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Home , Axiom of union, Equinumerosity, Ideal (set theory)

Salem State College MAT 420 Theory

L. Pedro Poitevin October 25, 2007

1 Foundations

Definition (Informal). We define a set A to be any unordered collection of objects, e.g., {3, 8, 5, 1} is a set. If x is an object, we say that x is an of A or x ∈ A if x lies in the collection; otherwise we say that x is not an element of A and write x∈ / A. For instance, 3 ∈ {1, 2, 3, 4, 5} but 7 ∈/ {1, 2, 3, 4, 5}. 1 (Sets are objects). If A is a set, then A is also an object. In particular, given two sets A and B, it is meaningful to ask whether A is an element of B. Definition (Equality of sets). Two sets A and B are equal—in notation A = B—if and only if every element of A is an element of B and vice versa. Axiom 2 (). There exists a set ∅, known as the empty set, which contains no elements, i.e. for every object x we have x∈ / ∅. Exercise. Prove that there is only one empty set. Lemma 1 (Single choice). Let A be a nonempty set. Then there exists an object x such that x ∈ A. Axiom 3 (Singletons). If a is an object, then there exists a set {a} whose only element is a. Axiom 4 (Pairwise ). Given any two sets A and B, there exists a set A ∪ B called the union of A and B, whose elements are all objects which are elements of A or B. In symbols, x ∈ A ∪ B ⇐⇒ x ∈ A ∨ x ∈ B. Remark. This axiom allows us to define pair sets, triplet sets, quadruplet sets, and so forth. If a, b are two objects, we define {a, b} := {a} ∪ {b}; if a, b, c are three objects, then {a, b, c} = {a} ∪ {b} ∪ {c}, and so on. Lemma 2. The union operation above described is commutative and associative. Remark. Because of the associativity of the union operation, we do not need to use parentheses to denote multiple unions; thus, for instance, we can write A∪B ∪C instead of (A∪B)∪C or A∪(B ∪C). Similarly for unions of four sets A ∪ B ∪ C ∪ D, etc. Definition (). Let A and B be sets. We say that A is a of B and write A ⊆ B if every element of A is an element of B. We say that A is a proper subset of B and write A ⊂ B if A ⊆ B and A 6= B. Proposition 1 (Sets are partially ordered by set inclusion). Let A, B, C be sets. The following three conditions are satisfied: 1. If A ⊆ B and B ⊆ A, then A = B. 2. If A ⊆ B and B ⊆ C, then A ⊆ C. 3. If A ⊂ B and B ⊂ C, then A ⊂ C. Axiom 5 (Specification). Let A be a set, and for each x ∈ A, let P (x) be a property pertaining to x (i.e., P (x) is either a true statement or a false statement). Then there exists a set {x ∈ A : P (x) is true} (or simply {x ∈ A : P (x)} for short), whose elements are precisely the elements x in A for which P (x) is true. In symbols, for any object y,

y ∈ {x ∈ A : P (x)} ⇐⇒ y ∈ A ∧ P (y) is true.

Definition (Pairwise intersection). The intersection S1 ∩ S2 of two sets S1,S2 is defined to be the set

S1 ∩ S2 := {x ∈ S1 : x ∈ S2}. In other words, x ∈ S1 ∩ S2 ⇐⇒ x ∈ S1 ∧ x ∈ S2. Remark. One should be careful with the English word “and”: rather confusingly, it can mean either union or intersection, depending on the context. For instance, if one talks about a set of “boys and girls,” one means the union of a set of boys and a set of girls, but if one talks about the set of people who are single and male, then one means the intersection of the set of single people with the set of male people. (Can you work out the rule of grammar that determines when “and” means union and when “and” means intersection?) Another problem is that “and” is used in English to denote addition. So, for instance, one could say that “2 and 3 is 5,” while also saying that the elements of {2} and the elements of {3} form the set {2, 3}” and “the elements of {2} and {3} form the set ∅.” This can certainly get confusing! One reason to use mathematical symbols instead of English words such as “and” is that mathematical symbols always have a precise and unambiguous meaning, whereas one must often look very carefully at the context in order to work out what an English word means. Definition. Two sets A and B are said to be disjoint if A ∩ B = ∅. Remark. Note that being disjoint is not the same as being distinct. If A and B are disjoint, then they are possibly distinct (find an exception!) but if they are distinct, they may not be disjoint. Definition (Set difference). Given two sets A and B, we define the set A \ B to be the set {x ∈ A : x∈ / B}. Proposition 2 (Sets form a Boolean algebra). Let A, B, C be sets, and let X be a set containing A, B, C as subsets. (a) A ∪ ∅ = A and A ∩ ∅ = ∅. (b) A ∪ X = X and A ∩ X = A. (c) A ∩ A = A and A ∪ A = A. (d) A ∪ B = B ∪ A and A ∩ B = B ∩ A. (e) (A ∪ B) ∪ C = A ∪ (B ∪ C) and (A ∩ B) ∩ C = A ∩ (B ∩ C). (f) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) and A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) (g) A ∪ (X \ A) = X and A ∩ (X \ A) = ∅. (h) X \ (A ∪ B) = (X \ A) ∩ (X \ B) and X \ (A ∩ B) = (X \ A) ∪ (X \ B). Axiom 6 (Replacement). Let A be a set. For any object x ∈ A and any object y, suppose we have a statement P (x, y) pertaining to x and y, such that for each x ∈ A there is at most one y for which P (x, y) is true. Then there exists a set {y : P (x, y) is true for some x ∈ A}, such that for any object z, z ∈ {y :P (x, y) is true for some x ∈ A} ⇐⇒ P (x, z) is true for some x ∈ A. Axiom 7 (Infinity). There exists a set N whose elements are called natural numbers, as well as an object 0 ∈ N and an object n ++ assigned to every n ∈ N, such that the Peano hold. Before the development of modern axiomatic , logicians and mathematicians by and large accepted a single, unifying principle, called universal specification, which implies most of the axioms above described. This principle is stated as follows: “Suppose for every object x we have a property P (x) pertaining to x. Then there exists a set {x : P (x) is true} such that for every object y, y ∈ {x : P (x) is true} ⇐⇒ P (y) is true.” (1) The principle of universal specification is also known as the axiom of comprehension, and it was at the heart of the foundation of mathematics, as understood by brilliant philosophers and mathematicians at the onset of the XX century, particularly Gottlieb Frege. Unfortunately for Frege, and for many mathematicians of his time, this so-called axiom was erroneous, as the venerable philosopher, mathematician, peace-activist, Nobel-prize winning writer, and fantastic educator Lord found out. Bertie Russell, whose exploits make him the grandfather of modern mathematical logic, considered the statement P (x) that says “x is a set, and x∈ / x.” Such a statement looks innocent enough, and it is easy to verify that most sets satisfy that statement. The exceptions seem problematic, true enough, but he concentrated his attention not on the exceptions but on the innocent looking sets x satisfying P (x). (Example: P ({0, 1}) is true, because {0, 1} is not an element of {0, 1}. On the other hand, P ({x : x is a set}) is presumably not true, because a priori {x : x is a set} is itself a set, and so it is an element of itself.) Bertie used the principle of universal specification to produce the very large set Ω defined below: Ω := {x : P (x) is true} = {x : x is a set and x∈ / x}, i.e., the set of all sets which do not contain themselves (as elements). Bertie finally asked himself (and all of you) the following mind-boggling question, which effectively destroyed Frege’s axiomatic edifice for all of mathematics: Question (Russell’s paradox). Is Ω ∈ Ω? The problem with the principle of universal specification is, in some sense, that it creates sets that are “too large.” For example, one can talk about the set of all sets, and so one has some sets that contain themselves as elements, which is not an thing. Now that we have seen just how awful the situation can get pretty easily, we add one axiom below to ensure that we do not get any ugly paradoxes like Russell’s paradox above. Axiom 8 (Regularity). If A is a nonempty set, then there is at least one element x ∈ A which is either not a set or is disjoint from A. This last axiom is also referred to in the literature as the axiom of foundation.

2 Functions

Definition (Functions). Let X and Y be sets, and let P (x, y) be a property pertaining to an object x ∈ X and an object y ∈ Y , such that for every x ∈ X, there is exactly one y ∈ Y for which P (x, y) is true. (This is sometimes referred to as the vertical line test.) Then we define the f : X → Y defined by P on the domain X and range Y to be the object which, given any input x ∈ X, assigns an output f(x) ∈ Y , defined to be the unique object f(x) for which P (x, f(x)) is true. Thus, for any x ∈ X and y ∈ Y , y = f(x) ⇐⇒ P (x, y) is true. Remark. A word on notation. Parentheses have now several different uses in mathematics. We use them to clarify the order of operations, as in 3(1 + 2), and we are also using them to enclose the argument of a function (as in f(x)) or of a property such as P (x). We are supposed to be mindful of these uses, and disambiguate from context. Definition (Equality of functions). Two functions f : X → Y and g : X → Y with the same domain and range are said to be equal, in symbols f = g, if and only if f(x) = g(x) for all x ∈ X. Definition (Composition of functions). Let f : X → Y and g : Y → Z be two functions, such that the range of f is the same as the domain of g. We then define the composition g ◦ f : X → Z of the two functions g and f to be the function defined explicitly by the formula: (g ◦ f)(x) = g(f(x)). If the range of f does not match with the domain of g, we leave the composition g ◦ f undefined. Lemma 3. Let f : X → Y , g : Y → Z, and h : Z → W be functions. Then f ◦ (g ◦ h) = (f ◦ g) ◦ h. Definition (Injectivity). A function f is one-to-one or injective if different elements map to different elements, i.e. if x 6= x0 ⇒ f(x) 6= f(x0). Equivalently, f is one-to-one if and only if f(x) = f(x0) implies that x = x0. Definition (Surjectivity). A function f is onto or surjective if for every y ∈ Y , there exists x ∈ X such that f(x) = y. Definition (Bijectivity). A function f : X → Y that is both one-to-one and onto is also called bijective. Remark. If f : X → Y is bijective, then for every y ∈ Y there is exactly one x ∈ X such that f(x) = y (there is at least one because of the surjectivity, and at most one because of the injectivity). This value of x is denoted f −1(y); thus f −1 is a function from Y onto X. We call f −1 the inverse of f. Definition (Forward images). If f : X → Y is a function from X to Y , and S is a subset of X, then we define f(S) to be the set f(S) := {f(x): x ∈ S}. This set is a subset of Y , and it is sometimes called the of S under the map f. We sometimes call f(S) the forward image of S. Definition (Inverse images). If U is a subset of Y , and if f : X → Y is a function, then we define f −1(U) to be the set f −1(U) := {x ∈ X : f(x) ∈ U}. We call f −1(U) the inverse image of U under f. Remark. If f is a bijective function, then we have defined f −1 in three different ways: talk about confusing! Axiom 9 ( axiom). Let X and Y be sets. Then there exists a set, denoted Y X , which consists of all the functions from X to Y . Thus f ∈ Y X ⇐⇒ f is a function with domain X and range Y .

Lemma 4. Let X be a set. Then {Y : Y is a subset of X} is a set. Remark. The set {Y : Y is a subset of X} is known as the power set of X and is denoted 2X .

AxiomS 10 (Union). Let A be a set, all of whose elements are themselves sets. Then there exists a set A whose elements are precisely those objects which are elements of the elements of A. Thus for all objects x, [ x ∈ A ⇐⇒ (∃S ∈ A)(x ∈ S). The axiom of union, combined with the axiom of pair set, implies the axiom of pairwise union. (Why?) Another important consequence of this axiom is that if oneS has some set I, and for every element α ∈ I we have some set Aα, then we can form the union set α∈I Aα by defining [ [ Aα := {Aα : α ∈ I}, α∈I which is a set thanks to the axiom of replacement and the axiom of union. In situations like this, we often refer to I as an index set, and the elements α of this index set as labels; the sets Aα are then called a , and are said to be indexed by the labels α ∈ I. We can similarly form intersections of families of sets, as long as the index set is nonempty. More specifically, givenT any nonempty set I, and given an assignment of a set Aα to each α ∈ I, we can define the intersection α∈I Aα by first choosing some element β ∈ I (which we can do since I is nonempty), and setting \ Aα := {x ∈ Aβ :(∀α ∈ I)(x ∈ Aα)}, α∈I which is a set by the axiom of specification. This definition may look like it depends on the choice of β, but it does not. Remark. The axioms of set theory so far specified comprise what is known as the Zermelo-Fraenkel axioms of set theory. Together with the so-called , which we shall discuss later on, they form the powerful system denoted ZFC, within which most of mathematics takes place. 3 Cartesian products

In addition to the basic operations of union, intersection, and set difference, another fundamental operation of sets is that of the . Definition (Ordered pairs). If x and y are any objects (possibly equal), we define the (x, y) to be set {{x}, {x, y}. The key property of ordered pairs is that any two ordered pairs (x, y) and (x0, y0) are equal if and only if x = x0 and y = y0. Remark. We have now “overloaded” the parentheses symbols again; they now are not only used to denote grouping of operators and arguments of functions, but also to enclose ordered pairs. Definition (Pairwise Cartesian product). If X and Y are sets, then we define the Cartesian product X × Y to be the collection of ordered pairs specified as follows: X × Y := {(x, y): x ∈ X ∧ y ∈ Y }.

Definition (Finite Cartesian product). Let n be a natural number. An ordered n- (xi)1≤i≤n (also denoted (x1, . . . , xn)) is a collection of objects xi, one for each natural number i between 1 and n; we refer to xi as the ith component of the ordered n-tuple. Two ordered n- (xi)1≤i≤n and (yi)1≤i≤n are said to be equal if and only if xi = yi for all i ∈ {1, . . . , n}. If (Xi)1≤i≤n is an ordered n-tuple of Q Qn sets, we define their Cartesian product 1≤i≤n Xi (also denoted i=1 Xi or X1 × · · · × Xn) by Y Xi := {(xi)1≤i≤n : xi ∈ Xi for all i ∈ {1, . . . , n}. 1≤i≤n

Remark. It is possible (and relatively easy) to define the ordered n-tuple (xi)1≤i≤n as a set. I leave it to you to think about how to do this.

Remark. Strictly speaking, given sets X1,X2,X3, the sets X1 × X2 × X3, X1 × (X2 × X3), and (X1 × X2) × X3 are distinct from each other. However, they are clearly very tightly related, and it is common practice to neglect the minor distinctions between them. Lemma 5 (Finite choice). Let n ≥ 1 be a natural number, and for each natural number i ∈ {1, . . . , n}, let Xi be a nonempty set. Then there exists an n-tupleQ(xi)1≤i≤n such that xi ∈ Xi for all i ∈ {1, . . . n}. In other words, if each Xi is nonempty, then the set 1≤i≤n Xi is also nonempty.

4 of sets

Definition (Equinumerosity). We say that two sets X and Y have equal cardinality (or are equinumer- ous if and only if there exists a f : X → Y from X to Y . Proposition 3. Let X,Y,Z be sets. Then X and X are equinumerous. If X and Y are equinumerous, then Y and X are equinumerous. If X and Y are equinumerous and Y and Z are equinumerous, then X and Z are equinumerous. Definition (Cardinality). Let n be a natural number. As set X is said to have cardinality n if and only if it is equinumerous with {i ∈ N : 1 ≤ i ≤ n}. We also say that X has n elements if and only if it has cardinality n. Lemma 6. Suppose that n ≥ 1 and X has cardinality n. Then X is nonempty and if x ∈ X, then the set X \{x} has cardinality n − 1. Proposition 4. Let X be a set with some cardinality n. Then X cannot have any other cardinality, i.e., X cannot have cardinality m for any m 6= n. Definition (Finiteness). A set is finite if and only if it has cardinality n for some natural number n. Otherwise, the set is called infinite. If X is a set, we use #(X) to denote the cardinality of X. Theorem 1. The set N of natural numbers is infinite. Corollary 1. Any unbounded set is infinite. Proposition 5 (Finite cardinal arithmetic). (a) Let X be a finite set and let x be an object which is not an element of X. Then X ∪ {x} is finite and #(X ∪ {x}) = #(X) + 1. (b) Let X and Y be finite sets. Then X ∪ Y is finite and #(X ∪ Y ) ≤ #(X) + #(Y ). If in addition X and Y are disjoint, then #(X ∪ Y ) = #(X) + #(Y ). (c) Let X be a finite set, and let Y be a subset of X. Then Y is finite and #(Y ) ≤ #(X). If in addition Y 6= X, then we have #(Y ) < #(X). (d) If X is a finite set and f : X → Y is a function, then f(X) is a finite set with #(f(X)) ≤ #(X). If in addition f is one-to-one, then #(f(X)) = #(X). (e) Let X and Y be finite sets. Then X × Y is finite and #(X × Y ) = #(X) × #(Y ). (f) Let X and Y be finite sets. Then the set Y X is finite and #(Y X ) = #(Y )#(X).