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Algebraic Systems, Spring 2014, January, 2014 Edition Gabriel Kerr Algebraic Systems, Spring 2014, January, 2014 Edition Gabriel Kerr Contents Chapter 0. Peano Axioms for Natural Numbers - An Introduction to Proofs 5 0.1. Sets and Logic 5 Exercises 7 0.2. Peano Axioms 8 Exercises 15 0.3. Relations 15 Exercises 19 Chapter 1. Basic Arithmetic 21 1.1. The set Z 21 Exercises 23 1.2. The ring Z 23 Exercises 27 1.3. Factoring integers, part I 27 Exercises 30 1.4. Factoring integers, part II 30 Exercises 32 1.5. Modular Arithmetic 32 Exercises 34 1.6. The fields Q and R 34 Exercises 39 1.7. The field C 39 Exercises 40 3 CHAPTER 0 Peano Axioms for Natural Numbers - An Introduction to Proofs We begin this course with the construction of the natural number system. The Peano Axioms form the heart of much of mathematics and lay the foundation for algebra and analysis. Every attempt will be made to stay away from unnecessary abstraction. This should be a guiding principle when working out the exercises as well! 0.1. Sets and Logic This section should serve as a very short introduction to 20th century mathe- matics. That is, it is an introduction to proofs involving sets. Set theory is in fact a subject within itself that was initiated and studied by many in the early 20th century. This was in response to several paradoxes that had come up. For us, a set is a collection of things, called elements. For example, Atoms could be the set of all atoms in the universe while Adams could be the set of all people named Adam in your family. You say there is no Adam in your family? Then Adams is known as the empty set which is the set that contains no elements at all. The notation for this important set is Adams = ;: Unfortunately, this is just the beginning of new notation. Now that we know what sets are, we need a slew of symbols to describe how they work and interact with each other. First, we often write a set with only a few elements by enclosing the elements inside curly brackets. For example, if we wanted to write the set S of letters in the alphabet that occur before g we would write S = fa; b; c; d; e; fg: If you have a set with a lot of elements that are ordered, you can use the dot-dot- dot notation... For example the set of integers T between 3 and 600 and the set of integers T 0 greater than 5 can be written T = f3; 4; :::; 600g T 0 = f6; 7;:::g: In general, we will be interested in stating whether something is or is not an element of a given set. If a is in A (which is the same thing as saying a is an element of A), we write a 2 A while if it is not, we write a 62 A. For the above examples we could say a 2 S but a 62 T and 5 2 T but 5 62 T 0. Sometimes we want to define a set that has elements in another set which satisfy a property. For example, if we want to consider atoms H that have only one proton we can write H = fa 2 Atoms : a has one proton g: 5 6 0. PEANO AXIOMS FOR NATURAL NUMBERS - AN INTRODUCTION TO PROOFS It is clear that any element of the set H is also an element of Atoms, after all, that was how H was defined. This is precisely what it means for H to be a subset of Atoms. To express this relationship we write H ⊆ Atoms: If we have two sets A and B, we can define a whole lot of new sets. We will sum many of these constructions (and one property) up in the following definition. Definition 0.1.1. Suppose A and B are sets. (1) The set A \ B is the intersection of A and B. It consists of elements c such that c 2 A and c 2 B. (2) Two sets A and B are called disjoint if A \ B = ;. (3) The set A [ B is the union of A and B. It consists of elements c such that c 2 A or c 2 B. (4) The set A × B is the Cartesian product of A and B. It consists of elements c = (a; b) where a 2 A and b 2 B. Enough of the definitions. Let's try proving something algebraic. Proposition 0.1.1 (Commutativity of intersection). If A and B are sets then A \ B = B \ A You may think about this a second and say \What's the big deal? Of course this is true!", but us mathematicians really need something better than \of course"... we need a proof! The way to prove a statement like this is to go back to the definition and methodically show that the definitions force the statement to be correct. This way of proving something is called a direct proof. Proof. Suppose c 2 A \ B. Then, by definition, c 2 A and c 2 B which implies that c 2 B and c 2 A. But this means, again by definition, that c 2 B \ A. Thus A \ B ⊆ B \ A. Conversely, suppose c 2 B \ A. Then, by definition, c 2 B and c 2 A which implies that c 2 A and c 2 B. Again by definition, we get that c 2 A \ B. Thus B \ A ⊆ A \ B. So every element in A \ B is an element of B \ A and vice-versa. This means that these two sets consist of the same elements and are therefore equal. What should not be lost in this discussion is that a new term was introduced in the title of the proposition, namely commutativity. It simply means that a combined with b equals b combined with a for some way of combining things. It is one of the key ideas in algebra that can sometimes fail, and will come up repeatedly in the course. Leaving this fascinating stuff for later, let us return to sets. If you are generous and have a set that you would like to share with others, then you may be tempted to break it up into subsets and pass those subsets around. In fact, the idea of breaking up a set into subsets is a precise and important notion in mathematics whose definition is given below. Definition 0.1.2. A partition P of a set A is a collection of non-empty subsets, P = fAigi2I indexed by I such that (1) For any element a 2 A, there is an i 2 I such that a 2 Ai. 0 (2) For any two distinct elements i; i 2 I, the Ai and Ai0 are disjoint. EXERCISES 7 In this definition, the indexing set is arbitrary could be called J or pπ or S 2 anything you want. The important thing is that you have a set P of subsets of Asatisfying (1) and (2). We will encounter many partitions as we progress through this course, but for now, let's look at a simple example. Example 0.1.1. For our set Atoms, we can form the partition P = fAngn2f1;:::;103g where An = fatoms with n protonsg. Now let's return back to relationships between sets. One way of relating two sets is by defining a function or a map from one to another. Here is the mathematical definition. Definition 0.1.3. Suppose A and B are sets. A function f from A to B is a subset f ⊂ A × B such that for every a 2 A there exists exactly one element c = (a; b) 2 f. A function can be denoted f : A ! B There is a lot of notation that comes along with a function. For example, we write f(a) as the unique element b for which (a; b) 2 f. We also call A the domain of f and B the codomain. This latter term should be prevalent in secondary school but is frequently confused with the different notion of range. The range of f is defined as the subset range(f) = fb 2 B : there is an a 2 A such that (a; b) 2 fg: Connection 0.1.1. Most middle and high school texts (and too many college texts) content themselves with saying 'a function is an assignment'. This is a good description of what a function does, but perhaps not what a function is. Neverthe- less, a teacher can usually pull off this type of definition and use it successfully at the high school and early college level. A worse situation occurs when high school students are taught that functions always send real numbers to real numbers. This is a sad injustice that no student of Math 511 will perpetuate! However, a question from this type of thinking arises. Why does the vertical line test for graphs mean that a graph is defined by a function? Functions are most useful when they are combined and compared. The most common way of combining two functions f : A ! B and g : B ! C is by compo- sition. Definition 0.1.4. If f : A ! B and g : B ! C are functions then g◦f : A ! C is defined as the set g ◦ f = f(a; g(f(a))) : a 2 Ag: Exercises (1) Using the notation developed, write the set of vowels V and the set of integers I between −100 and 100.
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