Lecture (1) Set Theory PDF

CHAPTER 1 Set Theory 1.1 INTRODUCTION The concept of a set appears in all mathematics. This chapter introduces the notation and terminology of set theory which is basic and used throughout the text. The chapter closes with the formal deﬁnition of mathematical induction, with examples. 1.2 SETS AND ELEMENTS, SUBSETS A set may be viewed as any well-deﬁned collection of objects, called the elements or members of the set. One usually uses capital letters, A, B, X, Y,... , to denote sets, and lowercase letters, a, b, x, y,..., to denote elements of sets. Synonyms for “set” are “class,” “collection,” and “family.” Membership in a set is denoted as follows: a ∈ S denotes that a belongs to a set S a, b ∈ S denotes that a and b belong to a set S Here ∈ is the symbol meaning “is an element of.” We use ∈ to mean “is not an element of.” Specifying Sets There are essentially two ways to specify a particular set. One way, if possible, is to list its members separated by commas and contained in braces { }. A second way is to state those properties which characterized the elements in the set. Examples illustrating these two ways are: A = {1, 3, 5, 7, 9} and B = {x | x is an even integer, x > 0} That is, A consists of the numbers 1, 3, 5, 7, 9. The second set, which reads: B is the set of x such that x is an even integer and x is greater than 0, denotes the set B whose elements are the positive integers. Note that a letter, usually x, is used to denote a typical member of the set; and the vertical line | is read as “such that” and the comma as “and.” EXAMPLE 1.1 (a) The set A above can also be written as A = {x | x is an odd positive integer, x < 10}. (b) We cannot list all the elements of the above set B although frequently we specify the set by B = {2, 4, 6,...} where we assume that everyone knows what we mean. Observe that 8 ∈ B, but 3 ∈ / B. 1 Copyright © 2007, 1997, 1976 by The McGraw-Hill Companies, Inc. Click here for terms of use. 2 SET THEORY [CHAP. 1 (c) Let E = {x | x 2 − 3x + 2 = 0}, F = {2, 1} and G = {1, 2, 2, 1}. Then E = F = G. We emphasize that a set does not depend on the way in which its elements are displayed. A set remains the same if its elements are repeated or rearranged. Even if we can list the elements of a set, it may not be practical to do so. That is, we describe a set by listing its elements only if the set contains a few elements; otherwise we describe a set by the property which characterizes its elements. Subsets Suppose every element in a set A is also an element of a set B, that is, suppose a ∈ A implies a ∈ B. Then A is called a subset of B. We also say that A is contained in B or that B contains A. This relationship is written A⊆B or B⊇A Two sets are equal if they both have the same elements or, equivalently, if each is contained in the other. That is: A = B if and only if A ⊆ B and B ⊆ A If A is not a subset of B, that is, if at least one element of A does not belong to B, we write A ⊆ B. EXAMPLE 1.2 Consider the sets: A = {1, 3, 4, 7, 8, 9}, B = {1, 2, 3, 4, 5}, C = {1, 3}. Then C ⊆ A and C ⊆ B since 1 and 3, the elements of C, are also members of A and B. But B ⊆ A since some of the elements of B, e.g., 2 and 5, do not belong to A. Similarly, A ⊆ B. Property 1: It is common practice in mathematics to put a vertical line “|” or slanted line “/” through a symbol to indicate the opposite or negative meaning of a symbol. Property 2: The statement A ⊆ B does not exclude the possibility that A = B. In fact, for every set A we have A ⊆ A since, trivially, every element in A belongs to A. However, if A ⊆ B and A = B, then we say A is a proper subset of B (sometimes written A ⊂ B). Property 3: Suppose every element of a set A belongs to a set B and every element of B belongs to a set C. Then clearly every element of A also belongs to C. In other words, if A ⊆ B and B ⊆ C, then A ⊆ C. The above remarks yield the following theorem. Theorem 1.1: Let A, B, C be any sets. Then: (i) A ⊆ A (ii) If A ⊆ B and B ⊆ A, then A = B (iii) If A ⊆ B and B ⊆ C, then A ⊆ C Special symbols Some sets will occur very often in the text, and so we use special symbols for them. Some such symbols are: N = the set of natural numbers or positive integers: 1, 2, 3,... Z = the set of all integers:... , −2, −1, 0, 1, 2,... Q = the set of rational numbers R = the set of real numbers C = the set of complex numbers Observe that N ⊆ Z ⊆ Q ⊆ R ⊆ C. CHAP. 1] SET THEORY 3 Universal Set, Empty Set All sets under investigation in any application of set theory are assumed to belong to some ﬁxed large set called the universal set which we denote by U unless otherwise stated or implied. Given a universal set U and a property P, there may not be any elements of U which have property P. For example, the following set has no elements: S = {x | x is a positive integer, x 2 = 3} Such a set with no elements is called the empty set or null set and is denoted by ∅ There is only one empty set. That is, if S and T are both empty, then S = T , since they have exactly the same elements, namely, none. The empty set ∅ is also regarded as a subset of every other set. Thus we have the following simple result which we state formally. Theorem 1.2: For any set A, we have ∅ ⊆ A ⊆ U. Disjoint Sets Two sets A and B are said to be disjoint if they have no elements in common. For example, suppose A = {1, 2}, B = {4, 5, 6}, and C = {5, 6, 7, 8} Then A and B are disjoint, and A and C are disjoint. But B and C are not disjoint since B and C have elements in common, e.g., 5 and 6. We note that if A and B are disjoint, then neither is a subset of the other (unless one is the empty set). 1.3 VENN DIAGRAMS A Venn diagram is a pictorial representation of sets in which sets are represented by enclosed areas in the plane. The universal set U is represented by the interior of a rectangle, and the other sets are represented by disks lying within the rectangle. If A ⊆ B, then the disk representing A will be entirely within the disk representing B as in Fig. 1-1(a). If A and B are disjoint, then the disk representing A will be separated from the disk representing B as in Fig. 1-1(b). Fig. 1-1 4 SET THEORY [CHAP. 1 However, if A and B are two arbitrary sets, it is possible that some objects are in A but not in B, some are in B but not in A, some are in both A and B, and some are in neither A nor B; hence in general we represent A and B as in Fig. 1-1(c). Arguments and Venn Diagrams Many verbal statements are essentially statements about sets and can therefore be described by Venn diagrams. Hence Venn diagrams can sometimes be used to determine whether or not an argument is valid. EXAMPLE 1.3 Show that the following argument (adapted from a book on logic by Lewis Carroll, the author of Alice in Wonderland) is valid: S1 : All my tin objects are saucepans. S2 : I ﬁnd all your presents very useful. S3 : None of my saucepans is of the slightest use. S : Your presents to me are not made of tin. The statements S1 , S2 , and S3 above the horizontal line denote the assumptions, and the statement S below the line denotes the conclusion. The argument is valid if the conclusion S follows logically from the assumptions S1 , S2 , and S3. By S1 the tin objects are contained in the set of saucepans, and by S3 the set of saucepans and the set of useful things are disjoint. Furthermore, by S2 the set of “your presents” is a subset of the set of useful things. Accordingly, we can draw the Venn diagram in Fig. 1-2. The conclusion is clearly valid by the Venn diagram because the set of “your presents” is disjoint from the set of tin objects. Fig. 1-2 1.4 SET OPERATIONS This section introduces a number of set operations, including the basic operations of union, intersection, and complement. Union and Intersection The union of two sets A and B, denoted by A ∪ B, is the set of all elements which belong to A or to B; that is, A ∪ B = {x | x ∈ A or x ∈ B} Here “or” is used in the sense of and/or. Figure 1-3(a) is a Venn diagram in which A ∪ B is shaded. The intersection of two sets A and B, denoted by A ∩ B, is the set of elements which belong to both A and B; that is, A ∩ B = {x | x ∈ A and x ∈ B} Figure 1-3(b) is a Venn diagram in which A ∩ B is shaded. CHAP. 1] SET THEORY 5 Fig. 1-3 Recall that sets A and B are said to be disjoint or nonintersecting if they have no elements in common or, using the deﬁnition of intersection, if A ∩ B = ∅, the empty set. Suppose S =A∪B and A∩B =∅ Then S is called the disjoint union of A and B. EXAMPLE 1.4 (a) Let A = {1, 2, 3, 4}, B = {3, 4, 5, 6, 7}, C = {2, 3, 8, 9}. Then A ∪ B = {1, 2, 3, 4, 5, 6, 7}, A ∪ C = {1, 2, 3, 4, 8, 9}, B ∪ C = {2, 3, 4, 5, 6, 7, 8, 9}, A ∩ B = {3, 4}, A ∩ C = {2, 3}, B ∩ C = {3}. (b) Let U be the set of students at a university, and let M denote the set of male students and let F denote the set of female students. The U is the disjoint union of M of F ; that is, U=M ∪F and M ∩F =∅ This comes from the fact that every student in U is either in M or in F , and clearly no student belongs to both M and F , that is, M and F are disjoint. The following properties of union and intersection should be noted. Property 1: Every element x in A ∩ B belongs to both A and B; hence x belongs to A and x belongs to B. Thus A ∩ B is a subset of A and of B; namely A∩B ⊆A and A∩B ⊆B Property 2: An element x belongs to the union A ∪ B if x belongs to A or x belongs to B; hence every element in A belongs to A ∪ B, and every element in B belongs to A ∪ B. That is, A⊆A∪B and B ⊆A∪B We state the above results formally: Theorem 1.3: For any sets A and B, we have: (i) A ∩ B ⊆ A ⊆ A ∪ B and (ii) A ∩ B ⊆ B ⊆ A ∪ B. The operation of set inclusion is closely related to the operations of union and intersection, as shown by the following theorem. Theorem 1.4: The following are equivalent: A ⊆ B, A ∩ B = A, A ∪ B = B. This theorem is proved in Problem 1.8. Other equivalent conditions to are given in Problem 1.31. 6 SET THEORY [CHAP. 1 Fig. 1-4 Complements, Differences, Symmetric Differences Recall that all sets under consideration at a particular time are subsets of a ﬁxed universal set U. The absolute complement or, simply, complement of a set A, denoted by AC , is the set of elements which belong to U but which do not belong to A. That is, AC = {x | x ∈ U, x ∈ / A} Some texts denote the complement of A by A or Ā. Fig. 1-4(a) is a Venn diagram in which AC is shaded. The relative complement of a set B with respect to a set A or, simply, the difference of A and B, denoted by A\B, is the set of elements which belong to A but which do not belong to B; that is A\B = {x | x ∈ A, x ∈ / B} The set A\B is read “A minus B.” Many texts denote A\B by A − B or A ∼ B. Fig. 1-4(b) is a Venn diagram in which A\B is shaded. The symmetric difference of sets A and B, denoted by A ⊕ B, consists of those elements which belong to A or B but not to both. That is, A ⊕ B = (A ∪ B)\(A ∩ B) or A ⊕ B = (A\B) ∪ (B\A) Figure 1-4(c) is a Venn diagram in which A ⊕ B is shaded. EXAMPLE 1.5 Suppose U = N = {1, 2, 3,...} is the universal set. Let A = {1, 2, 3, 4}, B = {3, 4, 5, 6, 7}, C = {2, 3, 8, 9}, E = {2, 4, 6,...} (Here E is the set of even integers.) Then: AC = {5, 6, 7,...}, B C = {1, 2, 8, 9, 10,...}, E C = {1, 3, 5, 7,...} That is, E C is the set of odd positive integers. Also: A\B = {1, 2}, A\C = {1, 4}, B\C = {4, 5, 6, 7}, A\E = {1, 3}, B\A = {5, 6, 7}, C\A = {8, 9}, C\B = {2, 8, 9}, E\A = {6, 8, 10, 12,...}. Furthermore: A ⊕ B = (A\B) ∪ (B\A) = {1, 2, 5, 6, 7}, B ⊕ C = {2, 4, 5, 6, 7, 8, 9}, A ⊕ C = (A\C) ∪ (B\C) = {1, 4, 8, 9}, A ⊕ E = {1, 3, 6, 8, 10,...}. Fundamental Products Consider n distinct sets A1 , A2 , …, An. A fundamental product of the sets is a set of the form A∗1 ∩ A∗2 ∩... ∩ A∗n where A∗i = A or A∗i = AC CHAP. 1] SET THEORY 7 We note that: (i) There are m = 2n such fundamental products. (ii) Any two such fundamental products are disjoint. (iii) The universal set U is the union of all fundamental products. Thus U is the disjoint union of the fundamental products (Problem 1.60). There is a geometrical description of these sets which is illustrated below. EXAMPLE 1.6 Figure 1-5(a) is the Venn diagram of three sets A, B, C. The following lists the m = 23 = 8 fundamental products of the sets A, B, C: P1 = A ∩ B ∩ C, P3 = A ∩ B C ∩ C, P5 = AC ∩ B ∩ C, P7 = AC ∩ B C ∩ C, P2 = A ∩ B ∩ C , P4 = A ∩ B ∩ C , P6 = AC ∩ B ∩ C C, C C C P8 = AC ∩ B C ∩ C C. The eight products correspond precisely to the eight disjoint regions in the Venn diagram of sets A, B, C as indicated by the labeling of the regions in Fig. 1-5(b). Fig. 1-5 1.5 ALGEBRA OF SETS, DUALITY Sets under the operations of union, intersection, and complement satisfy various laws (identities) which are listed in Table 1-1. In fact, we formally state this as: Theorem 1.5: Sets satisfy the laws in Table 1-1. Table 1-1 Laws of the algebra of sets Idempotent laws: (1a) A ∪ A = A (1b) A ∩ A = A Associative laws: (2a) (A ∪ B) ∪ C = A ∪ (B ∪ C) (2b) (A ∩ B) ∩ C = A ∩ (B ∩ C) Commutative laws: (3a) A ∪ B = B ∪ A (3b) A ∩ B = B ∩ A Distributive laws: (4a) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) (4b) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) (5a) A ∪ ∅ = A (5b) A ∩ U = A Identity laws: (6a) A ∪ U = U (6b) A ∩ ∅ = ∅ Involution laws: (7) (AC )C = A (8a) A ∪ AC = U (8b) A ∩ AC = ∅ Complement laws: (9a) UC = ∅ (9b) ∅C = U DeMorgan’s laws: (10a) (A ∪ B)C = AC ∩ BC (10b) (A ∩ B)C = AC ∪ B C 8 SET THEORY [CHAP. 1 Remark: Each law in Table 1-1 follows from an equivalent logical law. Consider, for example, the proof of DeMorgan’s Law 10(a): (A ∪ B)C = {x | x ∈ / (A or B)} = {x | x ∈ / A and x ∈ / B} = AC ∩ BC Here we use the equivalent (DeMorgan’s) logical law: ¬(p ∨ q) = ¬p ∧ ¬q where ¬ means “not,” ∨ means “or,” and ∧ means “and.” (Sometimes Venn diagrams are used to illustrate the laws in Table 1-1 as in Problem 1.17.) Duality The identities in Table 1-1 are arranged in pairs, as, for example, (2a) and (2b). We now consider the principle behind this arrangement. Suppose E is an equation of set algebra. The dual E ∗ of E is the equation obtained by replacing each occurrence of ∪, ∩, U and ∅ in E by ∩, ∪, ∅, and U, respectively. For example, the dual of (U ∩ A) ∪ (B ∩ A) = A is (∅ ∪ A) ∩ (B ∪ A) = A Observe that the pairs of laws in Table 1-1 are duals of each other. It is a fact of set algebra, called the principle of duality, that if any equation E is an identity then its dual E ∗ is also an identity. 1.6 FINITE SETS, COUNTING PRINCIPLE Sets can be ﬁnite or inﬁnite. A set S is said to be ﬁnite if S is empty or if S contains exactly m elements where m is a positive integer; otherwise S is inﬁnite. EXAMPLE 1.7 (a) The set A of the letters of the English alphabet and the set D of the days of the week are ﬁnite sets. Speciﬁcally, A has 26 elements and D has 7 elements. (b) Let E be the set of even positive integers, and let I be the unit interval, that is, E = {2, 4, 6,...} and I = [0, 1] = {x | 0 ≤ x ≤ 1} Then both E and I are inﬁnite. A set S is countable if S is ﬁnite or if the elements of S can be arranged as a sequence, in which case S is said to be countably inﬁnite; otherwise S is said to be uncountable. The above set E of even integers is countably inﬁnite, whereas one can prove that the unit interval I = [0, 1] is uncountable. Counting Elements in Finite Sets The notation n(S) or |S| will denote the number of elements in a set S. (Some texts use #(S) or card(S) instead of n(S).) Thus n(A) = 26, where A is the letters in the English alphabet, and n(D) = 7, where D is the days of the week. Also n(∅) = 0 since the empty set has no elements. The following lemma applies. Lemma 1.6: Suppose A and B are ﬁnite disjoint sets. Then A ∪ B is ﬁnite and n(A ∪ B) = n(A) + n(B) This lemma may be restated as follows: Lemma 1.6: Suppose S is the disjoint union of ﬁnite sets A and B. Then S is ﬁnite and n(S) = n(A) + n(B) CHAP. 1] SET THEORY 9 Proof. In counting the elements of A ∪ B, ﬁrst count those that are in A. There are n(A) of these. The only other elements of A ∪ B are those that are in B but not in A. But since A and B are disjoint, no element of B is in A, so there are n(B) elements that are in B but not in A. Therefore, n(A ∪ B) = n(A) + n(B). For any sets A and B, the set A is the disjoint union of A\B and A ∩ B. Thus Lemma 1.6 gives us the following useful result. Corollary 1.7: Let A and B be ﬁnite sets. Then n(A\B) = n(A) − n(A ∩ B) For example, suppose an art class A has 25 students and 10 of them are taking a biology class B. Then the number of students in class A which are not in class B is: n(A\B) = n(A) − n(A ∩ B) = 25 − 10 = 15 Given any set A, recall that the universal set U is the disjoint union of A and AC. Accordingly, Lemma 1.6 also gives the following result. Corollary 1.8: Let A be a subset of a ﬁnite universal set U. Then n(AC ) = n(U) − n(A) For example, suppose a class U with 30 students has 18 full-time students. Then there are 30 − 18 = 12 part-time students in the class U. Inclusion–Exclusion Principle There is a formula for n(A ∪ B) even when they are not disjoint, called the Inclusion–Exclusion Principle. Namely: Theorem (Inclusion–Exclusion Principle) 1.9: Suppose A and B are ﬁnite sets. Then A ∪ B and A ∩ B are ﬁnite and n(A ∪ B) = n(A) + n(B) − n(A ∩ B) That is, we ﬁnd the number of elements in A or B (or both) by ﬁrst adding n(A) and n(B) (inclusion) and then subtracting n(A ∩ B) (exclusion) since its elements were counted twice. We can apply this result to obtain a similar formula for three sets: Corollary 1.10: Suppose A, B, C are ﬁnite sets. Then A ∪ B ∪ C is ﬁnite and n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(A ∩ C) − n(B ∩ C) + n(A ∩ B ∩ C) Mathematical induction (Section 1.8) may be used to further generalize this result to any number of ﬁnite sets. EXAMPLE 1.8 Suppose a list A contains the 30 students in a mathematics class, and a list B contains the 35 students in an English class, and suppose there are 20 names on both lists. Find the number of students: (a) only on list A, (b) only on list B, (c) on list A or B (or both), (d) on exactly one list. (a) List A has 30 names and 20 are on list B; hence 30 − 20 = 10 names are only on list A. (b) Similarly, 35 − 20 = 15 are only on list B. (c) We seek n(A ∪ B). By inclusion–exclusion, n(A ∪ B) = n(A) + n(B) − n(A ∩ B) = 30 + 35 − 20 = 45. In other words, we combine the two lists and then cross out the 20 names which appear twice. (d) By (a) and (b), 10 + 15 = 25 names are only on one list; that is, n(A ⊕ B) = 25. 10 SET THEORY [CHAP. 1 1.7 CLASSES OF SETS, POWER SETS, PARTITIONS Given a set S, we might wish to talk about some of its subsets. Thus we would be considering a set of sets. Whenever such a situation occurs, to avoid confusion, we will speak of a class of sets or collection of sets rather than a set of sets. If we wish to consider some of the sets in a given class of sets, then we speak of subclass or subcollection. EXAMPLE 1.9 Suppose S = {1, 2, 3, 4}. (a) Let A be the class of subsets of S which contain exactly three elements of S. Then A = [{1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4}] That is, the elements of A are the sets {1, 2, 3}, {1, 2, 4}, {1, 3, 4}, and {2, 3, 4}. (b) Let B be the class of subsets of S, each which contains 2 and two other elements of S. Then B = [{1, 2, 3}, {1, 2, 4}, {2, 3, 4}] The elements of B are the sets {1, 2, 3}, {1, 2, 4}, and {2, 3, 4}. Thus B is a subclass of A, since every element of B is also an element of A. (To avoid confusion, we will sometimes enclose the sets of a class in brackets instead of braces.) Power Sets For a given set S , we may speak of the class of all subsets of S. This class is called the power set of S , and will be denoted by P (S). If S is ﬁnite, then so is P (S). In fact, the number of elements in P (S) is 2 raised to the power n(S). That is, n(P (S)) = 2n(S) (For this reason, the power set of S is sometimes denoted by 2S.) EXAMPLE 1.10 Suppose S = {1, 2, 3}. Then P (S) = [∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, S] Note that the empty set ∅ belongs to P (S) since ∅ is a subset of S. Similarly, S belongs to P (S). As expected from the above remark, P (S) has 23 = 8 elements. Partitions Let S be a nonempty set. A partition of S is a subdivision of S into nonoverlapping, nonempty subsets. Precisely, a partition of S is a collection {Ai } of nonempty subsets of S such that: (i) Each a in S belongs to one of the Ai. (ii) The sets of {Ai } are mutually disjoint; that is, if Aj = Ak then Aj ∩ A k = ∅ The subsets in a partition are called cells. Figure 1-6 is a Venn diagram of a partition of the rectangular set S of points into ﬁve cells, A1 , A2 , A3 , A4 , A5.

Lecture (1) Set Theory PDF

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