# MATH 321 EQUIVALENCE RELATIONS, WELL-DEFINEDNESS, MODULAR ARITHMETIC, AND THE RATIONAL NUMBERS

Save this PDF as:

Size: px
Start display at page:

Download "MATH 321 EQUIVALENCE RELATIONS, WELL-DEFINEDNESS, MODULAR ARITHMETIC, AND THE RATIONAL NUMBERS"

## Transcription

1 MATH 321 EQUIVALENCE RELATIONS, WELL-DEFINEDNESS, MODULAR ARITHMETIC, AND THE RATIONAL NUMBERS ALLAN YASHINSKI Abstract. We explore the notion of well-definedness when defining functions whose domain is the set of all equivalence classes of an equivalence relation. Then we apply this to define modular arithmetic and the set Q of rational numbers. 1. Equivalence Classes We shall slightly adapt our notation for relations in this document. Let be a relation on a set X. Formally, is a subset of X X. Given two elements x, y X, we shall write x y to mean (x, y). From now on, we shall just use the notation x y, and not explicitly reference as a subset of X X. Recall that a relation on a set X is an equivalence relation if is (i) reflexive: ( x X)(x x), (ii) symmetric: ( x, y X)(x y y x), and (iii) transitive: ( x, y, z X) ( (x y y z) (x z) ). Suppose is an equivalence relation on X. When two elements are related via, it is common usage of language to say they are equivalent. Given x X, the equivalence class of x is the set x = {y X : x y}. In other words, the equivalence class x of x is the set of all elements of X that are equivalent to x. Be mindful that x is a subset of X, it is not an element of X. Typically, the set x contains much more than just x. The element x is called a representative of the equivalence class x. Any element of an equivalence class can serve as a representative. Exercise A. Given an equivalence relation on a set X, show that for any two elements x, y X, x y if and only if x = y. We shall use the notation X/ to mean the set of all equivalence classes with respect to. That is, X/ = { x : x X }. Notice that X/ is a set whose elements are subsets of X. We ve often referred to such a thing as a family of subsets of X. As shown in Theorem (i), X/ is actually a partition of X. 1

2 2 ALLAN YASHINSKI Example 1. Fix n N. Recall that we say two integers a, b Z are congruent modulo n when n (a b). As discussed in class, conguence modulo n is an equivalence relation. We shall write a b mod n to mean a is conguent to b modulo n. The set of equivalence classes of integers with respect to this equivalence relation is traditionally denoted Z/nZ. This is less cumbersome notationally than writing something like Z/( mod n). Exercise B. Consider the relation of congruence modulo 5. Explicitly describe the equivalence classes 0 and 7 from Z/5Z. 2. Functions whose domain is X/ It is common in mathematics (more common than you might guess) to work with the set X/ of equivalence classes of an equivalence relation. Issues arise when one attempts to define functions f : X/ Y whose domain is X/. When defining any function, one usually describes what the function does to a typical element of the domain. In this case, a typical element of the domain is an equivalence class x, which is represented by some element x X. One typically wants to define what f(x) is in terms of the representative x. This is where one can get into trouble. Consider the following example. Example 2. Let us attempt to define a function f : Z/5Z Z by the formula f(m) = m. Then we have f(0) = 0, f(2) = 2, f(3) = 3, f(7) = 7, and so on. But there is a major problem here. Since 2 7 mod 5, it follows that 2 = 7. That is, 2 and 7 are the exact same element of the set Z/5Z. However, the function we ve defined above has the property that f(2) f(7). This is a clear violation of the definition of a function, and so f is not really a function at all. Mathematicians would say that the function f is not well-defined, which really just means that f is not a function. The issue is that the rule for f maps different representatives of the same equivalence class to different elements of the codomain. Example 3. Let us attempt to define another function with domain Z/5Z. Given an integer m Z, let r 5 (m) denote the remainder, as in the Division Algorithm, after m is divided by 5. So r 5 (m) {0, 1, 2, 3, 4}. For example, Let s define a function r 5 (13) = 3, because 13 = f : Z/5Z {0, 1, 2, 3, 4}, by f(m) = r 5 (m). In this case, this function is well-defined (we ll explain how one checks this below.) At first glance, the definition of f(m) appears to depend on the choice of the representative m of the equivalence class, but it actually does not.

3 EQUIVALENCE RELATIONS AND WELL-DEFINEDNESS 3 We shall now explore how to ensure that we are defining actual functions out of X/. In what follows, let π : X X/ be the function defined by π(x) = x. Notice this is an honest function. It is surjective, but in general is not injective, because any two equivalent elements will have the same image. Exercise C. Suppose that f : X/ Y is a function, that is an honest welldefined function. Let g : X Y be the function g = f π. Prove that g(x) = g(x ) whenever x x. The point of this exercise is that in order to have a well-defined function f : X/ Y, then it must be the case that f(x) = f(x ) whenever x x. If not, then our attempt at making a function is not really a function at all. The following theorem says that this is the only issue we need to confront. Theorem 1. Let be an equivalence relation on X and let g : X Y be a function with the property that g(x) = g(x ) whenever x x. Then there exists a (well-defined) function given by the formula f(x) = g(x). f : X/ Y Before we prove this theorem, let us return to Example 3. We started with the function r 5 : Z {0, 1, 2, 3, 4}. Here, r 5 is going to play the role of g in the statement of the theorem. To use the theorem to verify that f(m) = r 5 (m) is well-defined, we must show that Assume m m r 5 (m) = r 5 (m ) whenever m m mod 5. mod 5. Then there is some k Z such that m = m + 5k. Applying the Division Algorithm to m gives us integers q and r such that m = 5q + r. Now r {0, 1, 2, 3, 4} and r 5 (m ) = r by definition of r 5. Combining the above equations shows that m = m + 5k = (5q + r) + 5k = 5(q + k) + r. From this, it follows that r 5 (m) = r. Thus we have shown that By the above theorem, the function is well-defined. r 5 (m) = r 5 (m ) whenever m m mod 5. f : Z/5Z {0, 1, 2, 3, 4}, f(m) = r 5 (m)

4 4 ALLAN YASHINSKI This argument is typical of how one verifies that a function f : X/ Y is well-defined. One takes arbitrary elements x, x X for which x x, and then one shows that the proposed expressions for f(x) and f(x ) give the same element of Y. In this case we would say that the expression f(x) depends only on the equivalence class of x, not on the representative x itself. To prove the theorem, we must take a slight digression into the foundations of what a function actually is. 3. The Definition of a Function We have been dealing with functions for quite some time now, but we never actually gave them a proper definition. According to the text: A function f from X to Y, written f : X Y, is a rule that pairs an element x X with an element y Y, written f(x) = y, such that the following property holds. ( x X)(!y Y )f(x) = y. The central issue here is that the term rule is undefined. This makes for an ambiguous definition. This is really unacceptable in mathematics. Everything must always have a precise, unambiguous definition. This deficiency (hopefully) did not impede our understanding of functions. Instead, it is making it quite awkward to prove the above theorem. In order to prove that something is a function, we must be clear about what a function is. Modern mathematics is built upon the foundations of set theory. What this means, is that everything in mathematics is really a set. This includes numbers, ordered pairs, relations (as we saw in class), and functions. We often don t think of these objects as sets, but they must be defined as sets so that we can be clear about what they are when we need to be. Here is the definition of a function: Definition. A function f from a set X to a set Y, is a subset of X Y with the following property: ( x X)(!y Y ) (x, y) f. The set X is the domain of f and the set Y is the codomain of f. For each x X, the unique value y Y such that (x, y) f is denoted by f(x) and is called the image of x. Notice that f is a set. By definition, the statement (x, y) f is equivalent to the statement f(x) = y. So whenever we say f(x) = y, we really mean (x, y) f. The collection of ordered pairs in f completely encode the rule of f. On your exam, the set f was called the graph of f. Let us now define function composition. Definition. Suppose f X Y and g Y Z are functions. The composition of g with f is the set g f = { (x, z) X Z : ( y Y ) (x, y) f (y, z) g }. Exercise D. Prove that g f, as defined above, is a function. (Hint: for each x X, prove existence and uniqueness of z Z for which (x, z) g f separately. To prove uniqueness, suppose (x, z 1 ), (x, z 2 ) g f, and show that z 1 = z 2.)

5 EQUIVALENCE RELATIONS AND WELL-DEFINEDNESS 5 We can translate the definitions of injectivity and surjectivity in terms of the set f. Definition. Let f X Y be a function. (i) We say f is injective if ( y Y )( x 1, x 2 X) ( (x 1, y) f (x 2, y) f (x 1 = x 2 ) ). (ii) We say f is surjective if ( y Y )( x X) (x, y) f. Exercise E. Suppose that f X Y is a function. Define f 1 Y X by f 1 = {(y, x) Y X : (x, y) f}. Prove that if f is both injective and surjective, then f 1 is a function. Now we shall prove Theorem 1. Proof. (Theorem 1) Suppose is an equivalence relation on X and g : X Y is a function such that Define f (X/ ) Y by g(x) = g(x ) whenever x x. f = {( x, g(x) ) : x X }. We shall prove that f is a function. Notice that every element of X/ is of the form x for some x X. So let x X/ be an arbitrary equivalence class, one of whose elements is x X. Let y = g(x). Then ( x, y ) = ( x, g(x) ) f. This shows that for every x X/, there is y Y such that ( x, y ) f. For the uniqueness assertion, suppose that ( x, g(x) ) f and ( x, g(x ) ) f are such that x = x. Then x x, which implies x x by definition of equivalence class. So g(x) = g(x ), by the assumed properties of g. We conclude that for every x X/, there is a unique y Y such that ( x, y ) f. By definition, f is a function. From now on, we shall no longer explicitly think of a function f : X Y as a subset of X Y, unless we need to. In fact, other people may look strangely at you if you say things like (x, y) f, rather than f(x) = y (remember, they are equivalent statements, by definition.) So now we can go about our lives treating functions as we used to, secretly knowing that we ve built a rigorous foundation of what a function really is. Example 4. Another situation in which we have been a little lax is in defining ordered pairs properly (and consequently Cartesian products). In light of the definition of a function, this is a loose end we should tie up. Everything must be a set, so we need to encode an ordered pair (x, y) into some set. Since sets are unordered, we cannot use {x, y} to represent (x, y), because {x, y} = {y, x} would then represent (y, x) as well. This is a problem because we want to distinguish between (x, y) and (y, x). Someone clever (Kazimierz Kuratowski) came up with

6 6 ALLAN YASHINSKI the following definition. Given x X and y Y, define the ordered pair (x, y) to be the set (x, y) = { {x}, {x, y} }. Notice that (x, y) P(P(X Y )). This turns out to be a good definition, because it satisfies what we want to be true about ordered pairs, namely that (x 1, y 1 ) = (x 2, y 2 ) if and only if (x 1 = x 2 ) (y 1 = y 2 ). Notice here that (x 1, y 1 ) = (x 2, y 2 ) is an equality of sets. The Cartesian Product of X and Y is defined to be the subset X Y P(P(X Y )) given by X Y = { (x, y) : x X y Y }. Exercise F. Using Kuratowski s definition of an ordered pair, prove that (x 1, y 1 ) = (x 2, y 2 ) if and only if (x 1 = x 2 ) (y 1 = y 2 ). (Hint: The forward ( ) direction is the nontrivial part. You may want to deal with the case where x 1 = y 1 first, and then you can assume x 1 y 1 for the rest of the proof. The reason for doing this is that (x, x) = { {x}, {x, x} } = { {x}, {x} } = { {x} } reduces to something much simpler when the two coordinates are the same.) Similarly to the case of functions, no one ever really explicitly thinks of (x, y) as the set { {x}, {x, y} }. For example, people will look strangely at you if you write things like {x} (x, y) or { {x, y} } (x, y), even if they are technically true. What s most important here is that we ve rigorously defined the object (x, y) in such a way that (x 1, y 1 ) = (x 2, y 2 ) if and only if (x 1 = x 2 ) (y 1 = y 2 ). Now, we can treat an ordered pair as a black box that satisfies the above property. If you think about it, that s what we ve been doing all along. 4. A Canonical Bijection Given a function f : X Y, let f be the relation on X defined by x f x if and only if f(x) = f(x ). Exercise G. Prove that f is an equivalence relation on X. Theorem 2. If f : X Y is a surjection, then there is a bijection f : X/ f Y, defined by f ( x ) = f(x). Proof. By definition of f, we know that f(x) = f(x ) whenever x f x. So by Theorem 1, the function f : X/ f Y, ) f( x = f(x)

7 EQUIVALENCE RELATIONS AND WELL-DEFINEDNESS 7 is well-defined. We shall prove f is a bijection. To see it is surjective, let y Y be arbitrary. Since f is surjective, there is some x X for which f(x) = y. Thus, the element x X/ f satisfies f ( x ) = f(x) = y. Thus, f is surjective. To see it is injective, suppose x, x X/ f are arbitrary classes for which f ( x ) = f ( x ). Then f(x) = f(x ) by definition of f. It follows that x f x by definition of f. As f is an equivalence relation, this implies x = x. Thus, f is surjective. As an example of this theorem, consider for n N the surjection r n : Z {0, 1, 2,..., n 1} which associates to an integer its remainder after division by n. Notice that the equivalence relation rn on Z is exactly congruence modulo n. By Theorem 2, the map ( ) r n : Z/nZ {0, 1, 2,..., n 1}, r n m = rn (m) is a bijection. We conclude that Z/nZ contains exactly n distinct equivalence classes, and more explicitly, Z/nZ = { 0, 1, 2,..., n 1 }. 5. Modular Arithmetic The familiar arithmetical operations of addition and multiplication can actually be carried out on the elements of Z/nZ = { 0, 1, 2,..., n 1 } Addition. Notice that the operation of addition of integers can be thought of as a function + : Z Z Z. The element +(a, b) Z is usually written as a + b. To define addition on Z/nZ, we will define a function + : Z/nZ Z/nZ Z/nZ. We shall write a + b to mean the element + ( a, b ) of Z/nZ. Definition. Define + : Z/nZ Z/nZ Z/nZ by a + b = a + b. Notice that the a + b on the right hand side is the sum of two integers, which is already defined. The sum a + b is defined the be the equivalence class of the sum a + b. Here, we run into the potential problem that this operation is not welldefined. The sum a + b is actually defined in terms of the representatives of the equivalence classes. We must make sure that if we choose different representatives, we still get the same result. Here s how a typical proof of well-definedness of this operation looks: Suppose a c mod n and b d mod n. Then a = c + nk, b = d + nl,

8 8 ALLAN YASHINSKI for some k, l Z. Then which shows that c + d = a nk + b nl = (a + b) (k + l)n, c + d a + b mod n. Thus, c + d = a + b. We conclude that when a = c and b = d, we have a + b = a + b = c + d = c + d. So our formula for addition on Z/nZ is well-defined. Example 5. When n = 2, Z/2Z = { 0, 1 }. We have = 0, = 1, = 1, = 2 = 0. Example 6. In Z/8Z = { 0, 1, 2, 3, 4, 5, 6, 7 }, = 5, = 8 = 0, = 10 = 2, etc. Once we know that the operation + is well-defined on Z/nZ, in inherits many of the properties of + on the integers Z, such as associativity and commutativity. Proposition 3. For all a, b, c Z/nZ, (i) (Associativity) ( a + b ) + c = a + ( b + c ). (ii) (Commutativity) a + b = b + a. (iii) (Additive identity) 0 + a = a = a + 0. (iv) (Additive inverses) a + a = 0 = a + a. Proof. For arbitrary a, b, c Z/nZ, ( a + b ) + c = a + b + c = (a + b) + c. By associativity of addition on Z, (a + b) + c = a + (b + c) = a + b + c = a + ( b + c ). This proves (i). For (ii), a b = ab = ba = b a, where in the middle we used commutativity of multiplication on Z. Similarly, we prove (iii) and (iv): 0 + a = 0 + a = a = a + 0 = a + 0, a + a = a + ( a) = 0 = ( a) + a = a + a. In the language of abstract algebra, this proposition says that the addition operation we ve defined makes Z/nZ into an abelian group. We won t define this term, and I won t expect you to know it, this is just for your information (you ll see it again if you take abstract algebra.)

9 EQUIVALENCE RELATIONS AND WELL-DEFINEDNESS Multiplication. We can also define a multiplication operation : Z/nZ Z/nZ Z/nZ. We shall write a b to mean the image of ( a, b ) under this function. Definition. Define : Z/nZ Z/nZ Z/nZ by Then a b = ab. Again, we must check that this is well-defined. Suppose a c mod n, b d mod n. a = c + nk, for some integers k, l Z. We see that b = d + nl cd = (a nk)(b nl) = ab nal nbk + n 2 kl = ab n(al + bk nkl). This calculation shows that cd ab mod n, and consequently, ab = cd. Thus, if a = c and b = d, we have a b = ab = cd = c d. We conclude that the element a b Z/nZ does not depend on the choice of representatives a and b. Our operation of multiplication is well-defined. Example 7. In Z/8Z = { 0, 1, 2, 3, 4, 5, 6, 7 }, 2 3 = 6, 4 6 = 24 = 0, 3 7 = 21 = 5, 5 5 = 25 = 1, etc. Multiplication on Z/nZ inherits several properties from multiplication on Z, as the next proposition shows. Proposition 4. For all a, b, c Z/nZ, (i) (Associativity) ( a b ) c = a (b c ). (ii) (Commutativity) a b = b a. (iii) (Distributive Property) a (b + c ) = ( a b ) + ( a c ). (iv) (Multiplicative Identity) 1 a = a = a 1. Proof. The proof is similar to that of Proposition 3. For example, to prove (iii), a (b + c ) = a b + c = a(b + c) = ab + ac, by the distributive property of Z. Continuing the calculation, ab + ac = ab + ac = ( a b ) + ( a c ). All these properties mean that the set Z/nZ with its two operations of addition and multiplication is a commutative ring. Again, you don t need to know this terminology, but you would see it in an abstract algebra course. Definition. The multiplicative inverse of an element a Z/nZ is an element b Z/nZ for which a b = 1. Multiplicative inverses do not always exist. For example, 0 Z/nZ never has a multiplicative inverse (assuming n > 1).

10 10 ALLAN YASHINSKI Example 8. In Z/5Z, the following are true: 1 1 = 1, 2 3 = 1, 4 4 = 1. Thus, 3 is the inverse of 2 (and vice-versa), 4 is its own inverse, and 1 is its own inverse. Example 9. In Z/4Z, the element 2 has no multiplicative inverse. We can prove this by exhaustion (checking its product with all possibilities): 2 0 = 0, 2 1 = 2, 2 2 = 0, 2 3 = 2. Thus, there is no element a such that 2 a = 1. Exercise H. (a) It is a fact that in Z/7Z = { 0, 1, 2, 3, 4, 5, 6 }, every element except for 0 has a multiplicative inverse. Find the inverse of each nonzero class. (b) Determine which elements of Z/9Z = { 0, 1, 2, 3, 4, 5, 6, 7, 8 } have a multiplicative inverse, and find the inverses. A commutative ring for which each nonzero element has a multiplicative inverse is called a field. Once again, this is more abstract algebra terminology that you don t have to know for this class. It is a fact, which we shall neither prove nor use, that Z/nZ is a field if and only if n is prime. 6. Building the Rational Numbers As discussed above, everything in mathematics must be defined, typically as a set, and this includes numbers. Defining our number systems N, Z, Q, R, and C is an issue we have largely avoided, and is one which may be more complicated than you would think. Notice that N Z Q R C. This chain of subset inclusions also represents how one constructs these number systems: Z is defined in terms of N. Q is defined in terms of Z. R is defined in terms of Q. C is defined in terms of R. This begs the question: How does one define N? The answer is that N must be defined using sets, and sets alone. We shall not explore the construction of N, but it is a fact that once we have defined the set N, we can build the other number systems as indicated above. Each of the four constructions differs from the others. Some are simpler than others: Defining the complex numbers C in terms of the reals R is one of the simplest, whereas defining R in terms of Q is much more difficult. Here, we shall explore only one of them: constructing the set Q of rational numbers from the set Z of integers. Here is the main idea in defining Q. We think of a rational number as a quotient a b of two integers a and b. We could attempt to represent this quotient as an ordered

11 EQUIVALENCE RELATIONS AND WELL-DEFINEDNESS 11 pair (a, b) Z Z (remember we are supposed to be using Z to define Q.) The central issue is this: several pairs of integers can be used to represent the same rational number. For example, we all know that 1 2 = 2 4 = 5 10 = However, none of the ordered pairs (1, 2), (2, 4), (5, 10), (150, 300) are equal, as elements on Z Z. We shall want to think of these ordered pairs as being the same. The mathematical solution to this problem is to put an equivalence relation on the set of ordered pairs of integers in such a way that the above pairs are equivalent. This is the idea that will guide the construction. Not every ordered pair of integers should represent a fraction, because we want to prohibit the situation in which the denominator is 0. Thus, the set we shall work with will be Z ( Z \ {0} ). An element (a, b) Z ( Z \ {0} ) will ultimately represent the number a b. We need to figure out when two such pairs (a, b) and (c, d) represent the same fraction. We ve known since grade school that the statement a b = c if and only if ad = bc d needs to be true. Teachers usually call this cross-multiplying. This motivates the following definition. Definition. Define a relation on Z ( Z \ {0} ) so that (a, b) (c, d) whenever ad = bc. Exercise I. Prove that is an equivalence relation on Z ( Z \ {0} ). (Hint: To prove transitivity, assume (a, b) (c, d) and (c, d) (e, f). Then show that this implies that d(af be) = 0. Then use the fact that if a product of integers is zero, then one of the factors must be zero. Definition. The set Q of rational numbers is defined the be the set of equivalence classes of elements of the set Z ( Z \ {0} ) with respect to the relation. Given a, b Z with b 0, se shall use the notation a b to mean (a, b) Q. That is, a b = (a, b), is a definition, where we are defining the left hand side to be the right hand side. Exercise J. Prove that 1 2 = 2 4 and 2 5 = Exercise K. Define a function f : Z Q by f(n) = (n, 1). Prove that this function is injective. (Be careful about what it means when two elements of Q are equal.) This allows us to identify Z as a subset of Q, namely its image f(z) Q, under the function f.

13 EQUIVALENCE RELATIONS AND WELL-DEFINEDNESS 13 whereas Thus, we see that For (iv), r + (s + t) = (a, b) + ( (c, d) + (e, f) ) = (a, b) + (cf + de, df) = (adf + b(cf + de), bdf) = (adf + bcf + bde, bdf). (r + s) + t = r + (t + s). r + ( r) = (a, b) + ( a, b) = (ab + ( ab), b 2 ) = (0, b 2 ). Notice that for any m Z \ {0}, we have (0, m) (0, 1). Indeed, it is true that 0 1 = 0 m. Thus, for any such m, (0, m) = 0 Q. Thus, r + ( r) = (0, b 2 ) = 0, as desired. Similarly (or by part (ii)), we have ( r) + r = 0. For (vi), r s = (a, b) (c, d) = (ac, bd) = (ca, db) = (c, d) (a, b) = s r. For (viii), r 1 = (a, b) (1, 1) = (a 1, b 1) = (a, b) = r. By part (vi) and the previous calculation, 1 r = r 1 = r. Exercise N. Prove parts (ii), (iii), (v), (vii) of Proposition 5. Using the notation a b = (a, b) Q, our operations satisfy a b + c ad + bc a =, and d bd b c d = ac bd. Thus, we have defined the arithmetic of Q and proved its familiar properties. Lastly, let s address the issue of multiplicative inverses. Given r = (a, b) Q such that r 0, define r 1 = (b, a). Notice that a 0 here (Why?). Exercise O. Show that for any r Q such that r 0, r r 1 = 1 = r 1 r. In the language of abstract algebra, Q is a field, because it is a commutative ring (by Proposition 5), in which every nonzero element has a multiplicative inverse. People don t outwardly treat rational numbers as equivalence classes of ordered pairs of integers. Once again, you will earn strange looks if you write things like (4, 8) 1 2, even though it is a true statement. The point is that we have rigorously defined Q and its operations in such a way that it behaves in the way that we ve always treated rational numbers. Now we can go back to our old habits, confidently knowing that everything has been put on a firm foundation.

### 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,

### 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

### 3. Equivalence Relations. Discussion

3. EQUIVALENCE RELATIONS 33 3. Equivalence Relations 3.1. Definition of an Equivalence Relations. Definition 3.1.1. A relation R on a set A is an equivalence relation if and only if R is reflexive, symmetric,

### 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.

### You know from calculus that functions play a fundamental role in mathematics.

CHPTER 12 Functions You know from calculus that functions play a fundamental role in mathematics. You likely view a function as a kind of formula that describes a relationship between two (or more) quantities.

### Lecture 16 : Relations and Functions DRAFT

CS/Math 240: Introduction to Discrete Mathematics 3/29/2011 Lecture 16 : Relations and Functions Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT In Lecture 3, we described a correspondence

### 11 Ideals. 11.1 Revisiting Z

11 Ideals The presentation here is somewhat different than the text. In particular, the sections do not match up. We have seen issues with the failure of unique factorization already, e.g., Z[ 5] = O Q(

### Math 319 Problem Set #3 Solution 21 February 2002

Math 319 Problem Set #3 Solution 21 February 2002 1. ( 2.1, problem 15) Find integers a 1, a 2, a 3, a 4, a 5 such that every integer x satisfies at least one of the congruences x a 1 (mod 2), x a 2 (mod

### Set theory as a foundation for mathematics

Set theory as a foundation for mathematics Waffle Mathcamp 2011 In school we are taught about numbers, but we never learn what numbers really are. We learn rules of arithmetic, but we never learn why these

### 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 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

### Problem Set 7 - Fall 2008 Due Tuesday, Oct. 28 at 1:00

18.781 Problem Set 7 - Fall 2008 Due Tuesday, Oct. 28 at 1:00 Throughout this assignment, f(x) always denotes a polynomial with integer coefficients. 1. (a) Show that e 32 (3) = 8, and write down a list

### Properties of Real Numbers

16 Chapter P Prerequisites P.2 Properties of Real Numbers What you should learn: Identify and use the basic properties of real numbers Develop and use additional properties of real numbers Why you should

### I. GROUPS: BASIC DEFINITIONS AND EXAMPLES

I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called

### 26 Ideals and Quotient Rings

Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 26 Ideals and Quotient Rings In this section we develop some theory of rings that parallels the theory of groups discussed

### CONTENTS 1. Peter Kahn. Spring 2007

CONTENTS 1 MATH 304: CONSTRUCTING THE REAL NUMBERS Peter Kahn Spring 2007 Contents 2 The Integers 1 2.1 The basic construction.......................... 1 2.2 Adding integers..............................

### 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

### Chapter 10. Abstract algebra

Chapter 10. Abstract algebra C.O.S. Sorzano Biomedical Engineering December 17, 2013 10. Abstract algebra December 17, 2013 1 / 62 Outline 10 Abstract algebra Sets Relations and functions Partitions and

### Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan

Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 3 Binary Operations We are used to addition and multiplication of real numbers. These operations combine two real numbers

### Math 223 Abstract Algebra Lecture Notes

Math 223 Abstract Algebra Lecture Notes Steven Tschantz Spring 2001 (Apr. 23 version) Preamble These notes are intended to supplement the lectures and make up for the lack of a textbook for the course

### Finite Fields and Error-Correcting Codes

Lecture Notes in Mathematics Finite Fields and Error-Correcting Codes Karl-Gustav Andersson (Lund University) (version 1.013-16 September 2015) Translated from Swedish by Sigmundur Gudmundsson Contents

### Homework until Test #2

MATH31: Number Theory Homework until Test # Philipp BRAUN Section 3.1 page 43, 1. It has been conjectured that there are infinitely many primes of the form n. Exhibit five such primes. Solution. Five such

### 3 1. Note that all cubes solve it; therefore, there are no more

Math 13 Problem set 5 Artin 11.4.7 Factor the following polynomials into irreducible factors in Q[x]: (a) x 3 3x (b) x 3 3x + (c) x 9 6x 6 + 9x 3 3 Solution: The first two polynomials are cubics, so if

### Notes on Algebraic Structures. Peter J. Cameron

Notes on Algebraic Structures Peter J. Cameron ii Preface These are the notes of the second-year course Algebraic Structures I at Queen Mary, University of London, as I taught it in the second semester

### 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,

### 5.1 Commutative rings; Integral Domains

5.1 J.A.Beachy 1 5.1 Commutative rings; Integral Domains from A Study Guide for Beginner s by J.A.Beachy, a supplement to Abstract Algebra by Beachy / Blair 23. Let R be a commutative ring. Prove the following

### 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.

### Chapter 7. Functions and onto. 7.1 Functions

Chapter 7 Functions and onto This chapter covers functions, including function composition and what it means for a function to be onto. In the process, we ll see what happens when two dissimilar quantifiers

### Mathematics for Computer Science

Mathematics for Computer Science Lecture 2: Functions and equinumerous sets Areces, Blackburn and Figueira TALARIS team INRIA Nancy Grand Est Contact: patrick.blackburn@loria.fr Course website: http://www.loria.fr/~blackbur/courses/math

### So let us begin our quest to find the holy grail of real analysis.

1 Section 5.2 The Complete Ordered Field: Purpose of Section We present an axiomatic description of the real numbers as a complete ordered field. The axioms which describe the arithmetic of the real numbers

### Foundations of Mathematics I Set Theory (only a draft)

Foundations of Mathematics I Set Theory (only a draft) Ali Nesin Mathematics Department Istanbul Bilgi University Kuştepe Şişli Istanbul Turkey anesin@bilgi.edu.tr February 12, 2004 2 Contents I Naive

### 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

### 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,

### Cartesian 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

### 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,

### Notes on Factoring. MA 206 Kurt Bryan

The General Approach Notes on Factoring MA 26 Kurt Bryan Suppose I hand you n, a 2 digit integer and tell you that n is composite, with smallest prime factor around 5 digits. Finding a nontrivial factor

### 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

### CHAPTER 3. Methods of Proofs. 1. Logical Arguments and Formal Proofs

CHAPTER 3 Methods of Proofs 1. Logical Arguments and Formal Proofs 1.1. Basic Terminology. An axiom is a statement that is given to be true. A rule of inference is a logical rule that is used to deduce

### a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.

Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given

### Applications of Fermat s Little Theorem and Congruences

Applications of Fermat s Little Theorem and Congruences Definition: Let m be a positive integer. Then integers a and b are congruent modulo m, denoted by a b mod m, if m (a b). Example: 3 1 mod 2, 6 4

### ABSTRACT ALGEBRA: A STUDY GUIDE FOR BEGINNERS

ABSTRACT ALGEBRA: A STUDY GUIDE FOR BEGINNERS John A. Beachy Northern Illinois University 2014 ii J.A.Beachy This is a supplement to Abstract Algebra, Third Edition by John A. Beachy and William D. Blair

### Computer Algebra for Computer Engineers

p.1/14 Computer Algebra for Computer Engineers Preliminaries Priyank Kalla Department of Electrical and Computer Engineering University of Utah, Salt Lake City p.2/14 Notation R: Real Numbers Q: Fractions

### 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.

### Revised Version of Chapter 23. We learned long ago how to solve linear congruences. ax c (mod m)

Chapter 23 Squares Modulo p Revised Version of Chapter 23 We learned long ago how to solve linear congruences ax c (mod m) (see Chapter 8). It s now time to take the plunge and move on to quadratic equations.

### 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

### Lecture 17 : Equivalence and Order Relations DRAFT

CS/Math 240: Introduction to Discrete Mathematics 3/31/2011 Lecture 17 : Equivalence and Order Relations Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT Last lecture we introduced the notion

### Chapter 7: Products and quotients

Chapter 7: Products and quotients Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 42, Spring 24 M. Macauley (Clemson) Chapter 7: Products

### CHAPTER 5. Number Theory. 1. Integers and Division. Discussion

CHAPTER 5 Number Theory 1. Integers and Division 1.1. Divisibility. Definition 1.1.1. Given two integers a and b we say a divides b if there is an integer c such that b = ac. If a divides b, we write a

### Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

### Chapter 11 Number Theory

Chapter 11 Number Theory Number theory is one of the oldest branches of mathematics. For many years people who studied number theory delighted in its pure nature because there were few practical applications

### An Interesting Way to Combine Numbers

An Interesting Way to Combine Numbers Joshua Zucker and Tom Davis November 28, 2007 Abstract This exercise can be used for middle school students and older. The original problem seems almost impossibly

### 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

### 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)

### PYTHAGOREAN TRIPLES KEITH CONRAD

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

### Alex, I will take congruent numbers for one million dollars please

Alex, I will take congruent numbers for one million dollars please Jim L. Brown The Ohio State University Columbus, OH 4310 jimlb@math.ohio-state.edu One of the most alluring aspectives of number theory

### 3. Applications of Number Theory

3. APPLICATIONS OF NUMBER THEORY 163 3. Applications of Number Theory 3.1. Representation of Integers. Theorem 3.1.1. Given an integer b > 1, every positive integer n can be expresses uniquely as n = a

### 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

### 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

### Geometric Transformations

Geometric Transformations Definitions Def: f is a mapping (function) of a set A into a set B if for every element a of A there exists a unique element b of B that is paired with a; this pairing is denoted

### 1.4 Factors and Prime Factorization

1.4 Factors and Prime Factorization Recall from Section 1.2 that the word factor refers to a number which divides into another number. For example, 3 and 6 are factors of 18 since 3 6 = 18. Note also that

### Chapter 13: Basic ring theory

Chapter 3: Basic ring theory Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 42, Spring 24 M. Macauley (Clemson) Chapter 3: Basic ring

### Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.

Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specialize

### Stupid Divisibility Tricks

Stupid Divisibility Tricks 101 Ways to Stupefy Your Friends Appeared in Math Horizons November, 2006 Marc Renault Shippensburg University Mathematics Department 1871 Old Main Road Shippensburg, PA 17013

### Math Review. for the Quantitative Reasoning Measure of the GRE revised General Test

Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important

### FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES

FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES CHRISTOPHER HEIL 1. Cosets and the Quotient Space Any vector space is an abelian group under the operation of vector addition. So, if you are have studied

### Factoring Polynomials

Factoring Polynomials Sue Geller June 19, 2006 Factoring polynomials over the rational numbers, real numbers, and complex numbers has long been a standard topic of high school algebra. With the advent

### 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

### 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 Complex Functions and the Cauchy-Riemann Equations

2 Complex Functions and the Cauchy-Riemann Equations 2.1 Complex functions In one-variable calculus, we study functions f(x) of a real variable x. Likewise, in complex analysis, we study functions f(z)

### Practice with Proofs

Practice with Proofs October 6, 2014 Recall the following Definition 0.1. A function f is increasing if for every x, y in the domain of f, x < y = f(x) < f(y) 1. Prove that h(x) = x 3 is increasing, using

### 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

### 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

### Appendix A. Appendix. A.1 Algebra. Fields and Rings

Appendix A Appendix A.1 Algebra Algebra is the foundation of algebraic geometry; here we collect some of the basic algebra on which we rely. We develop some algebraic background that is needed in the text.

### Mathematics of Cryptography Modular Arithmetic, Congruence, and Matrices. A Biswas, IT, BESU SHIBPUR

Mathematics of Cryptography Modular Arithmetic, Congruence, and Matrices A Biswas, IT, BESU SHIBPUR McGraw-Hill The McGraw-Hill Companies, Inc., 2000 Set of Integers The set of integers, denoted by Z,

### Discrete Mathematics: Solutions to Homework (12%) For each of the following sets, determine whether {2} is an element of that set.

Discrete Mathematics: Solutions to Homework 2 1. (12%) For each of the following sets, determine whether {2} is an element of that set. (a) {x R x is an integer greater than 1} (b) {x R x is the square

### U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra

U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory

### Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.

Algebra 2 - Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers - {1,2,3,4,...}

### Lecture Notes on Polynomials

Lecture Notes on Polynomials Arne Jensen Department of Mathematical Sciences Aalborg University c 008 Introduction These lecture notes give a very short introduction to polynomials with real and complex

### 13 Infinite Sets. 13.1 Injections, Surjections, and Bijections. mcs-ftl 2010/9/8 0:40 page 379 #385

mcs-ftl 2010/9/8 0:40 page 379 #385 13 Infinite Sets So you might be wondering how much is there to say about an infinite set other than, well, it has an infinite number of elements. Of course, an infinite

### GROUPS ACTING ON A SET

GROUPS ACTING ON A SET MATH 435 SPRING 2012 NOTES FROM FEBRUARY 27TH, 2012 1. Left group actions Definition 1.1. Suppose that G is a group and S is a set. A left (group) action of G on S is a rule for

### MA651 Topology. Lecture 6. Separation Axioms.

MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples

### Vieta s Formulas and the Identity Theorem

Vieta s Formulas and the Identity Theorem This worksheet will work through the material from our class on 3/21/2013 with some examples that should help you with the homework The topic of our discussion

### V55.0106 Quantitative Reasoning: Computers, Number Theory and Cryptography

V55.0106 Quantitative Reasoning: Computers, Number Theory and Cryptography 3 Congruence Congruences are an important and useful tool for the study of divisibility. As we shall see, they are also critical

### 2.1 Functions. 2.1 J.A.Beachy 1. from A Study Guide for Beginner s by J.A.Beachy, a supplement to Abstract Algebra by Beachy / Blair

2.1 J.A.Beachy 1 2.1 Functions from A Study Guide for Beginner s by J.A.Beachy, a supplement to Abstract Algebra by Beachy / Blair 21. The Vertical Line Test from calculus says that a curve in the xy-plane

### 1. The algebra of exponents 1.1. Natural Number Powers. It is easy to say what is meant by a n a (raised to) to the (power) n if n N.

CHAPTER 3: EXPONENTS AND POWER FUNCTIONS 1. The algebra of exponents 1.1. Natural Number Powers. It is easy to say what is meant by a n a (raised to) to the (power) n if n N. For example: In general, if

### Introduction to Modern Algebra

Introduction to Modern Algebra David Joyce Clark University Version 0.0.6, 3 Oct 2008 1 1 Copyright (C) 2008. ii I dedicate this book to my friend and colleague Arthur Chou. Arthur encouraged me to write

### Chapter 7 - Roots, Radicals, and Complex Numbers

Math 233 - Spring 2009 Chapter 7 - Roots, Radicals, and Complex Numbers 7.1 Roots and Radicals 7.1.1 Notation and Terminology In the expression x the is called the radical sign. The expression under the

### Lesson 6: Proofs of Laws of Exponents

NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 6 8 Student Outcomes Students extend the previous laws of exponents to include all integer exponents. Students base symbolic proofs on concrete examples to

### TOPIC 3: CONTINUITY OF FUNCTIONS

TOPIC 3: CONTINUITY OF FUNCTIONS. Absolute value We work in the field of real numbers, R. For the study of the properties of functions we need the concept of absolute value of a number. Definition.. Let

### 6 Proportion: Fractions, Direct and Inverse Variation, and Percent

6 Proportion: Fractions, Direct and Inverse Variation, and Percent 6.1 Fractions Every rational number can be written as a fraction, that is a quotient of two integers, where the divisor of course cannot

### The Mathematics School Teachers Should Know

The Mathematics School Teachers Should Know Lisbon, Portugal January 29, 2010 H. Wu *I am grateful to Alexandra Alves-Rodrigues for her many contributions that helped shape this document. Do school math

### Taylor Polynomials and Taylor Series Math 126

Taylor Polynomials and Taylor Series Math 26 In many problems in science and engineering we have a function f(x) which is too complicated to answer the questions we d like to ask. In this chapter, we will

### 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

### Linear Algebra I. Ronald van Luijk, 2012

Linear Algebra I Ronald van Luijk, 2012 With many parts from Linear Algebra I by Michael Stoll, 2007 Contents 1. Vector spaces 3 1.1. Examples 3 1.2. Fields 4 1.3. The field of complex numbers. 6 1.4.

### Solution to Homework 2

Solution to Homework 2 Olena Bormashenko September 23, 2011 Section 1.4: 1(a)(b)(i)(k), 4, 5, 14; Section 1.5: 1(a)(b)(c)(d)(e)(n), 2(a)(c), 13, 16, 17, 18, 27 Section 1.4 1. Compute the following, if

### Real Roots of Univariate Polynomials with Real Coefficients

Real Roots of Univariate Polynomials with Real Coefficients mostly written by Christina Hewitt March 22, 2012 1 Introduction Polynomial equations are used throughout mathematics. When solving polynomials

### MATH 361: NUMBER THEORY FIRST LECTURE

MATH 361: NUMBER THEORY FIRST LECTURE 1. Introduction As a provisional definition, view number theory as the study of the properties of the positive integers, Z + = {1, 2, 3, }. Of particular interest,

### The Method of Partial Fractions Math 121 Calculus II Spring 2015

Rational functions. as The Method of Partial Fractions Math 11 Calculus II Spring 015 Recall that a rational function is a quotient of two polynomials such f(x) g(x) = 3x5 + x 3 + 16x x 60. The method