Set theory Definition and 444 Threads

Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concerned with those that are relevant to mathematics as a whole.
The modern study of set theory was initiated by the German mathematicians Richard Dedekind and Georg Cantor in the 1870s. In particular, Georg Cantor is commonly considered the founder of set theory. The non-formalized systems investigated during this early stage go under the name of naive set theory. After the discovery of paradoxes within naive set theory (such as Russell's paradox, Cantor's paradox and Burali-Forti paradox) various axiomatic systems were proposed in the early twentieth century, of which Zermelo–Fraenkel set theory (with or without the axiom of choice) is still the best-known and most studied.
Set theory is commonly employed as a foundational system for the whole of mathematics, particularly in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beside its foundational role, set theory also provides the framework to develop a mathematical theory of infinity, and has various applications in computer science, philosophy and formal semantics. Its foundational appeal, together with its paradoxes, its implications for the concept of infinity and its multiple applications, have made set theory an area of major interest for logicians and philosophers of mathematics. Contemporary research into set theory covers a vast array of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals.

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  1. J

    Real analysis before set theory?

    Hey guys I signed up for a "set theory and topology" class for the fall and was planning on taking real analysis in the spring. Set theory got canceled and so i am taking real analysis instead and pushing set theory to the spring. Is this a wise idea? Taking real analysis before a formal set...
  2. J

    Basic set theory / mathematical notation

    I'm supposed to write the following intervals as sets in descriptive form: a. (t, infinity), t a fixed real number b. (0, 1/n), n a fixed natural number --- I think it is: a. (t, infinity) = {x: t < x < infinity} b. (0,1/n) = {x: 0 < x < 1/n} Is this correct? Also, how do you...
  3. P

    Proving Set Theory Equations with Nullspace Intersection

    Homework Statement 1)Prove for all sets A and B contained in a universe U, if A intersection B' = nullspace then P(A) − P(B) is a subset of P(A − B). 2)Prove for all sets A and B contained in a universe U, if A intersection B = nullspace then P(A) − P(B) is a subset of P(A − B)...
  4. C

    Understanding Causal Set Theory and Its Implications for the Universe

    Anyone heard of causal set theory before? Basically, it is a concept that our universe should be viewed solely as set of discrete events and the causal relations between them. I wrote a thesis where I described the Lagrangians of quantum fields. Please let me know what you think: arXiv:0905.2263
  5. N

    How Can Intersection of Indexed Family Sets Belong to Their Power Sets?

    Homework Statement Suppose {Ai| i \in I} is an indexed family of sets and I does equal an empty set. Prove that \bigcap i \in I Ai \in \bigcap i\in I P(Ai ) and P(Ai) is the power set of Ai Homework Equations none The Attempt at a Solution Suppose x \in {Ai| i \in I}...
  6. C

    Proving Set Theory Proof: (A-C) \cap (B-C) \cap (A-B) = ∅

    Homework Statement Prove that for all sets A, B, and C, (A-C) \cap (B-C) \cap (A-B) = ∅ Homework Equations The Attempt at a Solution Proof: Suppose A, B, and C are sets Let x \in (A-C) \cap (B-C) \cap (A-B) Since x \in (A-C), by definition of difference, x \in A and x \notin C...
  7. C

    Proof of Set Theory: A \subseteq B implies Bc \subseteq Ac

    Homework Statement For all sets A and B, if A \subseteq B then Bc \subseteq Ac. Homework Equations The Attempt at a Solution Proof: Suppose A and B are sets and A \subseteq B. Let x \in Bc By definition of complement, if x \in Bc then x \notin B Since x \notin B, x \notin A...
  8. A

    Can someone recommend a good Set Theory textbook?

    Ideally covers lots of content in depth with lots of exercises and doesn't skip anything in hardcover. The only bit of set theory I know is the most very basic that would occupy the first chapter in a book that would require it. Self study, very motivated. :) Thanks!
  9. S

    Basic Set Theory: Can You Accommodate Countable Guests?

    I found this question in a book. Q-Suppose you own a hotel with a countable number of rooms. One night a traveler wishes to stay in your hotel, but all the rooms are occupied. Can you give him a room without kicking anybody out of the hotel? What if a tour bus shows up with countably many...
  10. B

    Prove (A∩B)C=AC∩BC is FALSE: Counterargument Needed

    Homework Statement A,B and C are sets. Prove (A∩B)C = AC∩BC is FALSE That is, I have to give a counterargument for this statement. Homework Equations I can't find a counterargument directly. My professor suggest trying to prove the statement to find a problem and come up with the...
  11. V

    What Are the Events in a Sample Space of a Deck of Cards?

    Suppose that one card is to be selected from a deck of 20 cards taht cointains 10 red cards numbered from 1 to 10 and 10 blue cards numbered from 1 to 10. Let A be the event that a card with an even number is selected; let B be the event that the blue card is selected; and let C be the event...
  12. I

    Counting Equivalence Relations on N

    Homework Statement Find the cardinality of the set of all equivalence relations on N Homework Equations by what we have learned yet we only have to determine if it's countable or not The Attempt at a Solution I know that the set of all relations on N is equivalent to P(NXN) thus is...
  13. B

    Good books in Set theory and Mathematical Logic

    I am more precisely looking for a book on mathematical logic which presupposes only minimal exposure to set theory. Preferably something which includes an introductory chapter delineating relevant set theoretic principals. I am familiar with only basic set theory. More precisely this means...
  14. F

    Proving Compactness of Sets Using Open Covers

    Homework Statement Suppose X ⊂ R^n is a compact set, and U_1, U_2, U3, ... ⊂ R^n are open sets whose union contains X. Prove that for some n ∈ N (the natural numbers) we have X ⊂ U_1 ∪ ... ∪ U_n. Homework Equations A set is called compact if it is both closed and bounded. The Attempt at a...
  15. M

    Prove/Find Counterexample: Intro to Set Theory

    Homework Statement Prove or find counterexamples. For any sets A, B, C in a universe U: if A union C contained B union C then A contained B Homework Equations none. The Attempt at a Solution im just not sure if i did it right. id appreciate if you can check my work and let me...
  16. M

    How Do You Prove Set Theory Relations and Operations?

    -------------------------------------------------------------------------------- I tried to do the questions but I am just not sure if i did it right. id appreciate if you can check my work and let me know what changes i have to make. thanks the symbol "n" means "intersect" U for Union...
  17. M

    Set Theory Theorems: Solving for A in A ∩ B = C ∩ B and A ∩ B' = C ∩ B

    I need help on how to get started with this question: Im stocked and i just don't have a clue on how to figure this out. Prove: If A intersect B = C intersect B and A intersect B' = C intersect B' then A = C
  18. M

    Is My Set Theory Proof Correct?

    I tried to do the questions but I am just not sure if i did it right. id appreciate if you can check my work and let me know what changes i have to make. thanks the symbol "n" means "intersect" U for Union Question: Prove A contained B iff A n B = A Answer: (=>) Assume A contained B...
  19. R

    Set Theory Book Reviews: Halmos Edition

    I'm looking for a book on Set Theory, currently. I've found one by Halmos which looks good, but I'd like some input on it.
  20. P

    Proof Set Theory: A, B, C, X, Y in E

    Let A,B,C,X,Y be subsets of E,and A' MEAN the compliment of A in E i.e A'=E-A,and A^B = A \cap B Then prove the following: a) (A^B^X)U(A^B^C^X^Y)U(A^X^A') = A^B^X b) (A^B^C)U(A' ^ B^C)U B' U C' = E Thanks
  21. T

    Set theory and category theory

    They seem to be different fields but both try to underpin maths. There has been suggestions that set theory is problematic, where some paradoxes cannot be resolved. But how about Category theory? Any problems or paradoxes? Is it more promising then set theory?
  22. P

    What are the axioms in ZFC set theory?

    since a lot of talking is going on with sets, will somebody write down the axioms in ZFC theory as a point of reference , when a discussion is opened up. thanx
  23. D

    How Can I Clarify Set Theory Proof Ambiguity?

    Hi! I am having trouble constructing the sentences in this proof. Its very simple, proof that A \cup \left( B \cap C \right) = \left( A \cup B \right) \cap \left( A \cup B \right) So basically I need to show that if x \in A \cup \left( B \cap C \right) then x \in \left( A \cup B...
  24. S

    Philosophy of basic set theory proofs involving or .

    Philosophy of basic set theory proofs involving "or". Hey! I'm working through an Introduction to Analysis text, and I'm currently on the first chapter, which covers set theory. In one of the end-of-chapter problems, I'm asked to prove a basic theorem which leads to the following statement...
  25. V

    Constructing the real numbers, set theory

    My analysis text mentions in passing that the real numbers can be constructed rigorously starting from set theory. I was wondering if there were a resource on the web that might go over this and show the proofs of how this is done?
  26. M

    Proving Set Theory Statements for Beginners

    I've been working on these problems and unfortunately i can't make heads or tails of these two. Any insight where to start the proof would be great. 1)Let A, B and C be sets. Show that if A~B⊆C, then A~C⊆B holds. What I got so far: Is it correct to state that A~B = A⋂B' and A~C = A⋂C'...
  27. I

    Solving Set Theory Questions Using Natural Numbers and the Latin Alphabet

    I've been trying to wrap my brain around sets lately. Please bear with me, as I've been trying to teach myself. So, from what I've read, you can construct most everything in modern mathematics from sets. You can form the natural numbers from the successor function, you can construct the...
  28. sujoykroy

    Page 87 of Introduction to Set Theory

    Page 87 of "Introduction to Set Theory" Sorry, if this post doesn't fit into this forum, but i had no other choice. I got a electronic version of "Introduction to Set Theory" by "Karel Hrbacek" and "Thomas Jech". It is a magnificent book which opens up every window of understandings of...
  29. M

    Set Theory Regulatory Axiom and Ranks

    Homework Statement Assume that D is a transitive set. Let B be a set with the property that for any a in D, a is a subset of B implies a is an element of B. Show that D is a subset of B. The Attempt at a Solution My first step is to show that the empty set must be an element of D...
  30. M

    Proving Set Theory: Showing B is a Subset of S(n)

    Homework Statement Assume that S is a function with domain w such that S(n) is a subset of S(n^+) for each n in w. (Thus S is an increasing sequence of sets.) Assume that B is a subset of the union of S(n)'s for all n such that for every infinite subset B' of B there is some n for which B'...
  31. M

    Do you use set theory in Physics?

    I started reading a book on writing proofs (it is all about set theory), and I really enjoy it. If I do physics at uni, will I get to use things like set theory and to write proofs? And if so what specific applications does set theory have in physics?
  32. M

    Set Theory, One to One Correspondence

    Homework Statement Show that the equation f(m,n) = 2^m(2n+1)-1 defines a one-to-one correspondence between w x w to w. Where w (omega) is a symbol used to represent the 0,1,2,3,4,5,6... Question: The book defines a one to one correspondence as a one to one function from A onto B. Is...
  33. P

    Proving Set Theory Problem: Counterexample for (A-B)intersect(A-C)=empty set

    I was wondering if someone could please look over my proof of this set theory problem and let me know if I am doing it right or not and give me some help. Provide a counterexample for the following: If (A-B)intersect(A-C)=empty set, then B intersect C = empty set. Proof: Assume...
  34. S

    Set Theory Proof: Proving lx-yl ≤2r for All x,y εA

    Homework Statement This is the problem stated verbatim. xo is supped to be x with a subscript o. Suppose that A is a set and there exists xo ε A for which lx-xol ≤ r. Is it necessarily true true that for all x,y εA, we will have lx-yl ≤2r? Homework Equations Well, this problem is just...
  35. I

    Really, really basic question in set theory

    Very simple question :smile: Are the Pairing Axiom and the Union axiom in the Zermelo–Fraenkel set theory the same? I have a book that states them as the following: Pairing Axiom: For any sets u and v, there is a set having as members just u and v. Union axiom: For any sets a and b there...
  36. I

    Anyone have an online set theory text?

    I'm looking for a book that can stand as an introduction to axiomatic set theory (if it contains basic logic even better). Only thing I need it to be in the public domain and freely available online, anyone know of anything? Thanks in advance!
  37. A

    Are there any standout books on Set Theory and what research is left to be done?

    Any books that really stand out? Currently, I'm reading "Set Theory and Logic" by Stoll. I'm not interested in the axiomatic type of set theory, like Godel's theory and all those unreadable symboic proofs. I'm more interested in stuff like the axiom of choice proofs and such. Also, is there...
  38. D

    Research Topic in Graph Theory or Non-Well-Founded Set Theory

    I'm doing to come up with a subject in either of them to do either an "independent study" or "project" on, the former is a course which simply requires you to learn the subject and the latter is "independent study" + a x-page paper. Unfortunately I don't know either subject too well so I can't...
  39. F

    Set theory representation of material implication

    Just checking here. Propositional logic connectives like AND and OR have analogs or representations in set theory. For example, the logical connective AND is represented in set theory by intersection, an element of X AND Y is the element of the intersection of sets X and Y. And similarly, the...
  40. T

    Proving Set Theory Equation: Subsets of Size m = Subsets of Size n-m

    Homework Statement Prove that, for all n, for all m with 0 <= m <= n, the number of subsets of {1, . . . , n} of size m is the same as the number of subsets of {1, . . . , n} of size n − m. Homework Equations n/a The Attempt at a Solution My problem is that I don't know where to...
  41. quasar987

    Set theory problem (uncountable set)

    Homework Statement I would like to show that if we have a non-negative real valued function f defined on f a set X, and that the set of points where f is non-vanishing is uncountable, then for any M > 0, I can find a sequence {x_n} of points in X such that \sum_n f(x_n)>MHomework Equations...
  42. D

    Can a Set be Well-Ordered Without the Axiom of Choice?

    Homework Statement Show that if every total order of a set x is a well-order, then there is no bijection between x and x\cup\{ x\} = Sx.The Attempt at a Solution Suppose there was, then you can have a total order on x and an induced total order on Sx. But this induced order on Sx is a total...
  43. D

    Set theory in Munkres Topology

    In Munkres' Topology he defines a Cartesian product AxB to be all (a,b) such that a is in A and b is in B. He says that this is a primative way of looking at things. And then defines it to be {{a},{a,b}} He says that if a = b then {a,b} will just be {a,a} = {a} and therefore will only be...
  44. P

    Does this make sense in set theory?

    Homework Statement Let X be any set, f a function. Let f:X->Y Does f(A) make sense for A in X? I know f^(-1)(B) makes sense for B in Y. The Attempt at a Solution I can't see why not
  45. E

    Proving Discreteness of M: Set Theory Problem

    Homework Statement THIS PROBLEM IS DRIVING ME INSANE! HELP! Let M be a metric space in which the closure of every open set is open. Prove that M is discrete. Homework Equations The Attempt at a Solution
  46. C

    Proving Set Theory Basics: A \subseteq C

    Homework Statement Trying to prove some of the basic laws in set theory, and would like any opinions on 1 of my proofs (eg hints on how can I improve it, is it even a valid proof). Thanks in advance. (A \subseteq B \wedge B \subseteq C) \rightarrow (A \subseteq C) Homework Equations...
  47. N

    Proving Subset Relation: A⊂B ⇒ B'⊂A

    Homework Statement I have to prove that if A blis a subset of B then B' is a subset of A'. Homework Equations The Attempt at a Solution I did: Let x belongs to B but x does not belong to A =>x does not belong to B' but x belongs to A' Hence proved. please tell me if I am...
  48. S

    Set theory - Cardinality of P(X)

    Homework Statement Let X be a finite set with n elements. Prove that P(X) has 2^n elements. <This is an extra credit problem for a summer class I'm taking.> Homework Equations P(X) is the power set of X, the set of all possible subsets of X. The principle of induction. The...
  49. honestrosewater

    Is My Approach to Understanding Set Theory Beneficial?

    Hey, I'm feeling very shaky for some reason. I'd like to run a few things by you guys. I can do formal if needed, but I'm trying to build up a better model in my head in which I can eventually reason more flexibly and quickly without making mistakes. I'm starting from the beginning. This is ZF...
  50. A

    Set Theory to Show Transcendental Numbers Exists

    Does anyone know how Cantor showed the existence of Transcendental numbers. How can he say that most numbers are transcendental? Is that why everyone critised it? Cheers Ash
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