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. K

    Good Book on Set Theory: An Introduction

    Price is irrelevant. I'm looking for an introductory though rigorous treatment of set theory. I'm about half-way through a text of mathematical logic (propositional, first order predicate, computability theory, etc). But the text doesn't cover set theory. Thanks.
  2. B

    Book Recommendation [Set Theory]

    Would you guys recommend the following Book By Paul Cohen as a good (and cheap) intro to set theory and the Continuum Hypothesis. Set Theory and the Continuum Hypothesis (Dover Books on Mathematics) Some reviewers attacked Mr. Cohen as being a poor logician. Maybe people were just mad...
  3. A

    Basic basic set theory. Please help. Simple answer will suffice.

    Hello my name is Andy I'm in high school, and I have a bit of a confusion or lack of information. Ok so I have been reading a book on set theory, and I keep encountering … ,but up more in the use of a set. Like A,B … but again up more to the middle of the sentence. I feel dumb, I googled it...
  4. D

    Basic set theory question about complement

    Homework Statement Hi I could use some help getting an explanation that kinda twists my head a bit. I want to know what I'm missunderstanding so I can get this right from the beginning. ^c = complement U = union I want to simplify this set A U (A U B^c)^c intersect (A U C ) to...
  5. K

    Recommended Set Theory Textbooks for Studying Topology and Beyond

    I'm a physics undergraduate and I'll starting learning topology from Munkres next semester. But first I want to learn set theory to feel more comfortable. Do you know any good textbook? A friend of mne from the math department said I should go with Kaplansky's "Set Theory and Metric Spaces".
  6. M

    Understanding Set Theory and the + Symbol: Solving Equations with Sets

    Set Theory "+" symbol 1. X, Y and Z are sets. Does X × (Y + Z) = X × Y + X × Z? The solution starts like so: X × (Y + Z) = {(x,(y,0)) | x \in X, y \in Y}\cup{(x,(z,1)) | x \in X, z \in Z} I don't understand how the "+" symbol works. Why does it equate to this (x,(y,0)) (x,(z,1))...
  7. U

    Understanding some more set theory for statistics

    Hope this is the right forum for my question. I'm into statistics and quite often see assumptions involving set theory. I know some set theory but am having trouble understanding it for any application. I would like to narrow this gap, maybe because this type of mathematics seems most...
  8. A

    Set Theory Help: Proving E C ]0,2] and Exploring Its Relationship with ]0,2]

    we have E=(a+b)/ab (a,b)EN* 1/ prove that E C ]0,2] (i already duid that ) 2/ is ]0,2] E E? shelp me in this one! Homework Equations The Attempt at a Solution x \in ]0,2] \Rightarrow x \in E
  9. A

    Prove A C B for Set Theory: Help with Pi and Integers

    Homework Statement A = { pi + 2k pi / k \in Z } B = {(- pi / 3) + (2k pi / 3 ) / k \in A } Prove that A C B Homework Equations A C B = \forallXE E : x \ni A \Rightarrow X \ni B The Attempt at a Solution \ni[k E Z ]: x = pi + 2k pi \ni[k E Z ]: x = pi ( 1 + 2k) I'm sure i need to...
  10. W

    Understanding the Limsup Operation in Set Theory

    Sorry if the title of my question is wrong, lim sup_n A_n = ∩^∞ U^∞ A_k does it mean that first we are taking the union of A_n and then we are taking intersection? actually I am confused, what is it actually?
  11. J

    Set Theory Question. Trouble defining a function precisely.

    Let A={1,...,n}. Show that there is a bijection of P(A) with the cartesian product Xn, where X is the two element set X={0,1} and P(A) is the power set of A. Below is the start of my proof. I just want to make sure that my function "makes sense." Proof: Let A={1,...n}, and X={0,1}. Define f...
  12. J

    Set Theory Proof. Inductive sets.

    Claim: If A is an inductive set of postive integers, then A is Z+. I tried to prove this two different ways for the fun of it. I would like to get some feedback concerning the correctness of both. Thank you. :-p Proof: By definition, Z+ is the intersection of all inductive subsets of ℝ. Since...
  13. B

    Proving Set Theory: A U B=(A-B) U (B-A)U (A∩B) | Homework Help"

    Homework Statement Let A and B be sets. Prove that A U B=(A-B) U (B-A)U (A∩B)Homework Equations The Attempt at a Solution If I want to prove the foward direction: A U B⊆(A-B) U (B-A)U (A∩B) then from my understanding I know that xεA or xεB. And I can assume wolog that xεA. But since I assume...
  14. J

    Solve Set Theory Question: Family of Subsets F of {1,2,..k}

    Homework Statement Hi everyone, i have the following problem, that for the moment i couldn't find the solution or even how to search about... This is it, we have a family of subsets F ={F_1,...,F_p} of the set {1,2,...,k}. We know that, for every i != j, F_i !=F_j and for every pair of...
  15. C

    Discrete Math Set Theory Question

    Let A, B, and C be sets. Show that a) (A-B) - C \subseteq A - C b) (B-A) \cup (C-A) = (B \cup C) - A I am using variable x to represent an element. Part A) I rewrote (A-B) - C as (x\inA ^ x\notinB) - C I think this could be rewritten as (x\inA ^ x\notinB) ^ x\notin C A-C can...
  16. N

    How Can a Set Both Belong to and Be a Subset of P(N)?

    Hi there, here's the question I am given, i will provide the answer that I think is correct, do you mind checking it and possibly pointing out where I am wrong if I am? Give an example of a set S such that: a) S is a subset P(N) b) S belongs to P(N) c) S belongs to P(N) and |S|=5 here...
  17. N

    Set Theory: P(P{1}) and its cardinality

    The notation has me a bit confused... Heres my logic for the P({1}) on the inside {EmptySet, {{1}}} reason being, you always include the empty set, {1} is a part of the set. The cardinality is two You have the set: {EmptySet, {{1}}}, and now you have to consider the outer "P" the...
  18. S

    Set Theory Problems: S1 U S2 = (S1' ∩ S2')' and S1 U S2 - (S1 ∩ S2') = S2

    Homework Statement show S1 U S2 = (S1' ∩ S2')' The Attempt at a Solution I'm pretty sure I have this right or I'm close Let x ∈ S1 U S2 x ∈ S1 or x ∈ S2 Since x ∈ S1 or S2, then x ∉ S1' and S2' If x ∉ S1' and S2', then x ∈ (S1' and S2')' Therefore, S1 U S2 = (S1' ∩ S2')' Homework Statement...
  19. C

    Relations, Set Theory, Reflexive, Symmetric, Transitive

    Homework Statement Determine whether the relations on three sets are Reflexive, Irrelfexive, Symmetric,, Asymmetric, Antisymmetric, Transitive, and Intransitive. The relation \subseteq on a set of sets. Homework Equations The Attempt at a Solution I am having trouble figuring out...
  20. B

    Basic Set Theory: Determining Relations: Reflexive, Symmetric, Transitive

    I am taking a philosophy course that covers basic set theory as part of the introduction. I’m not sure in which section of the forum set theory should be, but I think this is the right place. Homework Statement For each of the following relations, indicate whether it is Reflexive...
  21. H

    Set Theory Proof: Proving Identity (A U B) ∩ (B U C) ∩ (C U A)

    Hi, I have been trying for a very long time to prove the following set theory identity (A union B) intersect (B union C) intersect (C union A) = (A intersect B) union (B intersect C) union (C intersect A). I thought that I could simplify (A U B) intersect (B U C) intersect (C U A) as [B U...
  22. L

    Set Theory Basic Proof, showing two sets are equal

    Hello, I am trying to teach myself set theory...main problem is, as an engineer, mathematical proofs were never exactly stressed in my curriculum. (Scary, right?) The problem is stated as follows: "Prove the following, {x\inZ|for an integer y, x=6y}={x\inZ|for integers u and v, x=2u...
  23. T

    Does set theory serve as the foundation of ALL math?

    What, if anything, does set theory have to do with integrating x^2 or finding the center of symmetry for a polygon?
  24. S

    Discrete Mathematics - Basic Set Theory : Assignment review : Q2

    Question 2: -------------------- Homework Statement Consider the following sets, where U represents a universal set : U = {1, 2, 3, 4, ∅, {1}} A = {1, 3} B = {{1}, 1} C = {2 , 4} D = { ∅ , 1, 2 } Homework Equations A+D is the set : (Choose only one ) 1. {1, 3}...
  25. S

    Discrete Mathematics - Basic Set Theory : Assignment review : Q1

    Question 1 : -------------------- Homework Statement Consider the following sets, where U represents a universal set : U = {1, 2, 3, 4, ∅, {1}} A = {1, 3} B = {{1}, 1} C = {2 , 4} D = { ∅ , 1, 2 } Homework Equations Choose the correct option : D - B is the set ...
  26. A

    Can Strong Induction Prove an Element in Set S Matches Its Own Cardinality?

    I am interested in a little fun... It has been some time since I have done any set theory, and I am forgetting the symbols and ideas... So, I wanted to construct a strong inductive proof, and get a bit of help with how to concisely write the proof, and how to get TEX here at the forums to...
  27. S

    Set Theory : Discrete Mathematics

    Homework Statement The Question data is as follows : Consider the following sets, where U represents a universal set : U = {1, 2, 3, 4, ∅, {1}} A = {1, 3} B = {{1}, 1} C = {2 , 4} D = { ∅ , 1, 2 ) Homework Equations Which one of the following statements is true ? 1. The...
  28. G

    Applications of Set Theory in engineering?

    Hey everyone, I'm currently taking a mathematics course in set theory at my university as a general elective. I was wondering does set theory have any practical applications in engineering?
  29. M

    Does ZFC Imply the Power Set of Naturals?

    Is it true that for every standard formulation T of ZFC, T ⊢ the power set of {naturals}? After all, the empty set axiom and the pairing axiom are in T, and so we get N. Then by the power set axiom we get P(N).
  30. S

    Set theory problems from Hrbacek, Jech

    Homework Statement I'm attempting some self-study in set theory using the text mentioned above. The exercises here are quite different from those in previous texts which I've used, so I was hoping I could present some of my attempts (so far, only from the first problem set) and receive some...
  31. P

    Set Theory: Proving D has 2^d Subsets of Cardinality d

    Let D be a set that has cardinality d WTS that D has 2^{d} subsets of cardinal number d. So I was thinking about slitting D into two sets C_{1} and C_{2} both of cardinality d. From there I think that there are d^{d} subsets that contain C_{1}. Since d is an infinite cardinal d^{d}=2^{d}...
  32. S

    MHB Exploring Basic Set Theory: Cardinalities and Operations with Set Elements

    HI there, Just getting into set theory just had a few questions/clarifications I guess you could call it. if X = {Ø, a} Y={{Ø}, a} and Z = {a, {a}, {Ø}} So i understand X has 2 elements along with Y and Z has 3. I know what the cardinalities are of basic sets like {2, 3, 4, 5} etc but how do...
  33. C

    Graphing Cartesian Products and Unions: Set Theory Sketches

    So the book asks me to sketch out these graphs, and of course there are no examples. I was wondering how this is done. (a) [0,1] X [1, 2] // The X here stands for the Cartesian product. (b) ([0,1] U {2}) X [1,2] // How can I graph this? The U stands for Union and the X here stands for...
  34. C

    Discrete math set theory sum problem

    Homework Statement Prove that if k>1 then, 5/(k-1)-3/k-2/(k+2) = (9k+6)/(k-1)k(k+2) Hence simplify Ʃ of k=2 to n {(3k+2)/(k-1)k(k+2)} Homework Equations The Attempt at a Solution Ok so the first part is ok I just multiplied the denominators with the numerators and...
  35. R

    Is there a theory of probability based on fuzzy set theory?

    One thing I've wondered for quite some time, and looked for, but not found anything I consider adequate is whether there is a theory of probability based on Zadeh's fuzzy set theory. The closest I've found is Zadeh's Perception Probability Theory, but this doesn't quite seem to cut it...
  36. C

    Set theory: Largest number in the set

    Homework Statement How to write "the largest number in a set of real numbers A" using the appropriate set theory notation? Homework Equations - The Attempt at a Solution Tried Googling and searching on Wikipedia with no relevant results.
  37. I

    Can A be a subset of C if it's disjoint from B?

    Let A, B and C be sets. Prove that if A\subseteqB\cupC and A\capB=∅, then A\subseteqC. My attempted solution: Assume A\subseteqB\cupC and A\capB=∅. Then \veex (x\inA\rightarrowx\inB\cupx\inc). I'm not sure where to start and how to prove this. Any help would be greatly appreciated. Thank you.
  38. teroenza

    (0,0) Empty Interval? Basic Set Theory

    Homework Statement Give an example of an indexed family of sets such that the intersection of any finite subfamily is not empty, but the intersection when the index=infinity, is empty. The Attempt at a Solution The family I came up with is the exclusive interval (-1/k , 1/k) where...
  39. J

    Is This the Correct Power Set for X={S,{S}}?

    Homework Statement Is this the right power set for x? Homework Equations X={S,{S}} The Attempt at a Solution P(x)={ø, {S}, {{S}}, {S,{S}}}
  40. G

    Proving A ⊆ B using Set Theory

    How would I prove A \subseteq B \Leftrightarrow A \cap B^{c} = \emptyset ?
  41. B

    Understanding Subset Relations in Set Theory

    Homework Statement if A is a subset of B, then Bc \ C is a subset of Ac \ C for any set C Homework Equations A is a subset of B = for all elements in A are also elements in B A\B = the complement of a set B in a set A A\B= A and Bc The Attempt at a Solution I tried...
  42. D

    What is the difference between an element and a subset in set theory?

    I'm attempting to teach myself topology from a textbook. I'm on the first chapter and came into some trouble with some of the set theory. Here is what the textbook says. We make a distinction between the object a, which is an elemant of a set A, and the one-element set {a}, which is a...
  43. P

    Understanding Set Theory: Explanation and Examples for Beginners

    Hi guys, I someone help me to set my logic straight and help me understand the following situation. For three I was given the following definition: P(A ∪ B ∪ C) = P(A) + P(B) + P(C) − P(A ∩ B) − P(A ∩ C) − P(B ∩ C) + P(A ∩ B ∩ C). I don't understand why we need to + P(A ∩ B ∩ C)...
  44. L

    What is the best way to prove basic set theory statements?

    Homework Statement I'm working on some set theory stuff to prepare for Topology next semester. I'm actually working out of a Topology book from Dover Publications. I could really use some direction/correction. 1. If S ⊂ T, then T - (T - S) = S. 2. If S is any set, then ∅ ⊂ S. The...
  45. T

    I can't think of a counterexample to disprove this set theory theorem

    I can't think of a counterexample to disprove this set theory "theorem" Assume F and G are families of sets. IF \cupF \bigcap \cupG = ∅ (disjoint), THEN F \bigcap G are disjoint as well.
  46. P

    Student Learning Set Theory Independently

    Hi, I am a high school student, and I am trying to learn set theory (it's not for school - independent study; I love it). I have a book (Takeuti/Zaring Introduction to Axiomatic Set) Theory and I am going through it, but I feel like I'm going way to slowly (after trying to go through it for...
  47. J

    Set Theory vs. EM2: Deciding My Courses

    Hello all, I'm trying to decide my courses for next semester and I'm all set except for a choice between EM2 and set theory. I've had both teachers before, and the set theory teacher is a lot better. I need EM2 for my major, but I'm afraid I'll be wasting the opportunity to take a fun class...
  48. B

    Are All Non-Women Engineers? Investigating the Validity of a Subset Statement

    Homework Statement Given the following four statements concerning the student body at CU: ... b) There are no women engineering students at CU ... Homework Equations n/a The Attempt at a Solution Let W be the set of all women Let E be the set of all engineering students...
  49. K

    Would it have been possible to discover calculus from set theory?

    Or does calculus rely heavily on graphs for it's discovery to occur? Would it be possible to have looked at the functions on the graph as sets mapping from one A --->B? Or would a mathematician have to have insane intuition and crazy in them to discover this?
  50. U

    Understanding some basic set theory stuff.

    This is more to see if I understand it or not. There are four statements and I need to explain why they are true. (they all are) I understand it why some of they are, but my answers just don't feel accurate/formal enough. Homework Equations 1) \mathbb{R}^3 \subseteq \mathbb{R}^3 2)...
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