p. 1 Math 490 Notes 3 Continuing our overview of set theory, there are set operations which parallel all of the usual operations in arithmetic. We previously mentioned the formation of a quotient set A/S = {S(x) x ∈ A} relative to an equivalence relation S on A. If A has n elements and each equivalence class (under S) has k elements, then A/S has n/k elements. Unions are es- sentially set “addition”; when A and B are disjoint sets with n and k elements, respectively, then A ∪ B has n + k elements. Set differences corresponds to subtraction. To form set exponents, it is useful to first extend the definition of set products. Let {Xi i ∈ I} be an arbitrary collection of sets, and let X = {f : I → X f(i) ∈ X for all i ∈ I}. i i ∈ i I X is called the Cartesian product of the X , denoted by X = X . Note that each func- i i i∈I tion f ∈ X is a function which chooses a representative element, f(i), from each set Xi. Such a function is often referred to as a “choice function”, and its existence is a consequence of the Axiom of Choice. Also note that X = φ iff Xi = φ for some i ∈ I. In dealing with set prod- ucts, it is common to write f ∈ ∈ Xi as f =(xi)i∈I , where f(i)= xi. If I = {1, 2,...,n}, i I then f would be the n-tuple (x1, x2,...,xn). If I = N, then f would be the sequence (xn)n∈N. Set exponentiation is simply a special case of set product. If A and I are sets, and Xi = A for all i, then ∈ Xi can be thought of as the product of A with itself ”I times”, and we write i I I I ∈ Xi = ∈ A = A . By the earlier definition of ∈ Xi, it follows that A is simply i I i I i I the set of all functions from I into A. If A and I are finite sets with n and k elements, respectively, then AI contains nk elements. Note that Rn = RI , where I = {1, 2,...,n}, and RN is the set of all (infinite) sequences of real numbers. p. 2 The power set P(X) of all subsets of X is often written via exponential notation. Let ”2” denote the set {0, 1}, and for any subset A of X, let χA denote the characteristic function of A: 1, x ∈ A χ A = . 0, x ∈ X − A Then every f ∈ 2X is the characteristic function of some subset of X, and each subset of X is uniquely defined by its characteristic function. Thus P(X) can be identified with 2X . Cardinal Numbers Cardinal and ordinal numbers are not considered in Munkres, but both are frequently used in topology, so we’ll discuss them briefly here, beginning with cardinal numbers. Let A and B be sets. Define A . B to mean that there exists a bijection between A and a subset of B. Also define A ∼ B to mean that there is a bijection between A and B. Then ∼ is an equivalence relation, and A ∼ B =⇒ A . B and B . A. Schr¨oder-Bernstein Theorem If A and B are sets, then A ∼ B iff A . B and B . A. It is natural to ask, ”On what set are the relations . and ∼ defined?”. The answer would seem to be the ”set of all sets”, but that answer is not consistent. If there were a set S of all sets, consider the subset A of S defined by A = {B ∈ S B ∈ B}. Then by definition, if A is an element of itself, then A ∈ A, and if A is not an element of itself, then A ∈ A. This contradiction is known as Russel’s Paradox. We can avoid this paradox by saying that . and ∼ are relations on the class of all sets, denoted here by S. p. 3 The equivalence classes into which S is partitioned by ∼ are called cardinal numbers. Like S itself, the cardinal numbers are a class which we’ll call C. The equivalence class consisting only of φ is called ”0”; the equivalence class consisting of all singleton sets is called ”1”, etc. The equivalence class containing N is called ℵ0, and the equivalence class containing R we’ll denote by 2ℵ0 . If α and β are cardinal numbers, we define α ≤ β to mean ∃A ∈ α and ∃B ∈ β such that A . B. One can check that ≤ is a well-defined partial order on C. Note that the anti-symmetry of ≤ follows from the Schroder-Bernstein Theorem. Indeed, it can be shown that ≤ is a simple order using another equivalent version of the Axiom of Choice called the Well Ordering Theorem; more to come about that. The smallest infinite cardinal is ℵ0. A set is countable iff its cardinal number is ≤ ℵ0, and is otherwise uncountable. The distinction between countable and uncountable sets is important in analysis, so you may have seen various proofs of the facts that sets like Z and Q are countable, whereas R, R − Q, and the Cantor Set are uncountable. It is also well-known that countable unions and finite products of countable sets are countable. We’ll conclude this discussion of cardinal numbers by briefly considering cardinal arithmetic. For any set A, let |A| denote the cardinal number of A. Given two cardinal numbers α and β, choose A ∈ α and B ∈ β such that A ∩ B = φ. It is not customary to define cardinal subtraction or division, but the other operations are defined as follows: α + β = |A ∪ B|; αβ = |A × B|; αβ = |AB|. For finite cardinals, cardinal arithmetic reduces to ordinary arithmetic. p. 4 Propostion N3.1 If α, β, γ are arbitrary cardinal numbers, then: (a) Cardinal addition and multiplication are both commutative and associative; (b) α(β + γ)=(αβ)+(αγ); (c) α(β+γ) = αβαγ; (d) (αβ)γ = αγβγ; γ (e) αβγ = αβ . Propostion N3.2 Let α, β be cardinal numbers with α ≤ β and β infinite. Then: (a) α + β = αβ = β; (b) If α ≥ 2, then αβ > β. Proof of (b): If α ≥ 2, then 2β ≤ αβ, so it suffices to show β < 2β. Then we know {0, 1}∈ 2, and we can choose some B ∈ β. As discussed previously, the set 2β can be regarded as (is in one-to-one correspondence with) the power set of B. The fact that |B| ≤ |2B| should be fairly clear; considering the injection from B to its singleton subsets gives us B . P(B) ∼ 2B. Now, we show that no bijection exists between B and 2B (i.e. B ∼ 2B). Suppose, on the contrary, that there is a bijection g : B → 2B. Let A = {b ∈ B b ∈ g(b)}. Then A ∈ 2B, so ∃ a ∈ B such that g(a)= A. Now, if a ∈ g(a), then a ∈ A, and g(a)= A (a contradiction). If a ∈ g(a), then a ∈ A, and g(a)= A (a contradiction). So no element of B can map to A, and consequently g can not be a bijection. ¥ Well Ordered Sets The theory of well ordered sets provides the foundation for the study of ordinal numbers. A well ordered set (w.o.set) is a poset in which every non-empty subset contains a least element. p. 5 Proposition N3.3 Every well-ordered set is simply ordered. Proof: Let X be a well ordered set, and let x, y ∈ X. Then the subset {x, y} must contain a least element, which is either x or y. If it is x, then x ≤ y. If it is y, then y ≤ x. Thus X is a poset which also satisfies the comparability axiom, and is therefore simply ordered. ¥ Every w.o.set contains a least element, but not every simply ordered set with a least el- ement is well ordered. For example, [0, 1] (with the usual ordering) has a least element 0, but the subset (0, 1) does not. Every finite simply ordered set is well ordered, as is the set N of natural numbers. If A and B are disjoint well ordered sets, then A ∪ B can be made into a w.o.set by stipulating that (a ∈ A and b ∈ B) =⇒ a < b, while leaving the orders within A and B unchanged. For instance, suppose A = {a0,a1,...,an,...} and B = {b0, b1, . , bn,...}, where the elements of these sets are listed in increasing order from left to right. Then A ∪ B = {a0,a1,...,an, . , b0, b1, . , bn,...}, with elements again listed in increasing order from left to right, is a w.o.set. Furthermore, for any w.o.sets A and B, the set A×B is well ordered relative to the dictionary order. Any subset of a well ordered set with the inherited order is also well ordered. In a simply ordered set X, b is the immediate successor of a iff a < b and a<c ≤ b =⇒ c = b. The immediate predecessor is defined similarly. Proposition N3.4 In a w.o.set X, every element which is not the greatest element of X has an immediate successor, but every element does not necessarily have an immediate pre- decessor. Proof: Let a be an element of X which is not the greatest element. And let b be the least element of {x ∈ A a < x}, which is is non-empty. Then b is the immediate successor of a. In the example mentioned previously involving A ∪ B, neither a0 nor b0 has an immediate predecessor.