Math 300 - Introduction to Mathematical reasoning Summer 2013 Lecture 1: Sets - June 24 Chapter 1 - Sets and Functions Definition and Examples Sets are a \well-defined” collection of objects. We shall not say what we mean by \well-defined” in this course. At an introductory level, it suffices to think of a Set as a collection of objects. Some examples of sets are - • Collection of students taking Math 300, • Collection of all possible speeds that your car can attain, • Collection of all matrices with real entries (Math 308). Examples We mention some mathematical examples which we shall use throughout the course. Keep these examples in mind. • N - the set of natural numbers. Here, N = f1; 2; 3;:::; g • Z - the set of integers. Here, Z = f;:::; −3; −2; −1; 0; 1; 2; 3;:::; g • Q - the set of rational numbers. The set of rational numbers can described as fractions, where the numerator and denominator of the fraction are both integers and the denominator is not equal to zero. • R - the set of real numbers. • R+ - the set of positive real numbers. The objects of the set are called elements, members or points of the set. Usually a set is denoted by a capital letter. The elements of the set are denoted by lowercase letters. The symbol 2 denotes the phrase \belongs to". Let A be a set and let x be an element of the set A. The statement x 2 A should be read as x belongs to the set A. For example, the statement \1 2 N" should be read as \1 belongs to the set of natural numbers". Recall that π is defined to be the ratio of the circumference of a circle to its diameter. The statement \π 2 R" should be read as \π belongs to the set of real numbers". Here are some more useful examples to keep in mind - • E - the set of even integers. Here, the set E = f;:::; −8; −6; −4; −2; 0; 2; 4; 6; 8;:::; g: • O - the set of odd integers. Here, the set O = f;:::; −9; −7; −5; −3; −1; 1; 3; 5; 7; 9;:::; g: 1-1 1-2 Lecture 1: Sets - June 24 Describing a set We mention various ways to describe a set. We will be slightly informal here. However, we hope that our examples are illustrative and convey our thoughts. Listing the elements One way of describing a set is to list all the elements of a set. We separate the elements of a set using commas and enclose the elements within curly brackets. For example, consider the statement A = f3:5; 6; π; 0g The statement says that there is a set A with 5 elements. The elements of this set A are 3; 5; 6; π and 0. The order in which you specify the elements is NOT important. For example, f0; π; 5; 3; 6g also denotes the same set A. Repeating an element is also NOT important. For example, fπ; π; 3; 3; 3; 0; 5; 6; 6; 6g also denotes the same set A. The symbol \2=" should be read as \does not belong to". In our example, 2 2= A, i.e. the element 2 does not belong to the set A. Listing the elements of a set can be done with infinite sets too. We wrote down the set of natural numbers as follows, N = f1; 2; 3;:::g. Here \:::" should be read as \and so on". Here, it is implicit what the phrase\and so on" should be interpreted as. We have given similar descriptions for the set of integers Z, the set of odd integers O and the set of even integers E. However, there might be some sets where it might not be possible to list down all the elements of the set. We need alternate ways to describe such sets. Think about it - Is there a possible way to list out all the elements of R - the set of real numbers ? The answer is NO. It is not possible to list the elements of the set of real numbers R as we had listed out the elements of the set of natural numbers N. Now, is there possible way to list out all the elements of Q ? This is a slightly subtle question. The answer is YES ! We hope to talk about both these questions more in Chapter 4. Conditional definition of a set An alternate way of describing a set is to specify some conditions, that allows us to pick elements from a larger set whose description is already available to us. We provide some illustrative examples. Throughout, we assume that we have already described the set of integers Z. We make use of the symbol \j" which should be read as \such that" Examples : • B = fn 2 Z j 0 < n < 4g. Intrepret the definition of B as follows \The set B is the set of integers n such that n is greater than 0 and less than 4". One can see that B can be written down as f1; 2; 3g. • The set of even integers, E = fa 2 Z j a is a multiple of 2g. Interpret this definition of E as follows - \The set E is the set of integers n such that n is a multiple of 2". • The set of odd integers O = fa 2 Z j a is not a multiple of 2g. Lecture 1: Sets - June 24 1-3 • The set of rational numbers Q = fx 2 R j x = a=b for some a; b 2 Zg. Constructive definition of a set The other systematic way of describing a set is to give a formula or an algorithm to construct the elements of the set. We provide some illustrative examples. Example : • The set of rational numbers Q = fa=b j a; b 2 Zg. • The set of even integers E = f2 · aj a 2 Zg. • The set of odd integers O = f2 · a + 1 j a 2 Zg Set operations Universe : Let us be slightly informal here. In most of our questions, there will be a special distinguished set called the \universe" hovering the background. The elements of the sets in question will belong to this special set - the universe. Most often, we will denote the \Universe" by U. In most problems, we will not mention what the universe is and the existence of the universe will be quite implicit. Venn Diagrams Read about Venn diagrams in Wikipedia ! Subset Suppose we have two sets A and B. We say that A is a subset of B - we write A ⊂ B - if every element of A is an element of B. For example, we have N ⊂ Z ⊂ Q ⊂ R. A B 1-4 Lecture 1: Sets - June 24 Equality of sets We say that two sets A and B are equal to each other (and we write A = B) if every element of A is an element of B and every element of B is an element of A. Alternatively, two sets A and B are equal to each other if A ⊂ B and B ⊂ A. Example: Consider the two sets A = f2 · n + 1 j n 2 Zg and B = f2 · n − 1 j n 2 Zg We will prove that A = B. Proof: To prove that A = B, we need to show that A ⊂ B and B ⊂ A. 1. First we show that A ⊂ B. Choose an arbitrary element x 2 A. We will try to show that x 2 B. Since x was an arbitrarily chosen element of A, this would imply that every element of A is an element of B and hence that A ⊂ B. Since x 2 A, there is some integer n such that x = 2 · n + 1. x = 2 · n + 1 = 2 · (n + 1 − 1) + 1 = 2 · (n + 1) − 2 + 1 = 2 · (n + 1) − 1 Substitute m = n + 1, we get that x 2 B. Note that the variable n used in the definition of the sets A and B is just a dummy variable. We could have defined B = f2 · m − 1 j m 2 Zg. We have shown that A ⊂ B. 2. Now we have to show that B ⊂ A. Let x 2 B, i.e. we choose an arbitrary element x in B. This means that there is an integer n such that x = 2 · n − 1. We proceed similarly. x = 2 · n − 1 = 2 · (n − 1 + 1) − 1 = 2 · (n − 1) + 2 − 1 = 2 · (n − 1) + 1 Substitute m = n − 1, we get that x = 2 · m + 1. m is also an integer. Hence we can conclude that x 2 A. Since x was an arbitrary element in B, we can conclude that B ⊂ A. Empty set Another distinguished set is the empty set. In some sense, it is the opposite of the set we called the \Universe". The empty set is the unique set with no elements. The empty set is denoted by ;. Lecture 1: Sets - June 24 1-5 Union Suppose we have two sets A and B. Then we define the union of the sets A and B to be the set that consists of those elements that are either in A or in B. We denote the set by A [ B. We provide the following conditional definition A [ B = fx j x 2 A or x 2 Bg: Example E [ O = Z. A non-example- Let A = f1; 2; 3g and B = f1; 2g be two sets. Consider another set C = fA; Bg is another set, whose elements are sets themselves.