I might get tired of writing out all of the set brackets, { }, and just write:


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1 About the course: The language and symbols of math. Math is an exact science in the sense that it is not subjective, but objective. A statement is either true or false. Solutions and proofs are either correct or incorrect. There is room for variation in the amount of detail that is given in the proof. We must communicate mathematics in verbal and written form both formally and informally. This is not easy to learn and there are many aspects to that process, which can be confusing. If you see the symbol R in math context, you are to understand it to mean the real numbers. There are conventions, like using the letter x to indicate a real number. However, it is never assumed that x is a real number. To indicate that, you would say, Let x be a real number, or Let x R. Another convention is to use the letter n to indicate an integer. It would not be wrong to say, Let n R, but it may make it a little harder on the reader who is used to having n represent integers. We use the symbol to indicate a sum, which can be indexed in many different ways. Examples include: n, i=1, j I. i k You may use the same symbol to mean something else, but in that case, the meaning must be clearly stated. One might say, Let be the set of all trigonometric functions. We would say that the default meaning of is that of a sum. We have words in math that mean different things in different contexts. The word tangent can be used to mean a trigonometric function, or a type of line. We have symbols that mean different things in different contexts. Using 1 as a superscript could mean the power of negative 1 or the inverse. We have learned about addition from our early years as a child. After examination, we realize that addition of real numbers has properties of closure, associativity, commutativity, an identity, and each number has a negative. Now we may use the symbol + to indicate an operation that has these same properties, but may be different from our usual understanding of the operation of addition that we learned in elementary school. To make matters worse, we love to use shorthand notation. If I want to indicate several sets, {1, 2, 3}, {1, 2}, {1, 3}, {2, 3}, {1}, {3}, {2},, 1
2 2 I might get tired of writing out all of the set brackets, { }, and just write: 123, 12, 13, 23, 1, 3, 2,. If I verbally explain this to the observer, the observer should know what I am talking about, but when one just looks at the second line, it could be shorthand for any of several different things. Some sentences have commonly used shorthand. For instance, We will show the implication is often shortened to ( ) and The following are equivalent to Tfae. These types of abbreviations are not found in many formal writings and should be avoided in your final portfolio. This course will get you started on the right path and sets the stage with the basics. I expect you to learn and practice the formal style of writing and verbal communication, as you would expect to find in textbooks and research articles. Note about typesetting. The notation for everything you hand in needs to be proper. For instance, when entering a formula on a graphing calculator, you might write: (2 p 2)\2 p. But this is a translation of the correct notation, which is: 2 p 2 2 p Subscripts need to be subscripted. For instance, if you mean A 1, A 2,..., A n, do not write A1, A2,..., An. Our goal is to learn proper symbolism and practice using it. So that when you see a symbol or term used in context, you learn to immediately recognize and understand it. You must first know what you are abbreviating before the abbreviations can make any sense. Proving theorems. A mathematical statement is either true or false. In this course, you will learn many standard proof techniques. Logic is the foundation of all of these techniques. Through precise definitions of words like and, or, implies, if, not, every, exists, and equivalent we will develop valid methods of proofs. Then we will practice all of these proof techniques to prove many theorems. In doing so, we will cover much of the foundational material that is used in every subject in mathematics. When you complete this course, you will know proper definitions of terms that you have used before, like f unction and limit and definitions of common terms that might be new to you like permutations and contrapositive.
3 Topic: Theorems in mathematics are based on facts, axioms, and definitions. We will start with something familiar. The Pthagorean Theorem. Theorem 1. Let T be a right triangle with hypotenuse of length c and two other sides of lengths a and b. Then c 2 = a 2 + b 2. Proof. Consider a square of side length a + b. On each side, mark a point between a segment of length a and one of length b in such a way that connecting the point on one side to the point on the next adjacent side gives a right triangle with legs of length a and b and hypotenuse c. After joining the 4 such pairs of points we see that the square has a square of side length c inscribed in it. (How do we know this is a square?) We see that the area of the square can be computed as either (a + b) 2 or c 2 + 2ab. Thus, and so Name some integer solutions. Fermat s Last Theorem. (a + b) 2 = c 2 + 2ab a 2 + b 2 = c 2. Theorem 2. Let n 3 be a natural number. There exist no triples of integers, a, b, and c such that a n + b n = c n. Conjecture. A conjecture is a mathematical statement, the truth value of which has not yet been determined. It is thought to be true by the person who first published it. 3
4 4 Sections 1.1 and 1.2. A set is a collection of objects called elements. We usually use capital letters to denote sets. x A  x belongs to A x A  x is not an element of A A = B  every element of A is in B and every element of B is in A. A B  every element of A is in B, A is a subset of B, B contains A. So A = B if and only if A B and B A. A B  A B, but A B. B is a proper subset of A. A B  A not contained in B. There is some element of B that is not in A.  The empty set. The set that has no elements. It is a subset of every set, even of itself. The power set of a set A is denoted, P(A), and is the set which contains all subsets of A. Naturals, Wholes, Integers, Cardinality of a finite set, collection of sets = family of sets. Indexed family of sets. Roster method of indicating a set. Set builder notation way of indicating a set. * S = {x R : x > 7} * T = {x R : a x b} Interval notation, for a, b R, a < b: (a, b) [a, b] [a, b) (a, b] (a, + ) (, b) [a, + ) (, b] [n], n N Evens Odds Welldefined Venndiagram There are many correct ways to denote a given set The set of odds equals {2k + 1 : k Z} and it also equals {2k 1 : k Z}. We give a proof. Proof. Let A be the set of integers expressible as 2k 1 for some k Z. So A = {2k 1 : k Z}. Let B be the set of integers expressible as 2s + 1 for some s Z. So B = {2s + 1 : s Z}. Let x A. Then we know that x = 2k 1 for some k Z. Then x = 2k 2+1 = 2(k 1)+1. Since s = k 1 Z, we have that x B. Therefore, A B.
5 Let x B. Then we know that x = 2s + 1 for some s Z. Then x = 2s = 2(s + 1) 1. Since k = s + 1 Z, we have that x A. Thus, B A. Therefore, A = B. Unions, intersections, and compliments. the union of A and B, A B the intersection of A and B, A B the complement of A, denoted A c a universal set the cartesian product of A and B, denoted A B. Also called the cross product. S 1 S 2 S n. 2 sets are called disjoint if their intersection is empty. A family of sets is called mutually disjoint or pairwise disjoint if every pair of sets in the family is disjoint. Two sets are called nested if one is contained in the other. A family of sets is called nested if given any two, one is contained in the other. set difference, denoted A\B, or A B, Distributive Laws: (A B) C = (A C) (B C) and (A B) C = (A C) (B C). Proof: 5 Theorem 3. (1.2.1:) S, T finite then a) S T = S + T S T, b) if S and T are disjoint then S T = S + T, c) if S T, then S T. Proof: Later.
6 6 Example 1. Let U = {0, 1,..., 10}, A = {0, 1, 2, 9}, and B = {2, 3, 4, 5, 6}. Describe in roster notation A B, A B, A c, A B, A\B, B c, A c B c, (A B) c Example 2. Use interval notation to describe: [ 3, 7] (5, 9], [ 3, 7] (5, 9], [ 3, 7] (5, 9] {x R : x.01}, {x R x > 2} De Morgan s Laws. The complement of union is the intersection of individual complements and The complement of intersection is the union of individual complements. Theorem 4. (De Morgan s Laws:) A, B sets a) (A B) c = A c B c, b) (A B) c = A c B c. Proof: The general version of De Morgan s Laws will be given later. Theorem 5. (1.2.6:)If A and B are nonempty finite then A B = A B. The general version of Theorem will be given later. (Theorem 1.2.9) Tree diagrams
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