mat.haus project Zurab Janelidze s Lectures on UNIVERSE OF SETS last updated 18 May 2017 Stellenbosch University 2017

Transcription

1 Zurab Janelidze s Lectures on UNIVERSE OF SETS last updated 18 May 2017 Stellenbosch University 2017 Contents 1. Axiom-free universe of sets 1 2. Equality of sets and the empty set 2 3. Comprehension and pairing 2 4. Intersection and union 3 5. Power set and cartesian product 4 6. Difference of sets 5 Index 6

2 Zurab Janelidze 1. Axiom-free universe of sets A universe of sets is a collection of objects, called sets, equipped with a binary relation between sets, called the element relation and denoted by, adhering to axioms that will be presented in this section. 1 When a set x is in the element relation with a set X, we write x X and we say that x is an element of X. 2 When there exists a set X whose elements are precisely those which are in a given collection of sets, we say that the collection forms a set, and we call X a set formed by the given collection. 3 Given a formula ψ(x) with a free variable x, the collection of all sets x such that ψ(x) is true is called the collection of sets determined by the formula ψ(x). A class of sets is any collection of sets determined by some formula. When the collection of sets determined by a formula ψ(x) forms a set, we denote this set as {x ψ(x)}, and we say that ψ(x) defines a set, calling the above set, one defined by ψ(x). More generally, given a term t over a list of variables, and a formula ψ in which these variables are free, if the collection of those sets that can be constructed using the term t from those values of variables for which ψ is true forms a set, then we write {t ψ} to denote this set. We call this set the set of those t s for which ψ is true. 1. Show that the collection of all elements of a given set X is a class of sets. 2 (Russel s paradox). Show that for the class of sets determined by the formula [x x] does not form a set. When every element of a set A is also an element of a set B, formally, x [[x A] [x B]], we say that A is a subset of B, and we write A B. 3. Prove that for arbitrary sets A, B, and C, if A B and B C, then A C. 4. Prove that every set is a subset of itself. 1 Extensive use of sets in mathematics has been emerging in the formalization of geometry in the work of Bernhard Riemann ( ) and the formalization of algebra in the work of Richard Dedekind ( ). The work of Georg Cantor ( ) on infinite sets led to formation of set theory as an independent subject. The first axioms for set theory were proposed by Ernst Zermelo ( ) in The axioms in this section follow those of Zermelo except two of his last axioms, which will be dealt with later on: the axiom of choice and the axiom of infinity. The precise formulation of these axioms in a first order language was given in 1922 by Thoralf Skolen ( ). 2 Any directed graph gives a model of the axiom-free universe of sets. For instance, consider the following directed graph: In this model there are four sets, 1, 2, 3 and 4. Of these, 1 does not have any elements, 2 and 3 both have 1 as the unique element and 4 has 2 and 3 as elements. As the axioms for a universe of sets are introduced in this section, the reader may find it a worthwhile exercise to apply them on such models and check their validity in individual models. 3 In some sense, mathematics is all about the structure of things of pure structure, without form or sensation. Perception of structure begins by isolating different components that make up the whole, as well as combining different components to create a whole. The subject of set theory is a study of these two processes. What is remarkable about this subject is that the language of set theory appears to be rich enough to express all mathematical phenomenon. In set theory, the isolated components are called elements, while their combinations are called sets. Any element can be further decomposed into elements, or in other words, any element is a set; and also, any set is an element of another set. The foundation of mathematics through set theory is achieved by defining any mathematical object to be a set. Different ways of forming sets out of elements provide the means for all constructions in mathematics and the method of proof in set theory can be used to express proofs of all theorems in mathematics, once these theorems are suitably modeled in the universe of sets. UNIVERSE OF SETS 1

3 2. Equality of sets and the empty set The axiom of equality of sets states that two sets X and Y are equal whenever they are each other s subsets, i.e. whenever X Y and Y X. 5. Explain how this axiom states that there cannot be two different sets having exactly the same elements, in other words, that every set is uniquely determined by the collection of its elements. It may happen that a set does not have any elements. Such set A is a subset of every set B. Indeed, when A does not have any elements, [x A] [x B] is true for any x, by the principle that a false statement implies any other statement. If two sets A and B both do not have elements, then we get that they are subsets of each other, and hence must be the same set by the axiom of equality of sets. Thus, there can only be at most one set which does not have any elements. Such set exists by the following axiom: The empty set axiom states that there exists a set which does not have any elements. We adopt a special name and notation for the set which does not have any elements: The set which does not have any elements is called the empty set and is denoted by. 6. Show that is the unique subset of. Deduce from this that is the only set which is a subset of every set, as well as that is the unique set which has only one subset. 7. Give a formula involving one free variable which defines the empty set. 3. Comprehension and pairing Consider a set X, and the class of all of its elements. Now select only some members of this class, which still constitute a class. The next axiom guarantees that there is a subset S of X such that what we have selected is precisely the class of all elements of S. The axiom of restricted comprehension states that any class of elements of a set defines a set. When a class of elements of a set X is specified by a formula ϕ(x), we write {x X ϕ(x)} for the corresponding subset of X. 8. Deduce the existence of the empty set from restricted comprehension and the assumption that there exists at least one set. 9. The unrestricted comprehension states that any first order formula ψ(x), with a free variable x, defines a set. Disprove unrestricted comprehension. 10. Write the set {x X ϕ(x)} in the form {x ψ(x)}. The axiom of pairing states that for any two sets a and b, there exists a set having a and b as its only elements. A set having a and b as its only elements is defined by the formula [x = a] [x = b]. For any two sets a and b, we write {a, b} for the set {a, b} = {x [x = a] [x = b]} and call it the the set of a and b. When a = b we write {a} = {a, a} and call it the the singleton of a. So far we only know the existence of one set the empty set. The operation of pairing of two sets can be used to construct other sets. 11. Use the axiom of pairing to construct several nonequal sets. Would it be possible to show that there exist at least two different sets, without the axiom of pairing? An ordered pair of sets a and b is defined as the set (a, b) = {{a}, {a, b}}. UNIVERSE OF SETS 2

4 12. Show that if two ordered pairs (x, y) and (a, b) are equal, then their respective components must be equal, i.e. x = a and y = b. Can you find other ways to define an ordered pair of two sets so that the same result will still hold? Define an ordered triple (a, b, c) of sets, and show that if two ordered triples are equal, then their respective components must be equal. For a set A, consider the formula a [[a A] [x a]]. The sets x for which this formula holds are precisely those sets which are elements of every element of A. When A is the empty set, any set x is such. 14. Show that the class of all sets does not form a set. When A has at least one element a, then it follows by restricted comprehension that the formula above does define a set. Indeed, in this case all sets x satisfying the formula will be elements of a, and so the formula will be defining a subset of a. For a nonempty set A, the intersection of elements of A is defined as the set A = {x a [[a A] [x a]]}. When A = {t ϕ}, we also write t ϕ to denote the intersection of elements of A. 15. Prove that for any nonempty set A we have: a = A. a A The intersection of two sets A and B, denoted by A B, is defined as the set 16. Prove that A B = {A, B}. A B = {x [x A] [x B]} for any two sets A and B. 17. Show that for any two sets A and B, we have A B = B A. 18. Show that for any two sets A and B, we have A B if and only if A B = A. Deduce from this that A = is the only set which has the property that A X = A for any set 4. Intersection and union X. Also, deduce that X X = X holds for any set X. 19. Show that for any three sets A, B and C, we have (A B) C = A (B C). The union axiom states that for any set A, all elements of elements of A form a set, i.e. the formula a [[x a] [a A]] defines a set. We adopt special notation and name for the construction in the above axiom. For any set A, we write A for the set A = {x a [[x a] [a A]]} and call it the union of elements of A. When A = {t ϕ}, we also write t ϕ to denote the union of elements of A. While intersection of the elements of the empty set is undefined, the union of the elements of the empty set is defined, and moreover, it is the empty set. 20. Prove that =. Union of two sets A and B is defined as the set A B = {A, B}. 21. Prove that A B = {x [x A] [x B]} 4 There are indeed other ways of defining an ordered pair. In every application of an ordered pair, what is used is just the principle established in this act. The definition given in this text is due to Kazimierz Kuratowski ( ). UNIVERSE OF SETS 3

5 for any two sets A and B. 22. Show that for any two sets A and B, we have A B = B A. 23. Show that for any two sets A and B, we have A B if and only if A B = B. Deduce from this that A = is the only set which has the property that A X = X for any set X. Also, deduce that X X = X holds for any set X. 24. Show that for any three sets A, B and C, we have (A B) C = A (B C). 25. Show that for any three sets A, B and C, we have A (B C) = (A B) (A C). Prove that the identity remains true if we interchange and in it. The power set axiom states that for any set X, the class of all subsets of X forms a set. The set of all subsets of a set X is denoted by P(X) = {S S X}. and is called the power set of X Compute the power sets P( ) and P(P( )). 27. Prove that ( ) P X = P(x) x X for any nonempty set X. 28. Prove that for any set X, P(X) = X. 29. Prove that for any set X, ( ( )) X P P X. Given two sets X and Y, and their elements x X and y Y, the sets {x, y} and {x} are both subsets of X Y, and therefore, they are elements of the powerset P(X Y ). This makes the ordered pair (x, y) = 5. Power set and cartesian product {{x, y}, {x}} a subset of P(X Y ), and hence an element of P(P(X Y )). Thus, by restricted comprehension, the class of all possible ordered pairs (x, y) where x X and y Y forms a set. For any two sets X and Y, the set of all possible ordered pairs (x, y) where x X and y Y is called the (cartesian) product of X and Y, and is denoted by X Y = {(x, y) [x X] [y Y ]}. 30. Prove that the empty set A = is the only set having the property that A X = A for any set X. Do the same for the identity X A = A. 31. Given two sets X and Y, prove that the class of all sets x y, where x X and y Y, forms a set. Then, prove that ( ) ( ) X Y = (x y). (x,y) X Y for any two nonempty sets X and Y. 32. Prove that ( ) ( ) X Y = (x y). (x,y) X Y 5 An interesting partial model of a universe of sets in concrete mathematics is given by the collection of natural numbers with the element relation being the strict inequality relation, i.e., for two natural numbers n and m, we have n m n < m. Here, the axiom of equality of sets, the empty set axiom, the union axiom and the powerset axiom hold (but not the other axioms). The subset relation in this model turns out to be the inequality relation, i.e., n m n m, while the union and power set are given by the formulas n = n 1, P(n) = n + 1. In this model, 0 is the empty set, 1 = {0}, 2 = {0, 1}, and so on. We could infinitely extend this model by adding to it any finite set of any finite set of (and so on, finitely) natural numbers. In the resulting model, all axioms will hold. Sets in this model can be conveniently represented by finite trees (i.e., connected graphs having exactly one less edge than the number of vertices) with a distinguished vertex, in which the vertices connected to the distinguished vertex represent elements of the tree, vertices connected to those further in the tree represent their elements, and so on. Some illustrations of such representation, where the distinguished node is always the bottom-most node, is given in Figure 1. UNIVERSE OF SETS 4

6 0 1 2 {1, 2} {{0}} Figure 1. Representing sets as trees. for any two sets X and Y. 33. Given two sets X and Y, prove that the class of all sets x y, where x X and y Y, forms a set. Then, prove that ( ) ( ) X Y = (x y) for any two sets X and Y. (x,y) X Y 6. Difference of sets The difference of two sets A and B is defined as A \ B = {a A [a B]}. 34. Prove that for any set X, we have X \ = X and X \ X =. 35. Prove that (A \ B) \ C = A \ (B C) for any three sets A, B and C. 36. Prove that for any two nonempty sets A and B, the difference ( A) \ ( B) can be computed as follows: ( ) ( ) A \ B = (a \ b) b B a A = (a \ b). a A b B 37. Find similar computations as above, for the differences ( A)\( B), ( A)\( B), and ( A) \ ( B). The difference operation is noncommutative, i.e. in general A \ B is different from B \ A. The following operation is a commutative version of the difference operation. The symmetric difference of two sets A and B is defined as A B = (A \ B) (B \ A). We establish some basic properties of the symmetric difference. 38. Prove that X Y = Y X for any two sets X and Y. 39. Prove that for any set X, we have X = X and X X =. 40. Prove that (A B) C = A (B C) for any three sets A, B, and C. 41. Prove that (A B) C = (A C) (B C) for any three sets A, B, and C. UNIVERSE OF SETS 5

7 (, ), sets, 2, sets, 5, sets, 3, sets, 3, sets, 1 P, sets, 4 \, sets, 5, sets, 1, sets, 4, sets, 2 {, }, sets, 2 { }, sets, 1, 2 {}, sets, 2 Index axiom of equality of sets, 2 axiom of pairing, 2 axiom of restricted comprehension, 2 axiom: empty set, 2 axiom: power set, 4 axiom: union, 3 class of sets, 1 collection of sets determined by a formula, 1 defining a set, 1 difference, 5 element, 1 element relation, 1 empty set, 2 forming a set, 1 intersection, 3 intersection of elements, 3 ordered pair, 2 power set, 4 product of sets, 4 Russel s paradox, 1 set defined by a formula, 1 set formed by a collection, 1 set of two elements, 2 sets, 1 singleton set, 2 subset of a set, 1 symmetric difference, 5 the set of..., 1 union of elements of a set, 3 universe of sets, 1 unrestricted comprehension, 2 6