Chapter 9 Finite and Infinite Sets 9.1 FiniteSets Preview Activity 1 (Equivalent Sets, Part 1) 1. Let A and B be sets and let f be a function from A to B. .f A B/. Carefully complete each of the following using appropriate quantifiers:W ! (If necessary, review the material in Section 6.3.) (a) The function f is an injection provided that :::: (b) The function f is not an injection provided that :::: (c) The function f is a surjection provided that :::: (d) The function f is not a surjection provided that :::: (e) The function f is a bijection provided that :::: Definition. Let A and B be sets. The set A is equivalent to the set B provided that there exists a bijection from the set A onto the set B. In thiscase, wewrite A B. When A B, we also say that the set A is in one-to-one correspondence with the set B and that the set A has the same cardinality as the set B. Note: When A is not equivalent to B, we write A B. 6 2. For each of the following, use the definition of equivalent sets to determine if the first set is equivalent to the second set. 452 9.1. Finite Sets 453 (a) A 1;2;3 and B a;b;c D f g D f g (b) C 1;2 and B a;b;c D f g D f g (c) X 1;2;3;:::;10 and Y 57;58;59;:::;66 D f g D f g 3. Let DC be the set of all odd natural numbers. Prove that the function f N DC defined by f .x/ 2x 1, for all x N, is a bijection andW hence! that N D . D 2 C 4. Let RC be the set of all positive real numbers. Prove that the function x g R RC defined by g.x/ e , for all x R is a bijection and hence, thatW R! R . D 2 C Preview Activity 2 (Equivalent Sets, Part 2) 1. Review Theorem 6.20 in Section 6.4, Theorem 6.26 in Section 6.5, and Ex- ercise (9) in Section 6.5. 2. Prove each part of the following theorem. Theorem 9.1. Let A, B, and C be sets. (a) For each set A, A A. (b) For all sets A and B, if A B, then B A. (c) For all sets A, B, and C , if A B and B C , then A C . Equivalent Sets In Preview Activity 1, we introduced the concept of equivalent sets. The motiva- tion for this definitionwas to have a formal method for determining whether or not two sets “have the same number of elements.” This idea was described in terms of a one-to-one correspondence (a bijection) from one set onto the other set. Thisidea may seem simple for finite sets, but as we will see, this idea has surprising conse- quences when we deal with infinite sets. (We will soon provide precise definitions for finite and infinite sets.) Technical Note: The three properties we proved in Theorem 9.1 in Preview Ac- tivity 2 are very similar to the concepts of reflexive, symmetric, and transitive re- lations. However, we do not consider equivalence of sets to be an equivalence 454 Chapter 9. Finite and Infinite Sets relation on a set U since an equivalence relation requires an underlying (universal) set U . In this case, our elements would be the sets A, B, and C , and these would then have to subsets of some universal set W (elements of the power set of W ). For equivalence of sets, we are not requiring that the sets A, B, and C be subsets of the same universal set. So we do not use the term relation in regards to the equivalence of sets. However, if A and B are sets and A B, then we often say that A and B are equivalent sets. Progress Check 9.2 (Examples of Equivalent Sets) We will use the definition of equivalent sets from in Preview Activity 1 in all parts of this progress check. It is no longer sufficient to say that two sets are equivalent by simply saying that the two sets have the same number of elements. 1. Let A 1;2;3;:::;99;100 and let B 351;352;353;:::;449;450 . D f g D f g Define f A B by f.x/ x 350, for each x in A. Prove that f is a bijection fromW ! the set A to theD set CB and hence, A B. 2. Let E be the set of all even integers and let D be the set of all odd integers. Prove that E D by proving that F E D, where F .x/ x 1, for W ! D C all x E, is a bijection. 2 3. Let .0; 1/ be the open interval of real numbers between 0 and 1. Similarly, if b R with b>0, let .0;b/ be the open interval of real numbers between 0 2 and b. Prove that the function f .0; 1/ .0;b/ by f .x/ bx, for all x .0; 1/, W ! D 2 is a bijection and hence .0; 1/ .0;b/. In Part (3) of Progress Check 9.2, notice that if b>1, then .0; 1/ is a proper subset of .0;b/ and .0; 1/ .0;b/. Also, in Part (3) of Preview Activity 1, we proved that the set D of all odd natural numbers is equivalent to N, and we know that D is a proper subset of N. These results may seem a bit strange, but they are logical consequences of the definition of equivalent sets. Although we have not defined the terms yet, we will see that one thing that will distinguish an infinite set from a finite set is that an infinite set can be equivalent to one of its proper subsets, whereas a finite set cannot be equivalent to one of its proper subsets. 9.1. Finite Sets 455 Finite Sets In Section 5.1, we defined the cardinality of a finite set A, denoted by card .A/, to be the number of elements in the set A. Now that we know about functions and bijections, we can define this concept more formally and more rigorously. First, for each k N, we define N to be the set of all natural numbers between 1 and k, 2 k inclusive. That is, N 1;2;:::;k : k D f g We will use the concept of equivalent sets introduced in Preview Activity 1 to define a finite set. Definition. Aset A is a finite set provided that A or there exists a natural D ; number k such that A Nk. Asetis an infinite set provided that it is not a finite set. If A Nk, we say that the set A has cardinality k (or cardinal number k), and we write card .A/ k. D In addition,we say that the empty set has cardinality 0 (or cardinal number 0), and we write card . / 0. ; D Notice that by this definition, the empty set is a finite set. In addition, for each k N, the identity function on N is a bijection and hence, by definition, the set 2 k Nk is a finite set with cardinality k. Theorem 9.3. Any set equivalent to a finite nonempty set A is a finite set and has the same cardinality as A. Proof. Suppose that A is a finite nonempty set, B is a set, and A B. Since A is a finite set, there exists a k N such that A N . We also have assumed that 2 k A B and so by part (b) of Theorem 9.1 (in Preview Activity 2), we can conclude thatB A. Since A N , we can use part (c) of Theorem 9.1 to conclude that k B Nk. Thus, B is finite and has the same cardinality as A. It may seem that we have done a lot of work to prove an “obvious” result in Theorem 9.3. The same may be true of the remaining results in this section, which give further results about finite sets. One of the goals is to make sure that the con- cept of cardinality for a finite set corresponds to our intuitive notion of the number of elements in the set. Another important goal is to lay the groundwork for a more rigorous and mathematical treatment of infinite sets than we have encountered be- fore. Along the way, we will see the mathematical distinction between finite and infinite sets. 456 Chapter 9. Finite and Infinite Sets The following two lemmas will be used to prove the theorem that states that every subset of a finite set is finite. Lemma 9.4. If A is a finite set and x A, then A x is a finite set and … [ f g card.A x / card.A/ 1. [ f g D C Proof. Let A be a finite set and assume card.A/ k, where k 0 or k N. Assume x A. D D 2 … If A , then card.A/ 0 and A x x , which is equivalent to N1. Thus, A Dx ; is finite with cardinalityD 1,[ which f g D equals f g card.A/ 1. [ f g C If A , then A N , for some k N. This means that card.A/ k, and ¤ ; k 2 D there exists a bijection f A Nk.