Math Project Group 2 Part 1. Vincent Duarte, Eduardo Trejo, & Darrick Gowens. Math 170. T/Th 10:00am 11:50am. Professor Stanica

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1 Math Project Group 2 Part 1 Vincent Duarte, Eduardo Trejo, & Darrick Gowens Math 170 T/Th 10:00am 11:50am Professor Stanica October 1, 2014

2 Summary: A predicate is a sentence that contains a finite number of variables and becomes a statement when specific values are substituted for the variables. The domain of a predicate variable is the set of all values that may be substituted in place of the variable. If P(x) is a predicate and x has domain D, the truth set of P(x) is the set of all elements of D that make P(x) true when they are substituted for x. The truth set of P(x) is denoted {x D P(x)}. Quantifiers are words that refer to quantities such as some or all and tell for how many elements a given predicate is true The symbol denotes for all and is called the universal quantifier. Let Q(x) be a predicate and D the domain of x. A universal statement is a statement of the form x D, Q(x). It is defined to be true if, and only if, Q(x) is true for every x in D. It is defined to be false if,and only if, Q(x) is false for at least one x in D. A value for x for which Q(x) is false is called a counterexample to the universal statement. a. Check that x2 x is true for each individual x in D. 12 1, 22 2, 32 3, 42 4, Hence x D,x2 x is true. The technique used to show the truth of the universal statement in Example 3.1.3(a) is called the method of exhaustion. It consists of showing the truth of the predicate separately for each individual element of the domain. This method can, in theory, be used whenever the domain of the predicate variable is finite. The symbol denotes there exists and is called the existential quantifier. Let Q(x) be a predicate and D the domain of x. An existential statement is a statement of the form x D such that Q(x). It is defined to be true if, and only if, Q(x) is true for at least one x in D. It is false if, and only if, Q(x) is false for all x in D.

3 It is important to be able to translate from formal to informal language when trying to make sense of mathematical concepts that are new to you. It is equally important to be able to translate from informal to formal language when thinking out a complicated problem. Another way to restate universal and existential statements informally is to place the quantification at the end of the sentence. For instance, instead of saying For any real number x, x2 is nonnegative, you could say x2 is nonnegative for any real number x. In such a case the quantifier is said to trail the rest of the sentence. A prime number is an integer greater than 1 whose only positive integer factors are itself and 1. Consider the statement If a number is an integer, then it is a rational number. As shown earlier, this statement is equivalent to a universal statement. However, it does not contain the telltale word all or every or any or each. The only clue to indicate its universal quantification comes from the presence of the indefinite article a. This is an example of implicit universal quantification. Existential quantification can also be implicit. For instance, the statement The number 24 can be written as a sum of two even integers can be expressed formally as even integers m and n such that 24=m+n. Notation: Let P(x) and Q(x) be predicates and suppose the common domain of x is D. The notation P(x) Q (x)means that every element in the truth set of P(x) is in the truth set of Q(x), or, equivalently, x, P(x) Q(x). The notation P(x) Q (x)means that P(x) and Q(x) have identical truth sets, or, equivalently, x, P(x) Q(x). The reciprocal of a real number a is a real number b such that ab=1. Negations of Multiply Quantified Statements ( x in D, y in E such that P(x, y)) x in D such that y in E, P(x, y). ( x in D such that y in E, P(x, y)) x in D, y in E such that P(x, y). Taken together, the symbols for quantifiers, variables, predicates, and logical connectives make up what is known as the language of first order logic.

4 Chapter 3.1 Problem #25 The statement The square of any rational number is ratio nal can be rewritten formally as For all rational numbers x, x2 is rational or as For all x, ifx is rational then x2 is rational. 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. b. The derivative of any polynomial function is a polyno mial function. c. The sum of the angles of any triangle is 180. d. The negative of any irrational number is irrational. e. The sum of any two even integers is even. f. The product of any two fractions is a fraction. Solutions: a) * nonzero fractions x, the reciprocal of x is a fraction * x, if x is a nonzero fraction, then the reciprocal of x is a fraction. b) * polynomial function x, the derivative of x is a polynomial function. * x, if x is a polynomial function, then the derivative of x is a polynomial function. c) * triangles x, the sum of the angles of x is 180 * x, if x is a triangle, then the sum of x angles is 180 d) * irrational number x, the negative of x is irrational. * x, if x is a irrational number, then the negative of x is irrational. e) * even integers x and y, the sum of x and y is even. * x and y, if x and y are even integers, then the sum of x and y is even. f) * fractions x and y, the product of x and y is a fraction. * x and y, if x and y are fractions, then the product of x and y is a fraction. Problem #26 Consider the statement All integers are rational numbers but some rational numbers are not integers. a. Write this statement in the form x, if then, but x such that. b. Let Ratl(x) be x is a rational number and Int(x) be x is an integer. Write the given statement formally using only the symbols Ratl(x), Int (x),,,,,, and. Solutions:

5 a) * x, if x integer, then x is a rational number, but rational numbers x such that are not integers. b) * (x)(int(x) Ratl(x1)) x(ratl(x) ~Int(x)) Chapter 3.1 Problem #27 Refer to the picture of Tarski s world given in Example Let Above(x, y) mean that x is above y (but pos sibly 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). a, c, e, g = blue b, h, j = black d, f, i, k = grey Solutions: a) The statement is false. All circle s are either grey or black. b) The statement is true. All gray objects are circles. c) The statement is false because there is no square objects above the letter d. d) The statement is true because g is a triangle and f is above it. Chapter 3.3 Problem #12

6 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 xy y. d. x in D such that y in E, x y. Solutions: a) ~( x) ϵ D ~( y) x + y ~(=) 1 = for all x ϵ D, y ϵ E x + y 1 = there exist Take x = y = y = 3, since y = 3 is not ϵ E, negation is true. b) x ϵ D, y ϵ E. x + y = y Take x = 1 x + y = y x = 2y y y 1 = 2y ½ = y, since ½ is not ϵ E. negation is true c) x ϵ D, y ϵ Exy is not greater than or equal to y. Take x = 2 y = x ( 2)( 2) 2 y = xy is squared which is always true. statement true. d) x ϵ D, y ϵ D, x is not equal to or less than y. Take Statement true x = y = x

7 Biography: Alfred Tarkski was born on January 14, 1901 in Warsaw, Poland into a middle class Jewish family. His family name at birth was Tajtelbaum. During his lifetime he became known as one of the great logicians of the 20th century. As a young man with great mathematical abilities who was seeking the opportunity to pursue an academic career, he had to change his identity to Tarkski in 1923, due to the conditions of Poland. "Poland, one of the new states formed after World War I, tended to treat its large Jewish population like an alien presence."(davis, 2005). Tarkski studied mathematics and philosophy at the University of Warsaw from 1918 to He received his doctoral degree in 1924, under the supervision of Lesniewski. By 1939 he was teaching mathematics at a high school and held minor positions at the University of Warsaw. During this period he published prolifically on logic and set theory, building a reputation for himself. By 1939 he found himself in the United States for a conference, at that time he accepted a position at the University of California, Berkeley. "By the time of his arrival in the United States in 1939, Tarski had some very significant accomplishments to his credit. Perhaps most notorious was the Banach Tarski "paradox"."(davis, 2005). As a professor Tarkski demanded perfection from his students and charmed his students with his knowledge. During his time at Berkeley, Tarkski built a prominent school of research in logic and the foundations of mathematics and science, centered around the prestigious graduate program in logic and methodology of science, which he was instrumental in creating. Tarkski continued to be part of the University of California, Berkeley until his death in Tarkski's work has influenced many researchers but his greatest achievement was convincing the university that logic was important enough to justify resources. Bibliography Davis, Martin.(2005, March April). The Man Who Defined Truth. Retrieved from: man who defined truth

8 Epp, Susanna S.(2010, August). Discrete Mathematics with Applications 4th Edition JOC/EFR.(2003, October). Alfred Tarkski. Retrieved from: groups.dcs.st and.ac.uk/~history/biographies/tarski.html