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Math 170 Project Section 3.1 #25,#26,#27 Section 3.3 #12 By: Marlene Reyna, Mayra Aguilar, Guadalupe Esquivel #25 Rewrite each of the following statements in the two forms “ _____ x, _____” and “ x, if _____, then _____” or in the two forms “ x and y, _____, _____” and “ x and y, if _____, then ______.” Symbols used: • Universal quantifier reads as “for all” Ex. Another way to express “All human beings are mortal”, is to write (“ human beings x, x is mortal. 1.Domain of x ___________ x, _________”) 2. Rewrite the statement but Or replace “all human beings” with “x” and treat x as if it was singular. x, if x is a human being, then x is mortal. (If and then statement) (“ x, if _______________, then _________”) Rewrite each of the following statements in the two forms “ _____ x, _____” and “ x, if _____, then _____” or in the two forms “ x and y, _____, _____” and “ x and y, if _____, then ______.” a. The reciprocal of any nonzero fraction is a fraction. • nonzero fractions x, the reciprocal of x is a fraction. 2. Rewrite original statement 1. What is the domain of x? • but replace “any nonzero fraction” with “x”. x, if x is a nonzero fraction, then the reciprocal of x is a fraction. 1. Say what x is based on the domain b. The derivative of any polynomial function is a polynomial function. • polynomial functions x, the derivative of x is a polynomial function. 2. Rewrite original statement but replace “any polynomial function” with “x”. 1. What is the domain of x? • x, if x is a polynomial function, then the derivative of x is a polynomial function. 1.Say what x is based on the domain. c. The sum of the angles of any triangle is 180°. • triangles x, the sum of the angles of x is 180°. 1.What is the domain of x? • 2. Rewrite original statement but replace “any triangle” with “x”. x, if x is a triangle , then the sum of the angles of x is 180°. 1. Say what x is based on the domain. d. The negative of any irrational number is irrational. • irrational numbers x, the negative of x is irrational. 1.What is the domain of x? • 2. Rewrite the original statement but replace “any rational number” with “x”. x, if x is an irrational number, then the negative of x is irrational. 1.Say what x is based on the domain. e. The sum of any two even integers is even. • even integers x and y, the sum of x and y is even. 1.What is the domain of x and y? • 2. Rewrite the original statement but replace “any two even integers” with “x and y” x and y, if x and y are even integers, then the sum of x and y is even. 1.Say what x and y are based on their domain. f. The product of any two fractions is a fraction. • fractions x and y, the product of x and y is a fraction. 1.What is the domain of x and y? • 2. Rewrite the original statement but replace “any two fractions” with “x and y.” x, if x and y are fractions, then the product of x and y is a fraction. 1. Say what x and y are based on their domain. Charles Sanders Peirce • Born on September 10,1839 in Cambridge, MA. • American Philosopher, logician, and engineer • Known as “the father of pragmatism” • Wrote on a wide range of topics which included mathematical logic, physics, astronomy, psychology, etc. • He introduced the formal concept of quantifier into symbolic logic in the late nineteenth century. • Died from cancer on April 19, 1914 in Pennsylvania Section 3.1 #27 Refer to the picture of Tarski’s world given in Example 3.1.13. Let Above(x, y) mean that x is above y (but possibly in a different column). Determine the truth or falsity of each of the following statements. Give reasons for your answers. a. ∀u, Circle(u) →Gray(u). b. ∀u, Gray(u) →Circle(u). c. ∃y such that Square(y) ∧ Above(y,d). d. ∃z such that Triangle(z) ∧ Above(f,z). #26 Consider the statement, “All integers are rational but some rational numbers are not integers.” • A. Write this statement in the form. • “∀ x, if___ then ____, but _____ ∃x such that ____.” • “∀ x, if it’s an integer then it is rational, but some rational numbers ∃x such that are not all integers.” Let Ratl(x) be “x”is a rational number” and Intl(x) be “x is an integer.” • Write the given statement formally using only the symbols Ratl(x), Intl(x), V, ∃, ^, ~, ->. • B. ∀x [Intl(x0 Ratl(x) ] ^∃x [Ratl(x) ^ ~Intl(x)] Tarski’s World a b c e g d f h i j k The program for Tarski’s World provides pictures of blocks of various sizes, shapes, and colors, which are located on a grid. To the left is a picture of an arrangement of objects in a two-dimensional Tarski world. The configuration can be described using logical operators and—for the two-dimensional version— notation a. ∀u, Circle(u) →Gray(u). All circles are gray a False b c e g d A counter example is b, which is a black circle. f h i j k b. ∀u, Gray(u) →Circle(u). All gray shapes are circles a True b c e g d All the gray shapes are circles f h i j k c. ∃y such that Square(y) ∧ Above(y,d). There exist a y that is a square and that square is above d (but possibly in a different column) a b c e g False d Look at the grid and find d. There are no squares above d’s row f h i j k d. ∃z such that Triangle(z) ∧ Above(f,z). There exist a z, which is a Triangle and f is above that triangle(but possibly in a different column) a b c e g True d f h i j k Look at the grid and find f. Then look for the triangle below it. You will see that f is above triangle g. Section 3.3 #12 Let D = E = {−2,−1,0,1,2}. Write negations for each of the following statements and determine which is true, the given statement or its negation. a. ∀x in D,∃y in E such that x + y = 1. b. ∃x in D such that ∀y in E, x + y = −y. c. ∀x in D, ∃y in E such that x y ≥ y. d. ∃x in D such that ∀y in E, x ≤ y. Negation Symbol: ~ The negation of a statement can be found by using De Morgan’s Law ~(p ∧ q) = ~p v ~q or ~(p v q) = ~p ∧ ~q “The negation of p and q is not p or not q” The negation of a quantifier can be found using the same Law. ~(∀x)= ∃x and ~(∃x)= ∀x And everything that follows will be negated the same way a statement is negated Let D = E = {−2,−1,0,1,2}. a. ∀x in D, ∃y in E such that x + y = 1. Determine which is true. Original or Negation? ~(∀x in D,∃y in E such that x + y = 1) Negation: ∃x in D, such that ∀y in E x + y ≠ 1. “There exists an x in D , such that all y’s in E make x + y ≠ 1” Let x=-2 When you plug x=-2 into the original statement you get: -2 + y = 1 y= 3 3 is not in E Hence, the original statement is False. That negation is true Let D = E = {−2,−1,0,1,2}. b. ∃x in D such that ∀y in E, x + y = −y. Determine which is true. Original or Negation? ~(∃x in D such that ∀y in E, x + y = −y) Negation: ∀ x in D, ∃y in E such that x + y ≠ −y “For all x in D, there exist a y in E such that x + y ≠ −y Let x= 1 and y= 1 When you plug x and y into the original statement you get: 1+ 1 = -1 2 ≠ -1 Hence, the original statement is false. The negation is true. Let D = E = {−2,−1,0,1,2}. c. ∀x in D, ∃y in E such that x y ≥ y. Determine which is true. Original or Negation? ~(∀x in D, ∃y in E such that x y ≥ y) Negation: ∃x in D, such that ∀y in E x y < y “There exists an x in D such that all y’s in E makes x y < y” Let x=1 and y= -2 When you plug x and y into the negation you get: (1)(-2)< -2 -2 < -2 Since the inequality is less than and not less than or equal to the statement is false The original statement is true Let D = E = {−2,−1,0,1,2}. d. ∃x in D such that ∀y in E, x ≤ y. Determine which is true. Original or Negation? ~(∃x in D such that ∀y in E, x ≤ y) Negation: ∀ x in D, ∃ y in E such that x >y “For all x in D, there exists a y in E such that x>y” Let x= 2 and y= 2 When we plug in x into the negation we get: 2>2 Since the inequality is less than and not less than or equal to the statement is false The original statement is true Bibliography • Charles Sanders Peirce- Biography. (n.d.). The European Graduate School- Media and Cmmunication- Graduate and Postgraduate Studies Program. Retrieved October 09, 2013 from • http:??www.egs.edu/library/charles-sanders-peirce/bibliography