# 2.1.1 Examples of Sets and their Elements

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Chapter 2 Set Theory 2.1 Sets The most basic object in Mathematics is called a set. As rudimentary as it is, the exact, formal definition of a set is highly complex. For our purposes, we will simply define a set as a collection of objects that is well-defined. That is, it is possible to determine if an object is to be included in the set or not. Frequently a set is denoted by a capital letter, like S. Objects in the set collection are known as elements of S. If x is one of these elements, then we write that x S. Similarly, if an object x is not a part of the set, then we write x S Examples of Sets and their Elements The most basic set is the collection of no objects. This set, known as the empty set or null set, is denoted by or {}, to indicate that it contains no elements. In Mathematics, the most frequently encountered sets are various collections of types of real numbers. The below such sets are of key importance. The set of natural numbers, denoted by N, is the set of all non-negative whole numbers. Thus, we can list off a few of the elements by writing N = {0, 1, 2, 3,...}. Note that some mathematicians have the natural numbers defined as the set of all positive whole numbers. The set of integers, denoted by Z, is the set of all whole numbers. Thus, we can list them off as Z = {..., 2, 1, 0, 1, 2, 3,...}. The set of all real numbers is denoted by R. It is important to note that sets are unordered objects. Thus, the set {a, b, c} and the set {b, c, a} are considered the same because they contain the same elements, even if they are listed in different orders. 1

2 2.1.2 Set-Builder Notation If we wish to efficiently describe sets of objects by some defining characteristic or by restricting a larger set via some conditions, then we may use set-builder notation. A set S described in set-builder notation has the form S = {x p(x)}, where p(x) is some statement about x. This set should be read as S is the set of all x such that p(x) is true. Below are some examples of sets described via set-builder notation. We can describe the natural numbers N by placing a restriction on the set of integers: N = {x Z x 0}. We can describe the set of rational numbers, denoted by Q, in set-builder notation as { } p Q = q p, q Z and q 0, where we have the understanding that two rational numbers p1 q 1 and p2 q 2 are equal if p 1 q 2 = p 2 q 1. Thus, the set of all rational numbers is equal to the set of all fractions where the numerator and denominator are integers with the denominator not being 0. The set of complex numbers, denoted by C, can be written in set-builder notation as C = {a + bi a, b R}. We can describe specific sets using set-builder notation as well. For instance, the set S = {n Z 2 n < 5} is just the set { 2, 1, 0, 1, 2, 3, 4}. The set S = {x R x 2 1 = 0} is simply the two element set { 1, 1}. The set {x R 2 x} can be written in interval notation as the closed ray [ 2, ) Subsets If we compare the two sets N and Z, it is clear that every single natural number in N is also an integer in Z. As such, we can think of the set N as being contained inside of Z. In fact, we would say that N is a subset of Z. More formally, we have that a set A is a subset of a set S if whenever x A, then x S. Below are some examples of subsets. The empty set is a subset of every set S. Notice that it is true that if x, then x S because there are no x for which we need to check this condition. Thus, S for every set S. 2

3 The integers Z are a subset of the rationals Q since, if n Z, then it can be written as n 1 as is thus an element of Q. Thus, Z Q. The real numbers R are a subset of the complex numbers C since, if a R, then it can be written as a + 0i, which is an element of C. Thus, R C Proofs Involving Subsets Since the definition of A S says that if x A, then x S, then we can prove that A is indeed a subset of S by assuming that we have an arbitrary element x A and then show that x S as well. In the above examples, this is precisely what we did. Below, we will see a slightly more complicated example of a proof of subset inclusion. Proposition. Let S = ( 3, 5) and T = ( 6, 5]. Then, S T. What we know: We can re-write the interval notation sets in set-builder notation as ( 3, 5) = {x R 3 < x < 5} and ( 6, 5] = {x R 6 < x 5}. What we want: S T. Thus, we will be taking an arbitrary element x S and then, using the definition of S and T, show that x T as well. Proof. Assume x S. Then, x satisfies 3 < x < 5. We will show that x T by showing that x satisfies 6 < x 5. Since 3 < x, we also know that 6 < 3 < x and thus 6 < x. Furthermore, since x < 5 and 5 5, then x 5. So, x satisfies 6 < x 5, as desired. So, x T and S T. We can also use the definition of a subset to show a fact that seems obvious: If A is a subset of B and B is a subset of C, then A is a subset of C. Notice that, in this case, we do not have specific sets, so we can only rely on the definition of a subset to prove the above statement. Proposition. Let A, B, and C be sets. If A B and B C, then A C. What we know: We know that A B; thus, whenever we know that x A, we can conclude that x B. B C; thus, whenever we know that x B, we can conclude that x C. What we want: We want to show that A C. Thus, we are going to assume that x A and will eventually conclude that x C. Proof. Let x A. Since A B and x A, we can conclude that x B. Since B C and x B, we can conclude that x C. Thus, A C. 3

4 2.1.5 Unions and Intersections If we have two sets S and T, then we can form new sets that are combinations of the elements of S and T. First we defined the union of our sets as S T = {x x S or x T }. Thus, S T is the set that contains all the elements in S as well as all the elements in T. We remember, of course, that, the or in this definition means that if x S and also x T, then it is also included as an element of the union S T. Notice that, since every element x S is also included in S T, then we can conclude that S S T. Similarly, we also know that T S T. If instead of using or, we use the conjunction and, then we arrive at the definition of an intersection of two sets: S T = {x x S and x T }. Thus, for an element to be in the intersection of two sets, it has to meet the two conditions that it be an element of S and also an element of T. Notice that if x S T, then, by definition x S and also x T. Thus, we can conclude that S T S and S T T. Two sets S and T are called disjoint if they share no elements in common. In other words, S and T are disjoint when S T =. Below are some examples of unions and intersections. Let D n be the set of all natural numbers that divide the natural number n. Thus, for example, D 20 = {1, 2, 4, 5, 10, 20} and D 24 = {1, 2, 3, 4, 6, 8, 12, 24}. Then, D 20 D 24 = {1, 2, 4}. Notice that the largest number in this intersection is exactly the greatest common divisor of 20 and 24. Let M n be the set of all positive multiples of the natural number n. Thus for example, M 12 = {12, 24, 36, 48, 60, 72, 84, 96, 108,...} M 18 = {18, 36, 54, 72, 90, 108,...}. The intersection of these sets is given by M 12 M 18 = {36, 72, 108,...}. Notice that the smallest of these numbers is exactly the least common multiple of 12 and 18. Consider the sets S and T given by S = {x R sin x = 0} T = {x R cos x = 0}. The set S is simply the roots of the function sin x and is thus equal to S = {πk k Z} = {..., 2π, π, 0, π, 2π,...}; 4

5 in other words, it is the same as all integers multiples of π. Similarly, T is the roots of cos x, which is equal to π 2 plus integers multiples of π and can thus be written as { π } T = 2 + πk k Z = {..., 3π 2, π 2, π 2, 3π 2,...}. So, the union of S and T is equal to and can thus be re-written as S T = {..., 3π 2, π, π 2, 0, π 2, π, 3π 2,...} S T = { kπ 2 } k Z. Notice that since S gave the roots of sin x and T gave the roots of cos x, that the union S T is exactly the roots of the product sin x cos x. Furthermore, note that the intersection S T = since the two sets of roots are disjoint (i.e., they share no common elements) Complements of Sets It is just as important to know which objects are elements of a set as to know which objects are not elements of a set. If A S, then we can define the complement of A in S as the set of all element in S that are not in A: A = {x S x A}. In other texts, complements may also be written as A c, but we will stick to the A convention. Notice that, when taking the complement of A, it is important that the larger set S is also understood. For example, If A = {0, 1} is the set of two elements 0 and 1, and we are taking its complement in N, then A = {2, 3, 4,...}. If A = {0, 1}, but now, we take its complement in the set of all reals R, then A = (, 0) (0, 1) (1, ). Many important examples of complements of sets exist. For example, if we consider Q R, then the complement of Q in R is given by Q and is the set of irrational numbers. We will now prove a statement about the complement of sets that seems believable. Consider A and B, which are subsets of a set S; furthermore, consider their complements in S. We will prove if A B, then B A. Since we will be asked to prove a statement about complements, we will be dealing with negative statements (for example, x A). As such, we will use the technique of proof by contradiction, where we will show something is true by assuming the negation of what we want and arriving at a contradiction. Since we arrived at this contradiction, our initial assumption (that the negation of what we want is true) must be false and thus what we want is true. Proposition. Let A, B S. If A B, then B A. In this proof, we will be using the technique of proof by contradiction. In particular, when we prove that B A, we will want to conclude 5

6 that x A, which is equivalent to x A. Thus, we will assume that x A and arrive at a contradiction. What we know: A B : if we ever know that x A, then we can conclude that x B. What we want: B A : We will assume that x B and our job is to conclude that x A. What we ll do: Since we wish to show that B A, we will assume that x B, which is equivalent to x B. Our job is to show that x A, which is equivalent to x A. Since we wish to show that x A, we will use proof by contradiction. So, we will assume x A and arrive at a contradiction. When we reach this contradiction, our initial assumption that x A will be false and thus x A will be true. Proof. Assume that x B. We will show that x A by showing that x A. Assume, to the contrary, that x A. Since x B, then x B. Since x A and since A B, we conclude that x B. However, we also know that x B, a contradiction. Thus, our initial assumption that x A is false and thus x A is true. So, x A, and we conclude that B A. It is now tempting to make various claims about complements of sets. For example, it seems very obvious that if A S, then A A = and A A = S. However, we have never formally defined what it means for two sets to be equal to each other. In the next section, we will investigate the formal definition of set equality. 2.2 Definition of Set Equality It seems reasonable to define the equality of two sets by saying something along the lines of S and T are equal if they contain exactly the same elements. In proofs, though, this does not give us much of a methodology by which to show that two sets are equal. Thus, to formalize the definition of set equality, we use the concept of subsets. Notice that every set is a subset of itself. Thus, if S and T are to be equal as sets, then S T and T S. Thus, in a proof, to show that S = T, we must show two subset inclusions: S T and T S. Below is an example of a proof that the two sets regarding the roots of sin x and cos x are indeed equal. Proposition. Consider the sets along with the set Then, S T = R. S = {x R sin x = 0} and T = {x R cos x = 0}, What we want: S T = R. S T R and R S T. R = {x R sin x cos x = 0}. Thus, we need to show two subset inclusions: What we ll do: For the first subset inclusion S T R, we will let x S T. Thus, x S or x T. Anytime that we know an or to be true, we will want 6

7 to use cases. In this case, we have two cases: x S or x T. For each case, we must conclude that x R. For the other subset inclusion R S T, we will assume that x R. Then, we must conclude that x S T. To do this, we can show that x S S T or that x T S T. Proof. To show that S T = R, we will show the two following set inclusions: S T R and R S T. For the first subset inclusion S T R, assume that x S T. Thus x S or x T. We will proceed with these two cases. In the first case, x S and thus sin x = 0. Thus, sin x cos x = 0 cos x = 0 and so x R. In the second case, x T and thus cos x = 0. Thus, sin x cos x = sin x 0 = 0 and so x R. In either case, we have shown that x R. Thus, S T R. For the second subset inclusion R S T, we will assume that x R. Thus, x has the property that sin x cos x = 0. This means that sin x = 0 or cos x = 0, leaving us with two cases. If sin x = 0, then x S S T and thus x S T. If cos x = 0, then x T S T and thus x S T. Either way, x S T and thus R S T. Since we have proven above both subset inclusions, we may conclude that S T = R. A key technique in the above proof that was used several times is that if we ever assume p or q, then we use two cases: first, arrive at your conclusion using p and then separately, arrive at your conclusion using q. Below, we will prove some of the more important set theory laws, which state the equalities of certain sets, by using the same technique of proving the two subset inclusions DeMorgan s Set Theory Laws In the previous chapter, we saw that DeMorgan s Logic Laws give us a way to negate conjunctions ( and statements) and disjunctions ( or statements). They were: (p q) p q and (p q) p q. In Set Theory, there are analogous relationships called DeMorgan s Set Theory Laws. These laws are exactly the analogues of the corresponding logic laws, where is replace by the complement, is replace by, and is replaced by. DeMorgan s Set Theory Laws are stated below: We will prove the latter of these below. S T = S T and S T = S T. Theorem. Let S and T be sets. Then, S T = S T. What we want: S T = S T. Thus, we will prove the following two subset inclusions: S T S T and S T S T. What we ll do: This proof will be significantly simplified because of DeMorgan s Logic Laws. For this proof, we will use the logic law: (p q) p q. Proof. To prove that S T = S T, we will prove the following two subset inclusions: S T S T and S T S T. 7

8 For the first inclusion, assume that x S T. Thus x S T. So, it is not true that x S or x T. By DeMorgan s Logic Laws, this is equivalent to x S and x T being true. Thus, since x S, then x S, and since x T, then x T. Since both x S and x T, then x is in the intersection: x S T. Thus, S T S T. For the second inclusion, assume that x S T. Then, x S and x T. Thus, we have that x S and x T. Since both x S and x T is true, we can use DeMorgan s Logic Law to conclude that x S or x T is not true. Thus, x S T, and so x S T. Thus, S T S T. Since we have shown both of these subset inclusions, we know that our two sets are equal: S T = S T The Distributive Laws The above example of DeMorgan s Set Theory laws clearly indicates that union and intersection, the two major operations of set theory, are related. There remains a question, though, of what occurs if you take some sets and take unions and then intersections. Similar to adding and then multiplying real numbers, a distributive-type property seems to hold. In particular, if S, T, and R are sets, then the following two distributive laws hold: S (T R) = (S T ) (S R) and S (T R) = (S T ) (S R). Below, we will prove the first of these two distributive laws. Theorem. Let S, T, and R be sets. Then, S (T R) = (S T ) (S R). What we want: S (T R) = (S T ) (S R). Thus, we want to show the following two subset inclusions: S (T R) (S T ) (S R) and (S T ) (S R) S (T R). What we ll do: For the first inclusion S (T R) (S T ) (S R), we will assume that x S (T R). Thus, because this is an or /union assumption, we will have cases: when x S and then x T R. In both cases, we must conclude that x (S T ) (S R). To show that x is in this intersection, we will need to show both that x S T and also that x S R. For the second inclusion (S T ) (S R) S (T R), we will assume that x (S T ) (S R). Thus, we are free to assume that both x S T and x S R are true. To show that x S (T R), we will show that x S S (T R) or that x (T R) S (T R). Proof. To prove that S (T R) = (S T ) (S R), we will prove the two subset inclusions: S (T R) (S T ) (S R) and (S T ) (S R) S (T R). For the first inclusion, we will assume that x S (T R). Thus, x S or x T R, giving us two cases. If x S, then x S S T and x S S R. Thus, since x is in both sets, it is in the intersection: x (S T ) (S R). In the second case, we let x T R. Thus, x T and x R are both true. Since x T, then x T S T. Since x R, then x R S R. Since x is in both sets, it is in the intersection: x (S T ) (S R). Thus, in either of our two cases, we conclude that x (S T ) (S R). So, we have obtained the subset inclusion: S (T R) (S T ) (S R). 8

9 For the second inclusion, we will assume that x (S T ) (S R). Thus, we know that x S T and also x S R. Since x S T, then x S or x T ; furthermore, since x S R, then x S or x R. So, at least one of x S or x T has to be true while, at the same time, at least one of x S or x R must be true. If we know that x S, then x S S (T R). If it s not true that x S, then it must be that x T and that x R. Thus, x T R and thus x (T R) S (T R). So, we have the subset inclusions (S T ) (S R) S (T R). Since we have shown both inclusions to be true, then we can conclude the two sets are equal: S (T R) = (S T ) (S R). 2.3 Product Sets Since we are interested in building new sets from old sets, we can look at the example of two-dimensional real space R 2. This set is geometrically represented by the xy-plane, and individual elements in the plane take on the form of the 2-tuple (x, y) for x, y R. Thus, we can write R 2 in set-builder notation as R 2 = {(x, y) x, y R}. Thus, R 2 can be thought of as two copies of R, and elements in this set have two entries: one for each real number Examples of Product Sets If we wish to generalize the example of R 2 to that for two different sets S and T, then we can form what is called the direct product or cartesian product of the sets S and T. This new, larger set is denoted by S T and is given by S T = {(s, t) s S and t T }. It is important to note that individual elements in the product S T look like the 2-tuple (s, t) and that we know that the first entry is an element of S and the second entry is an element of T : s S and t T. Below are some examples of products. If we take S = T = R, then we have that which is exactly the same as R 2. R R = {(x, y) x, y R}, If S = {a, b} is a 2-element set and T = {x, y, z} is a 3-element set, then the product space S T is set of all possible tuples (s, t) with s S and t T. Thus, S T contains a total of 6 elements: S T = {(a, x), (a, y), (a, z), (b, x), (b, y), (b, z)}. In the same sprit as R R, the product of two copies of Z is called an integer lattice and is given by Z Z = {(n, m) n, m Z}. Geometrically, this corresponds to the points in R 2 that have both of their coordinates integers and appears as a lattice of points in the plane. 9

10 Notice that, in general, the products S T and T S are different sets. For example, if we use the S and T from the second example above, we have that T S = {(x, a), (x, b), (y, a), (y, b), (z, a), (z, b)}. The sets S T and T S are not equal because, for example, the element (a, x) S T is completely different from the (x, a) T S. In fact, (a, x) T S since it is not true that a T (nor is it true that x S). If S is any set and is the empty set, then notice that S = and S =. This follows from the fact that, for example, S is the set of all element (s, t) such that s S and t. Since there are no elements in, there are no tuples in the product S. We can also form triple (or any number of) products. If we take three sets S, T, and R, then S T R = {(s, t, r) s S, t T, and r R}. For triple products, a general element looks like a 3-tuple (s, t, r) Proofs involving Product Sets Now that we have this new structure called direct product, we can ask how this interacts with the various other set theoretic operations. Below, we will prove that intersection and direct products behave nicely with respect to each other. In fact, if A, B, C and D are sets, then we will show that (A B) (C D) = (A C) (B D). Proposition. Let A, B, C, and D be sets. Then (A B) (C D) = (A C) (B D). What we want: (A B) (C D) = (A C) (B D). Thus, we will need to show the two subset inclusions: (A B) (C D) (A C) (B D) and (A C) (B D) (A B) (C D). What we ll do: When we use direct products in proofs, it s important to keep in mind that our arbitrary elements will look like 2-tuples (x, y). Thus, for example, when we do our first subset inclusion and take an arbitrary element in (A B) (C D), it will look like (x, y) such that x A B and y C D. Proof. To show that (A B) (C D) = (A C) (B D), we will show the following two subset inclusions: (A B) (C D) (A C) (B D) and (A C) (B D) (A B) (C D). For the first inclusion, assume that (x, y) (A B) (C D). Thus, x A B and y C D. Since x A B, then x A and x B. Since y C D, then y C and y D. So, since x A and y C, we know that (x, y) A C. Furthermore, since x B and y D, then we also know that (x, y) B D. Since (x, y) is in both sets, we can conclude that (x, y) (A C) (B D), and thus (A B) (C D) (A C) (B D). 10

11 For the second inclusion, assume that (x, y) (A C) (B D). Thus, (x, y) A C and (x, y) B D. Since (x, y) A C, we know that x A and y C. Furthermore, since (x, y) B D, we know that x B and y D. Since x A and x B, then x A B. Also, since y C and y D, then y C D. So, (x, y) (A B) (C D). Thus, (A C) (B D) (A B) (C D). Since we have shown both subset inclusions, we can conclude that (A B) (C D) = (A C) (B D). 11

### We now explore a third method of proof: proof by contradiction.

CHAPTER 6 Proof by Contradiction We now explore a third method of proof: proof by contradiction. This method is not limited to proving just conditional statements it can be used to prove any kind of statement

### The Language of Mathematics

CHPTER 2 The Language of Mathematics 2.1. Set Theory 2.1.1. Sets. set is a collection of objects, called elements of the set. set can be represented by listing its elements between braces: = {1, 2, 3,

### Applications of Methods of Proof

CHAPTER 4 Applications of Methods of Proof 1. Set Operations 1.1. Set Operations. The set-theoretic operations, intersection, union, and complementation, defined in Chapter 1.1 Introduction to Sets are

### Chapter 3. Cartesian Products and Relations. 3.1 Cartesian Products

Chapter 3 Cartesian Products and Relations The material in this chapter is the first real encounter with abstraction. Relations are very general thing they are a special type of subset. After introducing

### 5.1 Radical Notation and Rational Exponents

Section 5.1 Radical Notation and Rational Exponents 1 5.1 Radical Notation and Rational Exponents We now review how exponents can be used to describe not only powers (such as 5 2 and 2 3 ), but also roots

### 4.1. Definitions. A set may be viewed as any well defined collection of objects, called elements or members of the set.

Section 4. Set Theory 4.1. Definitions A set may be viewed as any well defined collection of objects, called elements or members of the set. Sets are usually denoted with upper case letters, A, B, X, Y,

### Mathematics for Computer Science/Software Engineering. Notes for the course MSM1F3 Dr. R. A. Wilson

Mathematics for Computer Science/Software Engineering Notes for the course MSM1F3 Dr. R. A. Wilson October 1996 Chapter 1 Logic Lecture no. 1. We introduce the concept of a proposition, which is a statement

### 2.1 Sets, power sets. Cartesian Products.

Lecture 8 2.1 Sets, power sets. Cartesian Products. Set is an unordered collection of objects. - used to group objects together, - often the objects with similar properties This description of a set (without

### CmSc 175 Discrete Mathematics Lesson 10: SETS A B, A B

CmSc 175 Discrete Mathematics Lesson 10: SETS Sets: finite, infinite, : empty set, U : universal set Describing a set: Enumeration = {a, b, c} Predicates = {x P(x)} Recursive definition, e.g. sequences

### MATH10040 Chapter 2: Prime and relatively prime numbers

MATH10040 Chapter 2: Prime and relatively prime numbers Recall the basic definition: 1. Prime numbers Definition 1.1. Recall that a positive integer is said to be prime if it has precisely two positive

### MODULAR ARITHMETIC. a smallest member. It is equivalent to the Principle of Mathematical Induction.

MODULAR ARITHMETIC 1 Working With Integers The usual arithmetic operations of addition, subtraction and multiplication can be performed on integers, and the result is always another integer Division, on

### LEARNING OBJECTIVES FOR THIS CHAPTER

CHAPTER 2 American mathematician Paul Halmos (1916 2006), who in 1942 published the first modern linear algebra book. The title of Halmos s book was the same as the title of this chapter. Finite-Dimensional

### Zeros of a Polynomial Function

Zeros of a Polynomial Function An important consequence of the Factor Theorem is that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we

### The Fundamental Theorem of Arithmetic

The Fundamental Theorem of Arithmetic 1 Introduction: Why this theorem? Why this proof? One of the purposes of this course 1 is to train you in the methods mathematicians use to prove mathematical statements,

### Sets and set operations

CS 441 Discrete Mathematics for CS Lecture 7 Sets and set operations Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square asic discrete structures Discrete math = study of the discrete structures used

### POWER SETS AND RELATIONS

POWER SETS AND RELATIONS L. MARIZZA A. BAILEY 1. The Power Set Now that we have defined sets as best we can, we can consider a sets of sets. If we were to assume nothing, except the existence of the empty

### Section 1.1 Real Numbers

. Natural numbers (N):. Integer numbers (Z): Section. Real Numbers Types of Real Numbers,, 3, 4,,... 0, ±, ±, ±3, ±4, ±,... REMARK: Any natural number is an integer number, but not any integer number is

### Set Theory Basic Concepts and Definitions

Set Theory Basic Concepts and Definitions The Importance of Set Theory One striking feature of humans is their inherent need and ability to group objects according to specific criteria. Our prehistoric

### Announcements. CompSci 230 Discrete Math for Computer Science Sets. Introduction to Sets. Sets

CompSci 230 Discrete Math for Computer Science Sets September 12, 2013 Prof. Rodger Slides modified from Rosen 1 nnouncements Read for next time Chap. 2.3-2.6 Homework 2 due Tuesday Recitation 3 on Friday

### 2. Methods of Proof Types of Proofs. Suppose we wish to prove an implication p q. Here are some strategies we have available to try.

2. METHODS OF PROOF 69 2. Methods of Proof 2.1. Types of Proofs. Suppose we wish to prove an implication p q. Here are some strategies we have available to try. Trivial Proof: If we know q is true then

### Quotient Rings and Field Extensions

Chapter 5 Quotient Rings and Field Extensions In this chapter we describe a method for producing field extension of a given field. If F is a field, then a field extension is a field K that contains F.

### 3. Mathematical Induction

3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)

### 1 Review of complex numbers

1 Review of complex numbers 1.1 Complex numbers: algebra The set C of complex numbers is formed by adding a square root i of 1 to the set of real numbers: i = 1. Every complex number can be written uniquely

### Mathematics Review for MS Finance Students

Mathematics Review for MS Finance Students Anthony M. Marino Department of Finance and Business Economics Marshall School of Business Lecture 1: Introductory Material Sets The Real Number System Functions,

### MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES Contents 1. Random variables and measurable functions 2. Cumulative distribution functions 3. Discrete

### Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

### MATHEMATICAL INDUCTION. Mathematical Induction. This is a powerful method to prove properties of positive integers.

MATHEMATICAL INDUCTION MIGUEL A LERMA (Last updated: February 8, 003) Mathematical Induction This is a powerful method to prove properties of positive integers Principle of Mathematical Induction Let P

### Sections 2.1, 2.2 and 2.4

SETS Sections 2.1, 2.2 and 2.4 Chapter Summary Sets The Language of Sets Set Operations Set Identities Introduction Sets are one of the basic building blocks for the types of objects considered in discrete

### Discrete Mathematics Set Operations

Discrete Mathematics 1-3. Set Operations Introduction to Set Theory A setis a new type of structure, representing an unordered collection (group, plurality) of zero or more distinct (different) objects.

### Solutions of Linear Equations in One Variable

2. Solutions of Linear Equations in One Variable 2. OBJECTIVES. Identify a linear equation 2. Combine like terms to solve an equation We begin this chapter by considering one of the most important tools

### CHAPTER 3 Numbers and Numeral Systems

CHAPTER 3 Numbers and Numeral Systems Numbers play an important role in almost all areas of mathematics, not least in calculus. Virtually all calculus books contain a thorough description of the natural,

### Handout #1: Mathematical Reasoning

Math 101 Rumbos Spring 2010 1 Handout #1: Mathematical Reasoning 1 Propositional Logic A proposition is a mathematical statement that it is either true or false; that is, a statement whose certainty or

### INTRODUCTORY SET THEORY

M.Sc. program in mathematics INTRODUCTORY SET THEORY Katalin Károlyi Department of Applied Analysis, Eötvös Loránd University H-1088 Budapest, Múzeum krt. 6-8. CONTENTS 1. SETS Set, equal sets, subset,

### Elementary Number Theory We begin with a bit of elementary number theory, which is concerned

CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,

### This is a square root. The number under the radical is 9. (An asterisk * means multiply.)

Page of Review of Radical Expressions and Equations Skills involving radicals can be divided into the following groups: Evaluate square roots or higher order roots. Simplify radical expressions. Rationalize

### Proof of Infinite Number of Fibonacci Primes. Stephen Marshall. 22 May Abstract

Proof of Infinite Number of Fibonacci Primes Stephen Marshall 22 May 2014 Abstract This paper presents a complete and exhaustive proof of that an infinite number of Fibonacci Primes exist. The approach

### CLASS 3, GIVEN ON 9/27/2010, FOR MATH 25, FALL 2010

CLASS 3, GIVEN ON 9/27/2010, FOR MATH 25, FALL 2010 1. Greatest common divisor Suppose a, b are two integers. If another integer d satisfies d a, d b, we call d a common divisor of a, b. Notice that as

### Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011

Basic Concepts of Point Set Topology Notes for OU course Math 4853 Spring 2011 A. Miller 1. Introduction. The definitions of metric space and topological space were developed in the early 1900 s, largely

### A Little Set Theory (Never Hurt Anybody)

A Little Set Theory (Never Hurt Anybody) Matthew Saltzman Department of Mathematical Sciences Clemson University Draft: August 21, 2013 1 Introduction The fundamental ideas of set theory and the algebra

### 2.1 The Algebra of Sets

Chapter 2 Abstract Algebra 83 part of abstract algebra, sets are fundamental to all areas of mathematics and we need to establish a precise language for sets. We also explore operations on sets and relations

### Introduction to Diophantine Equations

Introduction to Diophantine Equations Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles September, 2006 Abstract In this article we will only touch on a few tiny parts of the field

### not to be republishe NCERT SETS Chapter Introduction 1.2 Sets and their Representations

SETS Chapter 1 In these days of conflict between ancient and modern studies; there must surely be something to be said for a study which did not begin with Pythagoras and will not end with Einstein; but

### INCIDENCE-BETWEENNESS GEOMETRY

INCIDENCE-BETWEENNESS GEOMETRY MATH 410, CSUSM. SPRING 2008. PROFESSOR AITKEN This document covers the geometry that can be developed with just the axioms related to incidence and betweenness. The full

### SOLUTIONS TO ASSIGNMENT 1 MATH 576

SOLUTIONS TO ASSIGNMENT 1 MATH 576 SOLUTIONS BY OLIVIER MARTIN 13 #5. Let T be the topology generated by A on X. We want to show T = J B J where B is the set of all topologies J on X with A J. This amounts

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2 Proofs Intuitively, the concept of proof should already be familiar We all like to assert things, and few of us

### CONTINUED FRACTIONS AND PELL S EQUATION. Contents 1. Continued Fractions 1 2. Solution to Pell s Equation 9 References 12

CONTINUED FRACTIONS AND PELL S EQUATION SEUNG HYUN YANG Abstract. In this REU paper, I will use some important characteristics of continued fractions to give the complete set of solutions to Pell s equation.

### Math Workshop October 2010 Fractions and Repeating Decimals

Math Workshop October 2010 Fractions and Repeating Decimals This evening we will investigate the patterns that arise when converting fractions to decimals. As an example of what we will be looking at,

### COMPASS Numerical Skills/Pre-Algebra Preparation Guide. Introduction Operations with Integers Absolute Value of Numbers 13

COMPASS Numerical Skills/Pre-Algebra Preparation Guide Please note that the guide is for reference only and that it does not represent an exact match with the assessment content. The Assessment Centre

### The Rational Numbers

Math 3040: Spring 2011 The Rational Numbers Contents 1. The Set Q 1 2. Addition and multiplication of rational numbers 3 2.1. Definitions and properties. 3 2.2. Comments 4 2.3. Connections with Z. 6 2.4.

### 1. Prove that the empty set is a subset of every set.

1. Prove that the empty set is a subset of every set. Basic Topology Written by Men-Gen Tsai email: b89902089@ntu.edu.tw Proof: For any element x of the empty set, x is also an element of every set since

### 1 if 1 x 0 1 if 0 x 1

Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or

### 3.1. RATIONAL EXPRESSIONS

3.1. RATIONAL EXPRESSIONS RATIONAL NUMBERS In previous courses you have learned how to operate (do addition, subtraction, multiplication, and division) on rational numbers (fractions). Rational numbers

### Integer roots of quadratic and cubic polynomials with integer coefficients

Integer roots of quadratic and cubic polynomials with integer coefficients Konstantine Zelator Mathematics, Computer Science and Statistics 212 Ben Franklin Hall Bloomsburg University 400 East Second Street

### Math 4310 Handout - Quotient Vector Spaces

Math 4310 Handout - Quotient Vector Spaces Dan Collins The textbook defines a subspace of a vector space in Chapter 4, but it avoids ever discussing the notion of a quotient space. This is understandable

### POLYNOMIAL FUNCTIONS

POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a

### Stanford Math Circle: Sunday, May 9, 2010 Square-Triangular Numbers, Pell s Equation, and Continued Fractions

Stanford Math Circle: Sunday, May 9, 00 Square-Triangular Numbers, Pell s Equation, and Continued Fractions Recall that triangular numbers are numbers of the form T m = numbers that can be arranged in

### Homework 5 Solutions

Homework 5 Solutions 4.2: 2: a. 321 = 256 + 64 + 1 = (01000001) 2 b. 1023 = 512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = (1111111111) 2. Note that this is 1 less than the next power of 2, 1024, which

### CS 103X: Discrete Structures Homework Assignment 3 Solutions

CS 103X: Discrete Structures Homework Assignment 3 s Exercise 1 (20 points). On well-ordering and induction: (a) Prove the induction principle from the well-ordering principle. (b) Prove the well-ordering

### 2. THE x-y PLANE 7 C7

2. THE x-y PLANE 2.1. The Real Line When we plot quantities on a graph we can plot not only integer values like 1, 2 and 3 but also fractions, like 3½ or 4¾. In fact we can, in principle, plot any real

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note 11

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note Conditional Probability A pharmaceutical company is marketing a new test for a certain medical condition. According

### Math212a1010 Lebesgue measure.

Math212a1010 Lebesgue measure. October 19, 2010 Today s lecture will be devoted to Lebesgue measure, a creation of Henri Lebesgue, in his thesis, one of the most famous theses in the history of mathematics.

### The last three chapters introduced three major proof techniques: direct,

CHAPTER 7 Proving Non-Conditional Statements The last three chapters introduced three major proof techniques: direct, contrapositive and contradiction. These three techniques are used to prove statements

### Pythagorean Triples. Chapter 2. a 2 + b 2 = c 2

Chapter Pythagorean Triples The Pythagorean Theorem, that beloved formula of all high school geometry students, says that the sum of the squares of the sides of a right triangle equals the square of the

### Fractions and Decimals

Fractions and Decimals Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles December 1, 2005 1 Introduction If you divide 1 by 81, you will find that 1/81 =.012345679012345679... The first

### We give a basic overview of the mathematical background required for this course.

1 Background We give a basic overview of the mathematical background required for this course. 1.1 Set Theory We introduce some concepts from naive set theory (as opposed to axiomatic set theory). The

PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient

### Equations, Inequalities, Solving. and Problem AN APPLICATION

Equations, Inequalities, and Problem Solving. Solving Equations. Using the Principles Together AN APPLICATION To cater a party, Curtis Barbeque charges a \$0 setup fee plus \$ per person. The cost of Hotel

### Guide to SRW Section 1.7: Solving inequalities

Guide to SRW Section 1.7: Solving inequalities When you solve the equation x 2 = 9, the answer is written as two very simple equations: x = 3 (or) x = 3 The diagram of the solution is -6-5 -4-3 -2-1 0

### This unit will lay the groundwork for later units where the students will extend this knowledge to quadratic and exponential functions.

Algebra I Overview View unit yearlong overview here Many of the concepts presented in Algebra I are progressions of concepts that were introduced in grades 6 through 8. The content presented in this course

### Linear Programming Notes V Problem Transformations

Linear Programming Notes V Problem Transformations 1 Introduction Any linear programming problem can be rewritten in either of two standard forms. In the first form, the objective is to maximize, the material

### Likewise, we have contradictions: formulas that can only be false, e.g. (p p).

CHAPTER 4. STATEMENT LOGIC 59 The rightmost column of this truth table contains instances of T and instances of F. Notice that there are no degrees of contingency. If both values are possible, the formula

### 8 Primes and Modular Arithmetic

8 Primes and Modular Arithmetic 8.1 Primes and Factors Over two millennia ago already, people all over the world were considering the properties of numbers. One of the simplest concepts is prime numbers.

### Basic Proof Techniques

Basic Proof Techniques David Ferry dsf43@truman.edu September 13, 010 1 Four Fundamental Proof Techniques When one wishes to prove the statement P Q there are four fundamental approaches. This document

### a = bq + r where 0 r < b.

Lecture 5: Euclid s algorithm Introduction The fundamental arithmetic operations are addition, subtraction, multiplication and division. But there is a fifth operation which I would argue is just as fundamental

### 8 Divisibility and prime numbers

8 Divisibility and prime numbers 8.1 Divisibility In this short section we extend the concept of a multiple from the natural numbers to the integers. We also summarize several other terms that express

### Determinants, Areas and Volumes

Determinants, Areas and Volumes Theodore Voronov Part 2 Areas and Volumes The area of a two-dimensional object such as a region of the plane and the volume of a three-dimensional object such as a solid

### PART I. THE REAL NUMBERS

PART I. THE REAL NUMBERS This material assumes that you are already familiar with the real number system and the representation of the real numbers as points on the real line. I.1. THE NATURAL NUMBERS

### CHAPTER 5 Round-off errors

CHAPTER 5 Round-off errors In the two previous chapters we have seen how numbers can be represented in the binary numeral system and how this is the basis for representing numbers in computers. Since any

### Equations and Inequalities

Rational Equations Overview of Objectives, students should be able to: 1. Solve rational equations with variables in the denominators.. Recognize identities, conditional equations, and inconsistent equations.

### Trigonometric Functions and Triangles

Trigonometric Functions and Triangles Dr. Philippe B. Laval Kennesaw STate University August 27, 2010 Abstract This handout defines the trigonometric function of angles and discusses the relationship between

### Induction Problems. Tom Davis November 7, 2005

Induction Problems Tom Davis tomrdavis@earthlin.net http://www.geometer.org/mathcircles November 7, 2005 All of the following problems should be proved by mathematical induction. The problems are not necessarily

Section 5.4 The Quadratic Formula 481 5.4 The Quadratic Formula Consider the general quadratic function f(x) = ax + bx + c. In the previous section, we learned that we can find the zeros of this function

### Discrete Mathematics Lecture 3 Elementary Number Theory and Methods of Proof. Harper Langston New York University

Discrete Mathematics Lecture 3 Elementary Number Theory and Methods of Proof Harper Langston New York University Proof and Counterexample Discovery and proof Even and odd numbers number n from Z is called

### k, then n = p2α 1 1 pα k

Powers of Integers An integer n is a perfect square if n = m for some integer m. Taking into account the prime factorization, if m = p α 1 1 pα k k, then n = pα 1 1 p α k k. That is, n is a perfect square

### THE DIMENSION OF A VECTOR SPACE

THE DIMENSION OF A VECTOR SPACE KEITH CONRAD This handout is a supplementary discussion leading up to the definition of dimension and some of its basic properties. Let V be a vector space over a field

### Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics

Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights

### Propositional Logic. Definition: A proposition or statement is a sentence which is either true or false.

Propositional Logic Definition: A proposition or statement is a sentence which is either true or false. Definition:If a proposition is true, then we say its truth value is true, and if a proposition is

### Open and Closed Sets

Open and Closed Sets Definition: A subset S of a metric space (X, d) is open if it contains an open ball about each of its points i.e., if x S : ɛ > 0 : B(x, ɛ) S. (1) Theorem: (O1) and X are open sets.

### Discrete Mathematics

Discrete Mathematics Chih-Wei Yi Dept. of Computer Science National Chiao Tung University March 16, 2009 2.1 Sets 2.1 Sets 2.1 Sets Basic Notations for Sets For sets, we ll use variables S, T, U,. We can

### Infinite Algebra 1 supports the teaching of the Common Core State Standards listed below.

Infinite Algebra 1 Kuta Software LLC Common Core Alignment Software version 2.05 Last revised July 2015 Infinite Algebra 1 supports the teaching of the Common Core State Standards listed below. High School

### Problems on Discrete Mathematics 1

Problems on Discrete Mathematics 1 Chung-Chih Li 2 Kishan Mehrotra 3 Syracuse University, New York L A TEX at January 11, 2007 (Part I) 1 No part of this book can be reproduced without permission from

### 3.4 Complex Zeros and the Fundamental Theorem of Algebra

86 Polynomial Functions.4 Complex Zeros and the Fundamental Theorem of Algebra In Section., we were focused on finding the real zeros of a polynomial function. In this section, we expand our horizons and

### Math 3000 Section 003 Intro to Abstract Math Homework 2

Math 3000 Section 003 Intro to Abstract Math Homework 2 Department of Mathematical and Statistical Sciences University of Colorado Denver, Spring 2012 Solutions (February 13, 2012) Please note that these

### DIFFERENTIABILITY OF COMPLEX FUNCTIONS. Contents

DIFFERENTIABILITY OF COMPLEX FUNCTIONS Contents 1. Limit definition of a derivative 1 2. Holomorphic functions, the Cauchy-Riemann equations 3 3. Differentiability of real functions 5 4. A sufficient condition

### Chapter 17. Orthogonal Matrices and Symmetries of Space

Chapter 17. Orthogonal Matrices and Symmetries of Space Take a random matrix, say 1 3 A = 4 5 6, 7 8 9 and compare the lengths of e 1 and Ae 1. The vector e 1 has length 1, while Ae 1 = (1, 4, 7) has length

### CHAPTER 5. Product Measures

54 CHAPTER 5 Product Measures Given two measure spaces, we may construct a natural measure on their Cartesian product; the prototype is the construction of Lebesgue measure on R 2 as the product of Lebesgue

### 0.8 Rational Expressions and Equations

96 Prerequisites 0.8 Rational Expressions and Equations We now turn our attention to rational expressions - that is, algebraic fractions - and equations which contain them. The reader is encouraged to

### PRINCIPLES OF PROBLEM SOLVING

PRINCIPLES OF PROBLEM SOLVING There are no hard and fast rules that will ensure success in solving problems. However, it is possible to outline some general steps in the problem-solving process and to