalez 8 Bibliography Deka, D. L. (2014). Lecture Notes. Seaside, Ca, USA. Encyclopaedia Britannica. (2013, May 9). August De Morgan. Retrieved March 13, 2014, from Encyclopaedia Britannica: Morgan Epp, S. S. (2011). Discrete Mathematics with Applications 4th Edition. Chicago: Brooks/Cole Cengage Learning. Sinapova, L. (2013, 12 22). Negation of Quantifiers. Retrieved March 13, 2014, from Simposon University: NegationQ.htm

THE LOGIC OF QUANTIFIED STATEMENTS

CHAPTER 3 THE LOGIC OF QUANTIFIED STATEMENTS Copyright Cengage Learning. All rights reserved. SECTION 3.2 Predicates and Quantified Statements II Copyright Cengage Learning. All rights reserved. Negations