DOC PREVIEW
UCR MATH 144 - Infinite constructions in set theory

This preview shows page 1-2-3 out of 10 pages.

Save
View full document
View full document
Premium Document
Do you want full access? Go Premium and unlock all 10 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 10 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 10 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 10 pages.
Access to all documents
Download any document
Ad free experience

Unformatted text preview:

Exercises for Unit VI (Infinite constructions in set theory)VI.1 : Indexed families and set – theoretic operations(Halmos, §§ 4, 8 – 9; Lipschutz, §§ 5.3 – 5.4)Problems for study . Lipschutz : 5.3 – 5.6, 5.29 – 5.32, 9.14Exercises to work . 1. Generalize Exercise 12 from Section I I I.1 to unions and intersections of arbitrary indexed families of sets: Suppose that we have nonempty indexed families of sets { A j | j  J } and { C j | j  J } such that A j  C j for all j, Prove the following relationships:( j  J A j ) ( j  J C j )( j  J A j ) ( j  J C j )2. Generalize DeMorgan’s laws to unions and intersections of arbitrary indexed families of sets as follows:. Suppose that S is a set and we have a nonempty indexed families of subsets of S of the form { A j | j  J }. Prove the following identities:S –  j  J A j =  j  J ( S – A j )S –  j  J A j =  j  J ( S – A j )3. (Halmos, p. 35) (a) Given that { A j | j  X } and { B k | k  Y } are nonempty indexed families of sets, prove the following indexed distributive identities:( j  J A j ) ( k  K B k ) =  j, k ( A j B k )( j  J A j ) ( k  K B k ) =  j, k ( A j B k ) (b) Suppose that { I j | j  J } is an indexed family of sets, and write K =  { I j | j  J } .Suppose we are also given an indexed family of sets { A k | k  K } . Prove the following identities, assuming in the second case that each of the indexed families is nonempty: k K A k =  j J (  { A i | i  I j } ) k K A k =  j  J (  { A i | i  I j } )4. (Halmos, p. 37) (a) Let { A j | j  J } and { B k | k  K } be indexed familiesof sets. Prove that( j  J A j )  ( k  K B k ) =  j, k ( A j  B k )(another indexed distributive law) and that a similar formula holds for intersections provided that all the indexing sets are nonempty. (b) Let { X j | j  J } be an indexed family of sets. Prove that ( j  J X j )  X k  ( j  J X j )for all k  J. Furthermore, if M and N are sets such that M  X j  N for all j, prove that M ( j  J X j ) and ( j  J X j )  N.V I.2 : Infinite Cartesian products(Halmos, § 9; Lipschutz, §§ 5.4, 9.2)Problems for study . Lipschutz : 5.11Exercises to work . 1. (“A product of products is a product.”) Let X j be a family of nonempty sets with indexing set J, and let J =  { J k | k  K } be a partition of J. Construct a bijective map from  j X j to the set k K (  { X j | j  J k } ) .[ Hint : Use the Universal Mapping Property. ] 2. Let J be a set, and for each j  J let f j : X j  Y j be a set – theoretic map.Prove that there is a unique map F =  j f j :  j X j   j Y jdefined by the conditions p j Y  F = f j  p j Xwhere p j X and p j Y denote the j th coordinate projections for  j X j and  j Y j respectively. Also prove that this map is the identity map if each f j is an identity map. Finally, if we arealso given sets Z j with maps g j : Y j  Z j , and G =  j g j , then show that G  F = j (g j  f j) .Notation. The map of products  j f j constructed in the preceding exercise is frequently called the product of the maps f j . 3. Let { X j } and { Y j } be indexed families sets with the same indexing set J, and assume that for each j  J the mapping f j : X j  Y j is a bijection. Prove that the product map  j f j :  j X j   j Y j is also a bijection. [ Hint : What happens when one takes the product of the inverse maps? ]4. Suppose in the preceding exercise we only know that each mapping f j is an injection or each mapping f j is a surjection. Is the corresponding statement true for the product map? In each case either prove the answer is yes or find a counterexample.Coequalizers. Here is another fundamental example of a universal mapping property. Given two functions f, g : A  B, a coequalizer of f and g is defined to be a map p : B  C such that p f = p g which has the following universality property: Given an arbitrary map q : B  D such that q f = q g , then there exists a unique mapping h : C  D such that q = h p. — In geometrical studies, such constructions arise naturally if one tries to build an object out of two simpler pieces by gluing them together in some manner (say along their edges), and there are also numerous other mathematical situations where examples of this concept arise.5. Prove that every pair of functions f, g : A  B has a coequalizer. [ Hint : Consider the equivalence relation generated by requiring that f(x) be related to g(x) for all x in A. ] 6. In the setting of the previous exercise, suppose that p : B  C and r : B  E are coequalizers of f and g. Prove that there is a unique bijection H : C  E such that r = H p. [ Hint : Imitate the proof of the corresponding result for products. ] VI.3 : Transfinite cardinal numbers(Halmos, §§ 22 – 23; Lipschutz, §§ 6.1 – 6.3, 6.5) Problems for study . Lipschutz : 6.4, 6.12Exercises to work . 1. (Halmos, p. 92) Prove that the set F(S) of finite subsets of …


View Full Document

UCR MATH 144 - Infinite constructions in set theory

Download Infinite constructions in set theory
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Infinite constructions in set theory and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Infinite constructions in set theory 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?