Writing proofs for MATH 51H Section 2: Set theory, proofs of existential statements, proofs of uniqueness statements, proof by cases
|
|
- Chloe Dickerson
- 5 years ago
- Views:
Transcription
1 Writing proofs for MATH 51H Section 2: Set theory, proofs of existential statements, proofs of uniqueness statements, proof by cases September 22, 2018 Recall from last week that the purpose of a proof is to communicate an argument. Your writing alone must convey the argument without you being there to explain what you meant to say. Here are some general reminders/guidelines to follow which will make your proofs clearer and more readable: Write in complete, grammatically correct sentences. Signal to your reader what you are doing in your proof. If you are proving a universal statement about elements in some set (for example, a vector space), say that you are arguing with a general element of that set. Don t just start doing an argument with a vector v without saying what v is. If you are doing a proof by contradiction, say at the beginning of the proof that you are doing a proof by contradiction. If you are doing a proof by induction, say at the beginning of the proof that you are doing a proof by induction and clearly label the base case and induction step somehow. If at some point in the proof you want to break the argument up into cases, say that you are breaking the proof up into cases and label the cases. If your proof has multiple parts (for example, an proof or showing that two sets are equal), then before each of these parts state what you are proving. This week we ll cover some basic notions in set theory that you ll need in this class and your life afterwards. In particular, we ll discuss what it means to show that two sets are equal. We ll also cover proving existence results, proving uniqueness results, and the strategy of division into cases. 1 Basic set theory At this point in the course, you ve encountered the notion of two sets (probably the span of some vectors) being equal. We intuitively know what it means for two sets to be equal: they 1
2 have exactly the same elements. In contrast to the situation for proving that two numbers are equal, where you start with one and do some manipulations to turn it into the other, it s not 100% obvious how to go about proving set equality. Recall that given two sets A and B, we say that A is a subset of B and write A B if every object in A is contained in B, i.e. if x A, then x B as well. We say that A and B are equal if they have the same elements, i.e. every element in A is also in B and every element in B is also in A. Note that A = B is equivalent to A B and B A. The most common strategy in showing that two sets are equal is to show these two inclusions separately. So to show A = B, show that A B and then that B A. We will shortly see two examples of this sort of proof. Assume, for the purposes of this section, that all of the sets considered are contained in some fixed larger set, so that we can define complements. Recall that given two sets A and B, their union A B is the set of all elements which are in A or B, A B = {x : x A or x B}. Here or is inclusive, so elements which are in both A and B are in A B. Also recall that given A and B, their intersection A B is the set of all elements which are in both A and B, A B = {x : x A and x B}. We can also define the union and intersection of any number of sets, finite or infinite, in the same manner. Given a sequence of sets A 1, A 2, A 3,..., then their union is A n = {x : x A n for some n N} n=1 and their intersection is A n = {x : x A n for all n N}. n=1 Since we have fixed a universe, we define the complement of A, denoted A c, to be all of the elements in the universe which are not in A, A c = {x : x / A}. Here you should be careful: complements of sets depend on the universe, that is, the set with respect to which we take the complement. For example, the complement of {1, 2, 3,... } in Z is different than its complement in Q. We ll now present two examples which demonstrate the strategy discussed above for proving that two sets A and B are equal show A B and then B A. Here is a template showing the general form of one of these proofs: Proof. First we will show that A B. Let x A. 2
3 proof that x B Thus, since x A was arbitrary, A B. Now we will show that B A. Let x B. proof that x A Thus, since x B was arbitrary, B A. Putting the two inclusions together, we conclude that A = B. 1.1 Examples Our first example relates unions, intersections, and complements. Proposition (De Morgan s laws, part 1). For any sets A and B, (A B) c = A c B c. Proof. First we will show that (A B) c A c B c. Let x (A B) c. Then by definition, x / A B, which means that x / A and x / B. But what this means is that x A c and x B c. Hence, x A c B c. Thus, since x (A B) c was arbitrary, we see that (A B) c A c B c. Now we will show that A c B c (A B) c. Let x A c B c. Then by definition, x A c and x B c, which means that x / A and x / B. Hence, x / A B, which means that x (A B) c. Since x A c B c was arbitrary, we see that A c B c (A B) c. Since (A B) c A c B c and A c B c (A B) c, we conclude that (A B) c = A c B c. This second example is more in line with what you ve done in class. Proposition. Let v 1, v 2 R n. Then span{v 1, v 2 } = span{v 1, 2v 2 }. Proof. First we ll show that span{v 1, 2v 2 } span{v 1, v 2 }. Let v span{v 1, 2v 2 }. Then v = a 1 v 1 + a 2 (2v 2 ) for some a 1, a 2 R. But then v = a 1 v 1 + (2a 2 )v 2, a linear combination of v 1 and v 2. Hence, v span{v 1, v 2 }. Since v was arbitrary, we ve shown that span{v 1, 2v 2 } span{v 1, v 2 }. Now we ll show that span{v 1, v 2 } span{v 1, 2v 2 }. Let v span{v 1, v 2 }. Then v = a 1 v 1 + a 2 v 2 for some a 1, a 2 R. But note that we can write v = a 1 v 1 + a 2 v 2 = a 1 v 1 + a 2 2 (2v 2 ), a linear combination of v 1 and v 2. Hence, v span{v 1, 2v 2 }. Since v was arbitrary, we ve shown that span{v 1, v 2 } span{v 1, 2v 2 }. Hence, since span{v 1, 2v 2 } span{v 1, v 2 } and span{v 1, v 2 } span{v 1, 2v 2 }, we conclude that span{v 1, v 2 } = span{v 1, 2v 2 }. 1.2 Exercises Exercise 1. Prove the second part of De Morgan s laws: for two sets A and B, (A B) c = A c B c. Exercise 2. Prove that for any sets A, B, and C, A (B C) = (A B) (A C). Exercise 3. Show that for any v 1, v 2 R n, span{v 1, v 2 } = span{v 1 + v 2, v 1 v 2 }. 3
4 Exercise 4. Show that {10n : n Z} = {2n : n Z} {5n : n Z}. Exercise 5. Prove that [n, n + 1) = [0, ). n=0 2 Proving existential statements, Part 1 Suppose that you want to prove that there exists some object X (which may be an integer, a vector, a set, etc.) such that some property P(X) (X satisfies some equation, X is in some span, X has a certain number of elements satisfying some conditions, etc.) holds. There are several strategies for proving these sort of existence statements. The simplest one is to give an explicit example of an X for which P(X) is true. You did one of these proofs on the previous homework, where you had to find a polynomial that took certain values at certain points. The real work in these sort of results is in finding the example, and then the proof itself is usually just verifying that the example does indeed work. The general structure of the proof is to give a candidate object for X and then show that P(X) does, in fact, hold: Proof. Consider A. proof that P(A) is true Thus, there exists an X such that P(X) is true, namely A. 2.1 Examples Proposition. There exist three distinct positive integers a, b, and c such that 1 a + 1 b + 1 c = 1. Proof. Consider a = 2, b = 3, and c = 6. Then 1 a + 1 b + 1 c = = = = 1, and a b, b c, c a. Thus, there do indeed exist three such integers, namely a = 2, b = 3, c = 6. Proposition. There exist x, y R that solve the system of equations { 2x y = 1. x + y = 2 Proof. Consider x = 3 and y = 5. Then 2x y = = 6 5 = 1 and x+y = 3+5 = 2. Thus, there do indeed exist two such real numbers, namely x = 3 and y = 5. 4
5 2.2 Exercises Exercise 6. Show that if a, b, c, d Z with a c, then there is an r Q such that ar + b cr + d = 1. Exercise 7. Prove that there exist arbitrarily large gaps between prime numbers. That is, for each n N, there exist at least n consecutive positive integers which are not prime. 3 Proving existential statements, Part 2 Another type of existence proof is to show indirectly that an object satisfying the property P(X) exists. There are several ways that you could do this. You could invoke some theorem which tells you that such an object exists, such as the intermediate value theorem or the mean value theorem. Alternatively, you could do a proof by contradiction (which we discussed last week) by showing that if such an object didn t exist, then we d be led to something which is false. Here are examples of both of these types of indirect proofs of existence. 3.1 Examples Proposition. The polynomial p(x) = x x 8 39x x 4 5x 2 10 has a root in [0, 1]. Proof. Note that p(x) is a continuous function on [0, 1], and thus the intermediate value theorem from calculus applies. We have p(0) = 10 and p(1) = = 90. Thus, by the intermediate value theorem, p(x) = 0 for some x [0, 1]. That is, p(x) has a root in [0, 1]. Recall that an integer p 2 is prime if whenever p = ab, a, b N, then either a = 1 or b = 1. We ll use without proof the fact that any positive integer greater than 1 is divisible by some prime. This is, in fact, an existence statement: given any integer greater than 1, there exists a prime dividing it. The proof of this is left as an exercise. Proposition. There are infinitely many prime numbers. Proof. Suppose by way of contradiction that there were only finitely many prime numbers p 1, p 2,..., p k. Consider the integer n = p 1 p 2 p k + 1. Since there exists a prime number (for example, 2), n > 1, so that n is divisible by some prime p i, i = 1,..., k. So since the difference of two multiples of a number is still a multiple of that number, we have that p i (n p 1 p k ). But n p 1 p k = 1, so this says that p i 1, a contradiction since p i 2 by the definition of a prime number. Thus, there must be infinitely many prime numbers. 5
6 3.2 Exercises Exercise 8. Prove that there is an x R such that x = 2 x. Exercise 9. Show that every integer greater than 1 is divisible by a prime. Exercise 10. Recall that an integer n 2 which is not prime is called composite. Prove that if n is a composite integer, then n has a prime factor p such that p n. 4 Proving uniqueness, Part 1 One very useful strategy for proving that if there is an object satisfying some property, then it is unique, is to assume that there are two objects with this property and then show that they must be equal. You ve already seen this technique used in class several times, so we ll recap the proofs as examples. The general strategy to show that an object, if it exists, which satisfies some property P(X) is the unique object that satisfies P(X) is to show that if P(A) and P(B) are true for some objects A and B, then we must have A = B. Proof. Suppose that P(A) and P(B) are true. proof that A = B Thus, we have shown that any two objects satisfying P(X) are equal, which implies that the object satisfying P(X), if it exists, is unique. 4.1 Examples Proposition. The identity element in a group G is unique. Proof. Suppose that e 1 and e 2 are such that, for all g G, ge 1 = e 1 g = g and ge 2 = e 2 g = g. Then setting g = e 2 in the first equation, we see that e 1 e 2 = e 2. Similarly, setting g = e 1 in the second equation, we see that e 1 e 2 = e 1. Hence, e 1 = e 1 e 2 = e 2, and we see that any two identity elements are equal, which implies that the identity element in G is unique. Proposition. Let A R. Then the supremum of A, if it exists, is unique. Proof. Suppose that a 1, a 2 are both the supremum of A. By definition, this means that for all a A, a a 1 and a a 2. In particular, specializing a to be a 2, we see that a 2 a 1. Reversing the roles, when a = a 1, we see that a 1 a 2. Now, a 2 a 1 and a 1 a 2 taken together imply that a 1 = a 2. Thus, the supremum, if it exists, is unique, for any supremum of A equals a 1. The following example was not covered in class. Proposition (Division algorithm, uniqueness). Suppose that n, d N and we have found integers q, r Z such that n = dq + r and 0 r < d. Then this pair (q, r) is unique. 6
7 Proof. Suppose that q 1, r 1, q 2, r 2 Z are such that n = dq 1 + r 1 and n = dq 2 + r 2 with 0 r 1, r 2 < d. Then we have dq 1 + r 1 = dq 2 + r 2, and moving the q terms to one side and r terms to the other, this is equivalent to d(q 1 q 2 ) = r 2 r 1. By the definition of divisibility, we see that d (r 2 r 1 ). However, 0 r 1, r 2 < d, so that r 1 r 2 < d. Since d can t divide any nonzero integers with absolute value less than d, we conclude that r 1 = r 2. Plugging this back in to the equation above, since d > 0, q 1 q 2 = 0, i.e. q 1 = q 2. Hence, (q 1, r 1 ) = (q 2, r 2 ), and thus we see that the pair (q, r) in the division algorithm is unique. 4.2 Exercises Exercise 11. Show that the infimum of a set of real numbers, if it exists, is unique. Exercise 12. For any r R, show that there is at most one integer in the interval (r, r ). 5 Proving uniqueness, Part 2 The next strategy is a way to show that the solution to some equation or system of equations is unique. You start with the equation or system of equations and show through manipulations what the solution must be if it exists. If all of the steps you ve used to get to that point are reversible, as in the case of Gaussian elimination, then this sort of proof also shows existence. The general form of the proof is as follows. If your aim is to prove that some solution to the equation f(x) = 0 is unique, then you should start your proof by assuming that x is such that f(x) = 0 holds, and then try to solve for x. If you get that x must equal some constant, then this shows that the solution in x, if it exists, is unique. Here are two examples demonstrating this strategy. 5.1 Examples Proposition. The solution in x and y to the system of equations { x + y = 0, x 2y = 0 if it exists, is unique. Proof. Suppose that x and y are such that x + y = 0 and x 2y = 0. Then adding two times the first equation to the second, we see that 3x = 0. Thus, x = 0. Now, this implies that since x + y = 0 that y = 0 as well. We have shown that if x and y solve the above system of equations, then x = y = 0. Hence, the solution is indeed unique. It s obvious in the previous example that x = y = 0 is a solution without doing any manipulations. In addition, since every step we did was reversible, our manipulations also gave a proof that x = y = 0 does indeed solve the above system of equations. 7
8 Proposition. The solution in R of x + 4 = x 2, if it exists, is unique. Proof. Suppose that x R is such that x + 4 = x 2. Then squaring both sides, x + 4 = x 2 4x + 4, and simplifying gives x 2 5x = 0. This equation factors as x(x 5) = 0, and hence we see that either x = 0 or x = 5. However, x = 0 is not a solution to the original equation x + 4 = x 2, for plugging it in to the left hand side yields 4 = 2 and plugging it in to the right hand side yields 2, and 2 2. Hence, x is forced to be 5. Thus, we have shown that any two solutions to the above equation are equal, for we have shown that they are equal to 5. So, the solution to the equation in question is unique. Here, in contrast to the system of linear equations example above, our manipulations do not show that x = 0 or x = 5 is a solution to the equation x + 4 = x 2. This is because squaring both sides of an equation is not a reversible step, for y 2 = ( y) 2 for all y R, but y y whenever y 0, so we d have to consider both positive and negative square roots if we tried to undo the squaring. You have to plug x = 5 into the equation to check that it is a solution. 5.2 Exercise Exercise 13. Show that the solution (x, y, z) to the system of equations x + y z = 1 x + 2y + z = 0, 3x y z = 2 if it exists, is unique. 6 Proof by cases Sometimes it happens that you need to consider two different possibilities in your proof, and these cases need to be treated differently. In this case, you should label your cases as below, or in some other way signal to the reader what you are proving and when. Remember to inform the reader that you are splitting up the proof into cases. Proof. We split the proof up into k cases. Case 1 : Suppose that [conditions defining Case 1] proof of the result in Case 1 Case 2 : Suppose that [conditions defining Case 2] proof of the result in Case 2 Case k: Suppose that [conditions defining Case k]. 8
9 proof of the result in Case k Since these cases cover all possibilities, the result holds. 6.1 Examples Proposition. For all r R, r r r. Proof. We split the proof up into two cases: r 0 and r < 0. Case 1, r 0: Suppose that r 0. Then r = r. Certainly r r = r, and r 0 r = r. Hence, r r r in this case. Case 2, r < 0: Suppose now that r < 0. Then r = r. We have r < 0 < r = r, so that r r, and r = ( r) = r r. Hence, r r r in this case. So since these cases cover all possibilities for a real number, for all r R, r r r. Proposition. For all r R, r = r. Proof. We split the proof up into two cases: r 0 and r < 0. Case 1, r 0: Suppose that r 0. Then r 0, so that r = ( r) = r. But r = r, so we conclude that indeed r = r in this case. Case 2, r < 0: Suppose now that r < 0. Then r > 0, so that r = r. But r = r, so we conclude that indeed r = r in this case was well. So since these cases cover all possibilities for a real number, for all r R, r = r. 6.2 Exercises Exercise 14. Prove that for all x, y R, x y = xy. Exercise 15. Prove that if n Z is odd, then 8 (n 3 1). Exercise 16. Prove that if n Z is not a multiple of 5, then 5 (n 4 1). 9
Writing proofs for MATH 61CM, 61DM Week 1: basic logic, proof by contradiction, proof by induction
Writing proofs for MATH 61CM, 61DM Week 1: basic logic, proof by contradiction, proof by induction written by Sarah Peluse, revised by Evangelie Zachos and Lisa Sauermann September 27, 2016 1 Introduction
More informationHOW TO CREATE A PROOF. Writing proofs is typically not a straightforward, algorithmic process such as calculating
HOW TO CREATE A PROOF ALLAN YASHINSKI Abstract We discuss how to structure a proof based on the statement being proved Writing proofs is typically not a straightforward, algorithmic process such as calculating
More informationProofs. Chapter 2 P P Q Q
Chapter Proofs In this chapter we develop three methods for proving a statement. To start let s suppose the statement is of the form P Q or if P, then Q. Direct: This method typically starts with P. Then,
More information6.2 Deeper Properties of Continuous Functions
6.2. DEEPER PROPERTIES OF CONTINUOUS FUNCTIONS 69 6.2 Deeper Properties of Continuous Functions 6.2. Intermediate Value Theorem and Consequences When one studies a function, one is usually interested in
More information2 Lecture 2: Logical statements and proof by contradiction Lecture 10: More on Permutations, Group Homomorphisms 31
Contents 1 Lecture 1: Introduction 2 2 Lecture 2: Logical statements and proof by contradiction 7 3 Lecture 3: Induction and Well-Ordering Principle 11 4 Lecture 4: Definition of a Group and examples 15
More informationSome Review Problems for Exam 1: Solutions
Math 3355 Fall 2018 Some Review Problems for Exam 1: Solutions Here is my quick review of proof techniques. I will focus exclusively on propositions of the form p q, or more properly, x P (x) Q(x) or x
More information18.S097 Introduction to Proofs IAP 2015 Lecture Notes 1 (1/5/2015)
18.S097 Introduction to Proofs IAP 2015 Lecture Notes 1 (1/5/2015) 1. Introduction The goal for this course is to provide a quick, and hopefully somewhat gentle, introduction to the task of formulating
More information5.5 Deeper Properties of Continuous Functions
5.5. DEEPER PROPERTIES OF CONTINUOUS FUNCTIONS 195 5.5 Deeper Properties of Continuous Functions 5.5.1 Intermediate Value Theorem and Consequences When one studies a function, one is usually interested
More informationProofs. Chapter 2 P P Q Q
Chapter Proofs In this chapter we develop three methods for proving a statement. To start let s suppose the statement is of the form P Q or if P, then Q. Direct: This method typically starts with P. Then,
More informationA Harvard Sampler. Evan Chen. February 23, I crashed a few math classes at Harvard on February 21, Here are notes from the classes.
A Harvard Sampler Evan Chen February 23, 2014 I crashed a few math classes at Harvard on February 21, 2014. Here are notes from the classes. 1 MATH 123: Algebra II In this lecture we will make two assumptions.
More informationMA103 STATEMENTS, PROOF, LOGIC
MA103 STATEMENTS, PROOF, LOGIC Abstract Mathematics is about making precise mathematical statements and establishing, by proof or disproof, whether these statements are true or false. We start by looking
More informationSets and Functions. (As we will see, in describing a set the order in which elements are listed is irrelevant).
Sets and Functions 1. The language of sets Informally, a set is any collection of objects. The objects may be mathematical objects such as numbers, functions and even sets, or letters or symbols of any
More informationSolutions to Homework Set 1
Solutions to Homework Set 1 1. Prove that not-q not-p implies P Q. In class we proved that A B implies not-b not-a Replacing the statement A by the statement not-q and the statement B by the statement
More informationCarmen s Core Concepts (Math 135)
Carmen s Core Concepts (Math 135) Carmen Bruni University of Waterloo Week 2 1 Divisibility Theorems 2 DIC Example 3 Converses 4 If and only if 5 Sets 6 Other Set Examples 7 Set Operations 8 More Set Terminology
More informationMATHEMATICAL INDUCTION
MATHEMATICAL INDUCTION MATH 3A SECTION HANDOUT BY GERARDO CON DIAZ Imagine a bunch of dominoes on a table. They are set up in a straight line, and you are about to push the first piece to set off the chain
More informationMath 300 Introduction to Mathematical Reasoning Autumn 2017 Proof Templates 1
Math 300 Introduction to Mathematical Reasoning Autumn 2017 Proof Templates 1 In its most basic form, a mathematical proof is just a sequence of mathematical statements, connected to each other by strict
More informationMath 40510, Algebraic Geometry
Math 40510, Algebraic Geometry Problem Set 1, due February 10, 2016 1. Let k = Z p, the field with p elements, where p is a prime. Find a polynomial f k[x, y] that vanishes at every point of k 2. [Hint:
More informationSolutions for Homework Assignment 2
Solutions for Homework Assignment 2 Problem 1. If a,b R, then a+b a + b. This fact is called the Triangle Inequality. By using the Triangle Inequality, prove that a b a b for all a,b R. Solution. To prove
More information3 The language of proof
3 The language of proof After working through this section, you should be able to: (a) understand what is asserted by various types of mathematical statements, in particular implications and equivalences;
More informationIn N we can do addition, but in order to do subtraction we need to extend N to the integers
Chapter The Real Numbers.. Some Preliminaries Discussion: The Irrationality of 2. We begin with the natural numbers N = {, 2, 3, }. In N we can do addition, but in order to do subtraction we need to extend
More informationIn N we can do addition, but in order to do subtraction we need to extend N to the integers
Chapter 1 The Real Numbers 1.1. Some Preliminaries Discussion: The Irrationality of 2. We begin with the natural numbers N = {1, 2, 3, }. In N we can do addition, but in order to do subtraction we need
More informationMATH CSE20 Homework 5 Due Monday November 4
MATH CSE20 Homework 5 Due Monday November 4 Assigned reading: NT Section 1 (1) Prove the statement if true, otherwise find a counterexample. (a) For all natural numbers x and y, x + y is odd if one of
More information1 Predicates and Quantifiers
1 Predicates and Quantifiers We have seen how to represent properties of objects. For example, B(x) may represent that x is a student at Bryn Mawr College. Here B stands for is a student at Bryn Mawr College
More information. As the binomial coefficients are integers we have that. 2 n(n 1).
Math 580 Homework. 1. Divisibility. Definition 1. Let a, b be integers with a 0. Then b divides b iff there is an integer k such that b = ka. In the case we write a b. In this case we also say a is a factor
More informationShow Your Work! Point values are in square brackets. There are 35 points possible. Some facts about sets are on the last page.
Formal Methods Name: Key Midterm 2, Spring, 2007 Show Your Work! Point values are in square brackets. There are 35 points possible. Some facts about sets are on the last page.. Determine whether each of
More informationInteger-Valued Polynomials
Integer-Valued Polynomials LA Math Circle High School II Dillon Zhi October 11, 2015 1 Introduction Some polynomials take integer values p(x) for all integers x. The obvious examples are the ones where
More informationMAT115A-21 COMPLETE LECTURE NOTES
MAT115A-21 COMPLETE LECTURE NOTES NATHANIEL GALLUP 1. Introduction Number theory begins as the study of the natural numbers the integers N = {1, 2, 3,...}, Z = { 3, 2, 1, 0, 1, 2, 3,...}, and sometimes
More informationSection 3.1: Direct Proof and Counterexample 1
Section 3.1: Direct Proof and Counterexample 1 In this chapter, we introduce the notion of proof in mathematics. A mathematical proof is valid logical argument in mathematics which shows that a given conclusion
More information2. Introduction to commutative rings (continued)
2. Introduction to commutative rings (continued) 2.1. New examples of commutative rings. Recall that in the first lecture we defined the notions of commutative rings and field and gave some examples of
More information4.1 Real-valued functions of a real variable
Chapter 4 Functions When introducing relations from a set A to a set B we drew an analogy with co-ordinates in the x-y plane. Instead of coming from R, the first component of an ordered pair comes from
More information1.3 Linear Dependence & span K
( ) Conversely, suppose that every vector v V can be expressed uniquely as v = u +w for u U and w W. Then, the existence of this expression for each v V is simply the statement that V = U +W. Moreover,
More informationLECTURE 1. Logic and Proofs
LECTURE 1 Logic and Proofs The primary purpose of this course is to introduce you, most of whom are mathematics majors, to the most fundamental skills of a mathematician; the ability to read, write, and
More informationMath 308 Spring Midterm Answers May 6, 2013
Math 38 Spring Midterm Answers May 6, 23 Instructions. Part A consists of questions that require a short answer. There is no partial credit and no need to show your work. In Part A you get 2 points per
More informationHOW TO WRITE PROOFS. Dr. Min Ru, University of Houston
HOW TO WRITE PROOFS Dr. Min Ru, University of Houston One of the most difficult things you will attempt in this course is to write proofs. A proof is to give a legal (logical) argument or justification
More informationThe cardinal comparison of sets
(B) The cardinal comparison of sets I think we can agree that there is some kind of fundamental difference between finite sets and infinite sets. For a finite set we can count its members and so give it
More informationPOL502: Foundations. Kosuke Imai Department of Politics, Princeton University. October 10, 2005
POL502: Foundations Kosuke Imai Department of Politics, Princeton University October 10, 2005 Our first task is to develop the foundations that are necessary for the materials covered in this course. 1
More information106 CHAPTER 3. TOPOLOGY OF THE REAL LINE. 2. The set of limit points of a set S is denoted L (S)
106 CHAPTER 3. TOPOLOGY OF THE REAL LINE 3.3 Limit Points 3.3.1 Main Definitions Intuitively speaking, a limit point of a set S in a space X is a point of X which can be approximated by points of S other
More informationSupremum and Infimum
Supremum and Infimum UBC M0 Lecture Notes by Philip D. Loewen The Real Number System. Work hard to construct from the axioms a set R with special elements O and I, and a subset P R, and mappings A: R R
More informationBasic Logic and Proof Techniques
Chapter 3 Basic Logic and Proof Techniques Now that we have introduced a number of mathematical objects to study and have a few proof techniques at our disposal, we pause to look a little more closely
More informationWe have seen that the symbols,,, and can guide the logical
CHAPTER 7 Quantified Statements We have seen that the symbols,,, and can guide the logical flow of algorithms. We have learned how to use them to deconstruct many English sentences into a symbolic form.
More informationMATH 117 LECTURE NOTES
MATH 117 LECTURE NOTES XIN ZHOU Abstract. This is the set of lecture notes for Math 117 during Fall quarter of 2017 at UC Santa Barbara. The lectures follow closely the textbook [1]. Contents 1. The set
More informationHyperreal Numbers: An Elementary Inquiry-Based Introduction. Handouts for a course from Canada/USA Mathcamp Don Laackman
Hyperreal Numbers: An Elementary Inquiry-Based Introduction Handouts for a course from Canada/USA Mathcamp 2017 Don Laackman MATHCAMP, WEEK 3: HYPERREAL NUMBERS DAY 1: BIG AND LITTLE DON & TIM! Problem
More informationConnectedness. Proposition 2.2. The following are equivalent for a topological space (X, T ).
Connectedness 1 Motivation Connectedness is the sort of topological property that students love. Its definition is intuitive and easy to understand, and it is a powerful tool in proofs of well-known results.
More informationHence, (f(x) f(x 0 )) 2 + (g(x) g(x 0 )) 2 < ɛ
Matthew Straughn Math 402 Homework 5 Homework 5 (p. 429) 13.3.5, 13.3.6 (p. 432) 13.4.1, 13.4.2, 13.4.7*, 13.4.9 (p. 448-449) 14.2.1, 14.2.2 Exercise 13.3.5. Let (X, d X ) be a metric space, and let f
More informationFoundations of Mathematics MATH 220 FALL 2017 Lecture Notes
Foundations of Mathematics MATH 220 FALL 2017 Lecture Notes These notes form a brief summary of what has been covered during the lectures. All the definitions must be memorized and understood. Statements
More informationMathematical Induction
Mathematical Induction James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University January 12, 2017 Outline Introduction to the Class Mathematical Induction
More informationChapter One. The Real Number System
Chapter One. The Real Number System We shall give a quick introduction to the real number system. It is imperative that we know how the set of real numbers behaves in the way that its completeness and
More informationCartesian Products and Relations
Cartesian Products and Relations Definition (Cartesian product) If A and B are sets, the Cartesian product of A and B is the set A B = {(a, b) : (a A) and (b B)}. The following points are worth special
More informationProof Techniques (Review of Math 271)
Chapter 2 Proof Techniques (Review of Math 271) 2.1 Overview This chapter reviews proof techniques that were probably introduced in Math 271 and that may also have been used in a different way in Phil
More informationPrinciples of Real Analysis I Fall I. The Real Number System
21-355 Principles of Real Analysis I Fall 2004 I. The Real Number System The main goal of this course is to develop the theory of real-valued functions of one real variable in a systematic and rigorous
More informationMath 300: Foundations of Higher Mathematics Northwestern University, Lecture Notes
Math 300: Foundations of Higher Mathematics Northwestern University, Lecture Notes Written by Santiago Cañez These are notes which provide a basic summary of each lecture for Math 300, Foundations of Higher
More informationInduction 1 = 1(1+1) = 2(2+1) = 3(3+1) 2
Induction 0-8-08 Induction is used to prove a sequence of statements P(), P(), P(3),... There may be finitely many statements, but often there are infinitely many. For example, consider the statement ++3+
More informationDirect Proof Universal Statements
Direct Proof Universal Statements Lecture 13 Section 4.1 Robb T. Koether Hampden-Sydney College Wed, Feb 6, 2013 Robb T. Koether (Hampden-Sydney College) Direct Proof Universal Statements Wed, Feb 6, 2013
More informationDivisibility = 16, = 9, = 2, = 5. (Negative!)
Divisibility 1-17-2018 You probably know that division can be defined in terms of multiplication. If m and n are integers, m divides n if n = mk for some integer k. In this section, I ll look at properties
More informationLecture 6: Finite Fields
CCS Discrete Math I Professor: Padraic Bartlett Lecture 6: Finite Fields Week 6 UCSB 2014 It ain t what they call you, it s what you answer to. W. C. Fields 1 Fields In the next two weeks, we re going
More informationM17 MAT25-21 HOMEWORK 6
M17 MAT25-21 HOMEWORK 6 DUE 10:00AM WEDNESDAY SEPTEMBER 13TH 1. To Hand In Double Series. The exercises in this section will guide you to complete the proof of the following theorem: Theorem 1: Absolute
More informationMathematical Reasoning & Proofs
Mathematical Reasoning & Proofs MAT 1362 Fall 2018 Alistair Savage Department of Mathematics and Statistics University of Ottawa This work is licensed under a Creative Commons Attribution-ShareAlike 4.0
More informationSection 2.1: Introduction to the Logic of Quantified Statements
Section 2.1: Introduction to the Logic of Quantified Statements In the previous chapter, we studied a branch of logic called propositional logic or propositional calculus. Loosely speaking, propositional
More informationProof by Contradiction
Proof by Contradiction MAT231 Transition to Higher Mathematics Fall 2014 MAT231 (Transition to Higher Math) Proof by Contradiction Fall 2014 1 / 12 Outline 1 Proving Statements with Contradiction 2 Proving
More informationLinear Algebra, Summer 2011, pt. 2
Linear Algebra, Summer 2, pt. 2 June 8, 2 Contents Inverses. 2 Vector Spaces. 3 2. Examples of vector spaces..................... 3 2.2 The column space......................... 6 2.3 The null space...........................
More informationTheorem 5.3. Let E/F, E = F (u), be a simple field extension. Then u is algebraic if and only if E/F is finite. In this case, [E : F ] = deg f u.
5. Fields 5.1. Field extensions. Let F E be a subfield of the field E. We also describe this situation by saying that E is an extension field of F, and we write E/F to express this fact. If E/F is a field
More informationHW 4 SOLUTIONS. , x + x x 1 ) 2
HW 4 SOLUTIONS The Way of Analysis p. 98: 1.) Suppose that A is open. Show that A minus a finite set is still open. This follows by induction as long as A minus one point x is still open. To see that A
More informationBasics of Proofs. 1 The Basics. 2 Proof Strategies. 2.1 Understand What s Going On
Basics of Proofs The Putnam is a proof based exam and will expect you to write proofs in your solutions Similarly, Math 96 will also require you to write proofs in your homework solutions If you ve seen
More informationWe want to show P (n) is true for all integers
Generalized Induction Proof: Let P (n) be the proposition 1 + 2 + 2 2 + + 2 n = 2 n+1 1. We want to show P (n) is true for all integers n 0. Generalized Induction Example: Use generalized induction to
More informationA Guide to Proof-Writing
A Guide to Proof-Writing 437 A Guide to Proof-Writing by Ron Morash, University of Michigan Dearborn Toward the end of Section 1.5, the text states that there is no algorithm for proving theorems.... Such
More informationClimbing an Infinite Ladder
Section 5.1 Section Summary Mathematical Induction Examples of Proof by Mathematical Induction Mistaken Proofs by Mathematical Induction Guidelines for Proofs by Mathematical Induction Climbing an Infinite
More informationImportant Properties of R
Chapter 2 Important Properties of R The purpose of this chapter is to explain to the reader why the set of real numbers is so special. By the end of this chapter, the reader should understand the difference
More informationPEANO AXIOMS FOR THE NATURAL NUMBERS AND PROOFS BY INDUCTION. The Peano axioms
PEANO AXIOMS FOR THE NATURAL NUMBERS AND PROOFS BY INDUCTION The Peano axioms The following are the axioms for the natural numbers N. You might think of N as the set of integers {0, 1, 2,...}, but it turns
More informationUnit 6 Study Guide: Equations. Section 6-1: One-Step Equations with Adding & Subtracting
Unit 6 Study Guide: Equations DUE DATE: A Day: Dec 18 th B Day: Dec 19 th Name Period Score / Section 6-1: One-Step Equations with Adding & Subtracting Textbook Reference: Page 437 Vocabulary: Equation
More informationMATH10040: Chapter 0 Mathematics, Logic and Reasoning
MATH10040: Chapter 0 Mathematics, Logic and Reasoning 1. What is Mathematics? There is no definitive answer to this question. 1 Indeed, the answer given by a 21st-century mathematician would differ greatly
More informationGeneralized eigenspaces
Generalized eigenspaces November 30, 2012 Contents 1 Introduction 1 2 Polynomials 2 3 Calculating the characteristic polynomial 5 4 Projections 7 5 Generalized eigenvalues 10 6 Eigenpolynomials 15 1 Introduction
More informationMath 38: Graph Theory Spring 2004 Dartmouth College. On Writing Proofs. 1 Introduction. 2 Finding A Solution
Math 38: Graph Theory Spring 2004 Dartmouth College 1 Introduction On Writing Proofs What constitutes a well-written proof? A simple but rather vague answer is that a well-written proof is both clear and
More informationSUPPLEMENT TO CHAPTER 3
SUPPLEMENT TO CHAPTER 3 1.1 Linear combinations and spanning sets Consider the vector space R 3 with the unit vectors e 1 = (1, 0, 0), e 2 = (0, 1, 0), e 3 = (0, 0, 1). Every vector v = (a, b, c) R 3 can
More informationConsequences of Continuity
Consequences of Continuity James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University October 4, 2017 Outline 1 Domains of Continuous Functions 2 The
More informationCHMC: Finite Fields 9/23/17
CHMC: Finite Fields 9/23/17 1 Introduction This worksheet is an introduction to the fascinating subject of finite fields. Finite fields have many important applications in coding theory and cryptography,
More informationMath 52: Course Summary
Math 52: Course Summary Rich Schwartz September 2, 2009 General Information: Math 52 is a first course in linear algebra. It is a transition between the lower level calculus courses and the upper level
More informationAnnouncements. Read Section 2.1 (Sets), 2.2 (Set Operations) and 5.1 (Mathematical Induction) Existence Proofs. Non-constructive
Announcements Homework 2 Due Homework 3 Posted Due next Monday Quiz 2 on Wednesday Read Section 2.1 (Sets), 2.2 (Set Operations) and 5.1 (Mathematical Induction) Exam 1 in two weeks Monday, February 19
More information3. Only sequences that were formed by using finitely many applications of rules 1 and 2, are propositional formulas.
1 Chapter 1 Propositional Logic Mathematical logic studies correct thinking, correct deductions of statements from other statements. Let us make it more precise. A fundamental property of a statement is
More informationMATH 221: SOLUTIONS TO SELECTED HOMEWORK PROBLEMS
MATH 221: SOLUTIONS TO SELECTED HOMEWORK PROBLEMS 1. HW 1: Due September 4 1.1.21. Suppose v, w R n and c is a scalar. Prove that Span(v + cw, w) = Span(v, w). We must prove two things: that every element
More informationHandout on Logic, Axiomatic Methods, and Proofs MATH Spring David C. Royster UNC Charlotte
Handout on Logic, Axiomatic Methods, and Proofs MATH 3181 001 Spring 1999 David C. Royster UNC Charlotte January 18, 1999 Chapter 1 Logic and the Axiomatic Method 1.1 Introduction Mathematicians use a
More informationVector Spaces. 9.1 Opening Remarks. Week Solvable or not solvable, that s the question. View at edx. Consider the picture
Week9 Vector Spaces 9. Opening Remarks 9.. Solvable or not solvable, that s the question Consider the picture (,) (,) p(χ) = γ + γ χ + γ χ (, ) depicting three points in R and a quadratic polynomial (polynomial
More informationSection Summary. Proof by Cases Existence Proofs
Section 1.8 1 Section Summary Proof by Cases Existence Proofs Constructive Nonconstructive Disproof by Counterexample Uniqueness Proofs Proving Universally Quantified Assertions Proof Strategies sum up
More informationChapter 2. Mathematical Reasoning. 2.1 Mathematical Models
Contents Mathematical Reasoning 3.1 Mathematical Models........................... 3. Mathematical Proof............................ 4..1 Structure of Proofs........................ 4.. Direct Method..........................
More informationMath 300: Final Exam Practice Solutions
Math 300: Final Exam Practice Solutions 1 Let A be the set of all real numbers which are zeros of polynomials with integer coefficients: A := {α R there exists p(x) = a n x n + + a 1 x + a 0 with all a
More informationHomework 3 Solutions, Math 55
Homework 3 Solutions, Math 55 1.8.4. There are three cases: that a is minimal, that b is minimal, and that c is minimal. If a is minimal, then a b and a c, so a min{b, c}, so then Also a b, so min{a, b}
More informationMathematical Induction
Mathematical Induction MAT30 Discrete Mathematics Fall 018 MAT30 (Discrete Math) Mathematical Induction Fall 018 1 / 19 Outline 1 Mathematical Induction Strong Mathematical Induction MAT30 (Discrete Math)
More informationAN INTRODUCTION TO MATHEMATICAL PROOFS NOTES FOR MATH Jimmy T. Arnold
AN INTRODUCTION TO MATHEMATICAL PROOFS NOTES FOR MATH 3034 Jimmy T. Arnold i TABLE OF CONTENTS CHAPTER 1: The Structure of Mathematical Statements.............................1 1.1. Statements..................................................................
More informationCOMP 182 Algorithmic Thinking. Proofs. Luay Nakhleh Computer Science Rice University
COMP 182 Algorithmic Thinking Proofs Luay Nakhleh Computer Science Rice University 1 Reading Material Chapter 1, Section 3, 6, 7, 8 Propositional Equivalences The compound propositions p and q are called
More informationMath 10850, fall 2017, University of Notre Dame
Math 10850, fall 2017, University of Notre Dame Notes on first exam September 22, 2017 The key facts The first midterm will be on Thursday, September 28, 6.15pm-7.45pm in Hayes-Healy 127. What you need
More informationMATH 115, SUMMER 2012 LECTURE 4 THURSDAY, JUNE 21ST
MATH 115, SUMMER 2012 LECTURE 4 THURSDAY, JUNE 21ST JAMES MCIVOR Today we enter Chapter 2, which is the heart of this subject. Before starting, recall that last time we saw the integers have unique factorization
More informationSupplementary Material for MTH 299 Online Edition
Supplementary Material for MTH 299 Online Edition Abstract This document contains supplementary material, such as definitions, explanations, examples, etc., to complement that of the text, How to Think
More informationLinear Algebra Handout
Linear Algebra Handout References Some material and suggested problems are taken from Fundamentals of Matrix Algebra by Gregory Hartman, which can be found here: http://www.vmi.edu/content.aspx?id=779979.
More informationChapter 1 The Real Numbers
Chapter 1 The Real Numbers In a beginning course in calculus, the emphasis is on introducing the techniques of the subject;i.e., differentiation and integration and their applications. An advanced calculus
More informationPractice Test III, Math 314, Spring 2016
Practice Test III, Math 314, Spring 2016 Dr. Holmes April 26, 2016 This is the 2014 test reorganized to be more readable. I like it as a review test. The students who took this test had to do four sections
More informationNotes on the Point-Set Topology of R Northwestern University, Fall 2014
Notes on the Point-Set Topology of R Northwestern University, Fall 2014 These notes give an introduction to the notions of open and closed subsets of R, which belong to the subject known as point-set topology.
More informationChapter 2. Polynomial and Rational Functions. 2.5 Zeros of Polynomial Functions
Chapter 2 Polynomial and Rational Functions 2.5 Zeros of Polynomial Functions 1 / 33 23 Chapter 2 Homework 2.5 p335 6, 8, 10, 12, 16, 20, 24, 28, 32, 34, 38, 42, 46, 50, 52 2 / 33 23 3 / 33 23 Objectives:
More informationTwitter: @Owen134866 www.mathsfreeresourcelibrary.com Prior Knowledge Check 1) Simplify: a) 3x 2 5x 5 b) 5x3 y 2 15x 7 2) Factorise: a) x 2 2x 24 b) 3x 2 17x + 20 15x 2 y 3 3) Use long division to calculate:
More informationIn this initial chapter, you will be introduced to, or more than likely be reminded of, a
1 Sets In this initial chapter, you will be introduced to, or more than likely be reminded of, a fundamental idea that occurs throughout mathematics: sets. Indeed, a set is an object from which every mathematical
More informationMath 535: Topology Homework 1. Mueen Nawaz
Math 535: Topology Homework 1 Mueen Nawaz Mueen Nawaz Math 535 Topology Homework 1 Problem 1 Problem 1 Find all topologies on the set X = {0, 1, 2}. In the list below, a, b, c X and it is assumed that
More informationMath 115A: Linear Algebra
Math 115A: Linear Algebra Michael Andrews UCLA Mathematics Department February 9, 218 Contents 1 January 8: a little about sets 4 2 January 9 (discussion) 5 2.1 Some definitions: union, intersection, set
More information