Chapter 3 Set Theory 3.1 What is a Set? A set is a well-defined collection of objects called elements or members of the set. Here, well-defined means accurately and unambiguously stated or de- scribed. Any given object must either be an element of the set, or not be an element of the set. There is no concept of partial membership, and there is no possibility of an being a member more than once. If x is a member of the set S, we write x 2 S, and if x is a not member of the set S, we write x 62 S. Sets are defined in terms of the objects they contain. We say sets A and B are equal, and write A = B if they have exactly the same elements. That is, A = B when x 2 A , x 2 B. The barber paradox gives an example of a set that is not well-defined: There is only one barber in a certain town. He is male. All of the men in the town are clean-shaven. The barber shaves all and only the men in the town who do not shave themselves. Who shaves the barber? Now, if the barber shaves himself, then since the barber only the men who do not shave themselves, he does not shave himself. And if he does not shave himself, then since he shaves all of the men who don't shave themselves, he shaves himself. Hmmm. One explanation for this paradox is that the set, S, of men in the town who are shaved by the barber is not well-defined, as the barber must 1 2 CHAPTER 3. SET THEORY simultaneously be a member of the set, and not be a member of the set. Sets can be described in several ways. One way to describe a set is to write a description of the set in words, as in \the set of all integers that can be written as the sum of two squares". There are three main ways of describing a set using mathematical notation. 1. Explicit listing: list the elements between brackets, as in f2; 3; 5; 7g. The elements of a set that's described by explicit listing are exactly the (different) objects in the list obtained when the outer brackets are erased. For example, the elements of fcar; π; Xg are car; π, and X. The elements of {−1; f3gg are -1 and f3g. (Sets can be members of other sets.) 2. Implicit listing: list enough its elements to establish a pattern and use an ellipsis \:::". Proper use of the ellipsis requires that at least two elements be listed so that the pattern is established. (It could be that more elements must be listed before the pattern is apparent.) For example, f0; 2; 4;:::; 120g is the set of non-negative even integers less than or equal to 120, while f::: − 3; −1; 1; 3;:::g is the set of odd integers. The elements of a set that's described by implicit listing are those that follow the pattern, and respect any limits set. For example, the elements of ff1g; f2g;:::; f6gg are f1g; f2g; f3g; f4g; f5g, and f6g. 3. Set-builder notation: specify the set of the collection of all objects of a particular type that satisfy a given condition. Examples are: the set of all prime numbers less than 10 is fx :(x is prime) ^ (x < 10)g = f2; 3; 5; 7g; and the set of all positive even integers is f2k : k = 1; 2;:::g = f2; 4; 6;:::g. The elements of a set described using set-builder notation are those objects of the given type that make the stated condition true. For example, the elements of the set fa=b : a and b are integers; and a=b = 0:25g are exactly the fractions whose numerical value is 0.25. There are infinitely many of these including 1=4; 3=12 and −5=(−20). 3.2. THE EMPTY SET 3 By the definition of equality of sets, it does not matter how a set is described; what matters is which elements it contains. Any particular object either belongs to the collection or it doesn't. All of f1; 2; 2; 3g; f1; 2; 3; 3g f3; 2; 3; 1g and f1; 2; 3g all describe the same set because they all have the same three elements: 1, 2, and 3. Some sets are well-known, and are denoted by special symbols. • The set of natural numbers is N = f1; 2; 3;:::g. Some people include 0 as an element of this set. It is always wise to check the definition that a particular author is using. • The set of integers Z = f:::; −2; −1; 0; 1; 2;:::g, The use of the symbol Z can be traced back to the German word z¨ahlen. • The set of rational numbers is Q = fa=b : a; b 2 Z; and b 6= 0g. The symbol Q is used because these are quotients of integers. • The set of real numbers, denoted by R, has as elements all numbers that have a decimal expansion. • The set of complex numbers is C = fa + bi : a; b 2 R; and i2 = −1g. 3.2 The Empty Set It is certainly possible for a collection to have nothing in it. A good example would be the collection of years after 1967 in which the Toronto Maple Leafs have won the Stanley Cup. The empty set is the set that has no elements, that is fg. It is commonly denoted by ;. 2 The following sets are all equalp to ;: fx 2 R : x + 1 = 0g, fn 2 Z : n2 − 1 = 7g and fa=b 2 Q : a=b = 2g. The empty set is a perfectly legitimate object, and as such can occur as an element of a set. Notice that ; is different from f;g. The former set has no elements, while the latter set has one element, ;. The set f;; f;g; ff;g; ;gg has three elements: ;; f;g, and ff;g; ;gg. 4 CHAPTER 3. SET THEORY 3.3 Subsets We say that a set A is a subset of a set B if every element of A is an element of B (i.e., x 2 A ) x 2 B). If A is a subset of B we write A ⊆ B, and otherwise we write A 6⊆ B. For example, N ⊆ Z, Z ⊆ Q, and Q ⊆ R. Also, f1; 3; 5g ⊆ f1; 3; 5g, and f2; 4g 6⊆ f4; 5; 6g. Notice that every set is a subset of itself (why?), that is X ⊆ X for every set X. A more subtle point is that ; is a subset of every set A. According to the definition, this is the same as the logical implication x 2 ; ) x 2 A which, in turn, is the same as the implication (x 2 ;) ! (x 2 A) being a tautology. The implication has only the truth value \true" because its hypothesis, x 2 ;, is false for any x. A different way to say it is that every element in the collection of members of the empty set { there aren't any { is a member of A. How many subsets does fa; bg have? Let's count the options. Any par- ticular subset either contains a or it does not. In both situations, there are two further options: the subset either contains b or it does not. Thus there are four possibilities fa; bg; fag; fbg; fg. The above reasoning can be extended to show that a set with n elements has exactly 2n subsets. Sometimes confusion arises in making the distinction between 2 and ⊆. The first one makes the assertion that a particular object belongs to a set; the second one says that every element of one set belongs to another set. In the following we show that the subset relation is transitive, that is, if A is a subset of B, and B is a subset of C, then A is a subset of C. (There is a more general meaning for the word \transitive". It will arise later in the course.) Before beginning the proof, it is useful to identify the statement to be proved, and the hypotheses that can be used in the argument. The statement to be proved is \A is a subset of C". That is, it needs to be argued that every element of A is an element of C. Equivalently, it needs to be argued that an arbitrary element of A is an element of C. The hypotheses that can be used in the argument are: \A is a subset of B", and \B is a subset of C". Constructing the proof involves using these to help argue that 3.4. PROPER SUBSETS 5 an arbitrary element of A must be an element of C. Proposition 3.3.1 Let A; B and C be sets. If A ⊆ B and B ⊆ C, then A ⊆ C. Proof. Take any x 2 A. Since A ⊆ B, the element x 2 B. Since B ⊆ C, the element x 2 C. Therefore, if x 2 A then x 2 C. That is, A ⊆ C. Recall that if p and q are statements, then the logical equivalence p , q is the same as the two logical implications p ) q and q ) p. The logical equivalence is proved once the two logical implications are proved. Proposition 3.3.2 Let A and B be sets. Then A = B if and only if A ⊆ B and B ⊆ A.