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.

View More On Wikipedia.org
Recent contents Top users
  1. Y

    Set Theory Question: a ∩ b ⊆ a

    Homework Statement I'm reviewing my powerpoints from class and see the formula A ∩ B ⊆ A. Is this a correct formula? I interpret this as all elements of set A intersected with set B is a subset of set A. I don't think this is a true statement, is it? Sorry it's been a while since I have...
  2. A

    Understanding the or in Set Theory

    Homework Statement So I was doing this problem in Munkres's Topology book: Determine whether the statement is true or false, If a double implication fails, determine whether one or the other of the possible implications holds: A ⊂ B or A ⊂ C ⇔ A ⊂ ( B ∪ C ) Homework Equations - The Attempt...
  3. Avatrin

    Overcoming Abstraction in Mathematics

    Hi As I am venturing in graduate level mathematics, I am having a recurring problem; I keep getting stuck in the abstraction of it. Usually it involved set theory; I never get "fluent" in it. However, the main problem is abstraction. For instance, this semester I had topology, and the...
  4. Math Amateur

    MHB Set Theory and ZFC - The Axiom of Replacement - Searcoid, Pages 6-7

    I am reading Micheal Searcoid's book: Elements of Abstract Analysis ( Springer Undergraduate Mathematics Series) ... I am currently focussed on Searcoid's treatment of ZFC in Chapter 1: Sets ... I am struggling to attain a full understanding of the Axiom of Replacement which reads as shown...
  5. Math Amateur

    MHB Set Theory and ZFC - The Subset Principal - Searcoid, Page 7

    I am reading Micheal Searcoid's book: Elements of Abstract Nalysis ( Springer Undergraduate Mathematics Series) ... I am currently focussed on Searcoid's treatment of ZFC in Chapter 1: Sets ... I am struggling to attain a full understanding of the Subset Principle which reads as shown below...
  6. mmarkov

    Markov chains: period of a state

    Hello, I am trying to understand the intuition of the definition of the period of a state in a Markov chain. Say for example we can go from state i to state i in 3,5,7,9,11... and so on steps. The gcd here is one. So is this aperiodic state or one with periodicity of 2. Thanks
  7. Gh. Soleimani

    I Are These Rules new Conjectures in Set Theory?

    We can easily find out below rules in set theory: 1. Let consider set “A” as follows: A = {a1, a2, a3, a4… an} and also power set of A is set C: C = {{}, {a1}, {a2}, {a3}, {a4}, {a1, a2}, {a1, a3},….{an}} Rule 1: To find the number of subsets with precise members number, we can use Binomial...
  8. J

    Looking for book on ZFC axiomatic set theory (I think)

    I am looking for a book that starts at the standard ZFC axioms and progresses to the point where some recognisable non-trivial mathematical statement is proved. By recognisable I mean something that you may encounter in school/early university level and is not purely set-theoretical (e.g...
  9. Z

    Real Analysis - Infimum and Supremum Proof

    Hi Guys, I am self teaching myself analysis after a long period off. I have done the following proof but was hoping more experienced / adept mathematicians could look over it and see if it made sense. Homework Statement Question: Suppose D is a non empty set and that f: D → ℝ and g: D →ℝ. If...
  10. W

    Set Theory, Rough Set Theory, Fuzzy Set Theory

    Hi. I hope this is not too far into philosophy. Set Theory is commonly accepted as the foundations of Mathematics. Is it possible to develop a different type of Mathematics by using Fuzzy sets or Rough sets instead?
  11. W

    Probability and Random Experiments

    Homework Statement Problem Consider a random experiment with a sample space S={1,2,3,⋯}. Suppose that we know: P(k) = P({k}) = c/(3^k) , for k=1,2,⋯, where c is a constant number. Find c. Find P({2,4,6}). Find P({3,4,5,⋯}) I am primarily interested in part 1, finding C. The rest...
  12. W

    Probability, Set Theory, Venn Diagrams

    Homework Statement Let A and B be two events such that P(A) = 0.4, P(B) = 0.7, P(A∪B) = 0.9 Find P((A^c) - B) 2. Homework Equations I can't think of any relevant equations except maybe the Inclusion Exclusion property. P(A∪B) = P(A) + P(B) - P(A∩B) This leads us to another thing P(A∩B^c)...
  13. DeldotB

    Show a functions inverse is injective iff f is surjective

    Hello all, Can anyone give me a pointer on how to start this proof?: f:E\rightarrow F we consider f^{-1} as a function from P(F) to P(E). Show f^(-1) is injective iff f is surjective.
  14. DeldotB

    Set Theory: Prove a function is injective

    Homework Statement Hello, I need some help on the following. I am BRAND new to set theory and this was in my first HW assignment and I am not sure where to start. The question is as follows: Let A and B be parts of a set E Let P(E)\rightarrow P(A) X P(B) be defined by f(X)=(A\cap X,B\cap X)...
  15. R

    Exploring the Controversy Around the Axiom of Choice

    Hi guys. So I've been wondering, what's so controversial about the axiom of choice? I heard it allows the Banach-Tarski Paradox to work. A little insight would be much appreciated, thanks.
  16. M

    Does Intersection Distribute Over Subset in Set Theory?

    Homework Statement $$ A \subset B \Rightarrow A \cap C \subset B \cap C $$ 2. Homework Equations [/B] $$ A \subset B \Leftrightarrow A \cup B \subset B$$ $$ A \cap C \Leftrightarrow A \cap C \subset A \wedge A \cap C \subset C$$ The Attempt at a Solution For sets A and C $$A \cap C...
  17. T

    Are set theory functions sets too?

    I read somewhere that mathematical functions can be implemented as sets by making a set of ordered tuples <a,b> where a is a member of A and b is a member of B. That should create a function that goes from the domain A to the range B. But set theory has functions too, could they be sets too...
  18. O

    What is the situation of relational algebra?

    Hi! I would like to know if my assumptions are right: Topology is the merging domain of analysis and algebra; Relational algebra use topological operators; Relational algebra is a specification of topology ?
  19. Ryanodie

    Check my work on these proofs? (basic set theory)

    Homework Statement [/B] I am going through Apostol's Calculus volume 1 and am working through I 2.5 #3. I'm not very familiar with doing proofs so I just wanted to make sure that I got the right idea here. Here's the question: Let A = {1}, B = {1,2} Prove: 1. ## A \subset B ## 2. ## A...
  20. B3NR4Y

    Simple Set Theory Proofs to Proving Set Identities - Homework Help

    Homework Statement 1. Prove that if A \cap B = A and A \cup B = A , then A = B 2. Show that in general (A-B) \cup B \neq A 3. Prove that (A-B) \cap C = (A \cap C) - (B \cap C) 4. Prove that \cup_{\alpha} A_{\alpha} - \cup_{\alpha} B_{\alpha} \subset \cup_{\alpha} (A_{\alpha} -...
  21. B

    Set theory, intersection of two sets

    Homework Statement We have the set D which consists of x, where x is a prime number. We also have the set F, which consists of x, belongs to the natural numbers (positive numbers 1, 2, 3, 4, 5..) that is congruent with 1 (modulo 8). What numbers are in the intersection of these two sets...
  22. phoenixthoth

    A method for proving something about all sets in ZFC

    I would appreciate any and all feedback regarding this document currently housed in Google docs. Basically, I generalize induction among natural numbers to an extreme in an environment regarding what I call grammatical systems. Then an induction principle is derived from that which holds in...
  23. C

    Solving the Relation: ##n((AXB) \cap (BXA)) = n(A \cap B)^2##

    Homework Statement If I am given ##n(A)## and ##n(B)## for two sets A and B, and also provided with ##n(A\cap B)^2##. We are supposed to find ##n((AXB) \cap (BXA))##. Homework Equations My teacher said that the formula for ##n((AXB) \cap (BXA)) = n(A \cap B)^2##. I am not sure how do you get to...
  24. heff001

    Pure Mathematics study - question

    I am planning to study the following pure mathematics areas (on my own) and wanted to know if this is the best sequence: 1- Formal Logic 2 -Philosophical Logic 3- Sentential Logic 4- Predicate Logic 5- Symbolic Logic 6 -Set Theory 7 -Pure Mathematics (Intro, Pure Math I and II and Hardy) -...
  25. UncertaintyAjay

    Prob/Stats Axiomatic Set Theory: Book Recommendations?

    So I just finished "Book of Proof" and I'm looking for a more rigourous book on axiomatic set theory, including Gödel's theorems.Any recommendations?
  26. K

    Are Projection Mappings considered Quotient Maps?

    The book I am using for my Introduction to Topology course is Principles of Topology by Fred H. Croom. Problem: Prove that if ##X=X_1\times X_2## is a product space, then the first coordinate projection is a quotient map. What I understand: Let ##X## be a finite product space and ##...
  27. M

    Researching Set Theory as an Undergrad

    I'm currently an undergrad in math who's going to graduate next year. I'm interested in doing research in set theory (not now of course, perhaps in grad school). Unfortunately, I'm at a liberal arts school and there are no set theorists in the math department. All they offer is a naiive set...
  28. AlephNumbers

    Solve Set Theory Question: Can Countably Infinite Set Have Uncountable Subsets?

    Lately I have been attempting (and failing miserably at) whatever sample Putnam questions I can find on the internet. Here is my latest endeavor. I found this question on the Kansas State University website, so I think I am allowed to post it. I must warn you that I know almost nothing about...
  29. xwolfhunter

    Question about empty sets in set theory

    So I'm reading Naive Set Theory by Paul Halmos. He asks: His response is that no ##x## fails to meet the requirements, thus, all ##x##es do. He reasons that if it is not true for a given ##x## that ##x \in X~ \mathrm{for ~ every} ~X~ \mathrm{in} ~ \phi##, then there must exist an ##X## in...
  30. xwolfhunter

    Question about "or" in set theory

    So I'm reading up on some set theory, and I came to the axiom of pairing. The book uses that axiom to prove/define a set which contains the elements of two sets and only the elements of those two sets. ##~~B## is the set which contains the elements and only the elements of sets ##a## and...
  31. M

    Initial development of set theory and determinism in QM

    I am considering the following question and I want you to agree (but perhaps you don’t):Rutherford wrote a letter to Bohr, as an answer to a previous letter from Bohr containing one of the first of Bohr’s descriptions of the atomic model, saying that he understood the atom model Bohr advocated...
  32. D

    Confusion over definition of relations in set theory

    I'm coming from a physics background, but find pure mathematics extremely interesting, so have decided to try and gain a more fundamental understanding of the subject. I've recently been reading up on relations and how one can define them as sets of ordered pairs. I am particularly interested in...
  33. Marco Lugo

    Help with Set theory, compund statements

    The class is called Math for EE and CE. The professor teaches from his own notes and doesn't give many examples. Any help checking my work would be appreciated and/or if you could point me in the direction of more examples like these. I've looked trough Set Theory and discrete math books but...
  34. D

    Difference between equivalence and equality

    Apologies if this is in the wrong forum, but I chose to post here as the question pertains to equivalence relations and classes. Sorry if it's such a trivial question, but what is the mathematical difference between equivalence and equality? My understanding is the following, but I'm a little...
  35. E

    Show that the set S is Closed but not Compact

    Homework Statement Show that the set S of all (x,y) ∈ ℝ2such that 2x2+xy+y2 is closed but not compact. Homework Equations set S of all (x,y) ∈ ℝ2such that 2x2+xy+y2 The Attempt at a Solution I set x = 0 and then y = 0 giving me [0,±√3] and [±√3,0] which means it is closed However, for it to...
  36. E

    Are the following Sets: Open, Closed, Compact, Connected

    Homework Statement Ok I created this question to check my thinking. Are the following Sets: Open, Closed, Compact, Connected Note: Apologies for bad notation. S: [0,1)∪(1,2] V: [0,1)∩(1,2] Homework Equations S: [0,1)∪(1,2] V: [0,1)∩(1,2] The Attempt at a Solution S: [0,1)∪(1,2] Closed -...
  37. E

    Closed/Open Sets and Natural Numbers

    Homework Statement I am trying to understand why ℕ the set of natural numbers is considered a Closed Set. 2. Relevant definition A Set S in Rm is closed iff its complement, Sc = Rm - S is open. The Attempt at a Solution I believe I understand why it is not an Open Set: Given that it...
  38. S

    Sample spaces, events and set theory intersection

    Homework Statement Problem: Given a regular deck of 52 cards, let A be the event {king is drawn} or simply {king} and B the event {club is drawn} or simply {club}. Describe the event A ∪ B Solution: A ∪ B = {either king or club or both (where "both" means "king of clubs")} Homework Equations...
  39. Prof. 27

    Understanding Nested Ordered Pairs: Properties & Examples

    Homework Statement One way of modeling tuples in set theory is through nested ordered pairs. A notation I'm not familiar with (I'm assuming it means that the following elements are nested into the last one) is used. (a1, a2, a2,... an) = (a1(a2, a3,..., an)). I have never seen the second "(" in...
  40. S

    Is the Subtraction of Power Sets Possible?

    I have two quick questions: With P being the power set, P(~A) = P(U) - P(A) and P(A-B) = P(A) - P(B) I'm told if it's true to prove it, and if false to give a counterexample. To be they're both false, since the null set is part of any power set, the subtraction of two power sets would get...
  41. Dethrone

    MHB Is this a well-formed set-builder notation?

    Are these three sets equivalent? $$A=\left\{(x,y):x,y\in\Bbb{R},y\ge x^2-1\right\}$$ $$B=\left\{x,y\in\Bbb{R}:y\ge x^2-1\right\}$$ $$C=\left\{(x,y)\in\Bbb{R}^2:y\ge x^2-1\right\}$$ I am thinking that $A$ and $C$ are, but not $B$ as it might be ambigious as to which dimension it is in, i.e it...
  42. Blackberg

    Introducing Set Theory: Proving Real #s Identical in Bases

    I'm introducing myself to set theory. My reference doesn't seem to address the fact that 1/1 = 2/2 = 1. If we make a correspondence between natural numbers and rational numbers using sequential fractions, should we just skip equivalent fractions so as to make it a bijection? In other words, does...
  43. S

    Proving least upper bound property implies greatest lower bound property

    Homework Statement Prove if an ordered set A has the least upper bound property, then it has the greatest lower bound property. Homework Equations Definition of the least upper bound property and greatest lower bound property, set theory. The Attempt at a Solution Ok, I think that my main...
  44. UncertaintyAjay

    Books on Set Theory: Recommendations & Reviews

    Could anyone recommend some good books on set theory?
  45. Schnurmann

    Least Squares Estimation - Problem with Symbols

    Hi folks, 1. Homework Statement I don't fully understand the question statement, how is it supposed to be read? Question: Give a formula for the minimizer x* (to be read as x-star) of the function ƒ:ℝn → ℝ, x → ƒ(x) = ||Ax-b||22, where A∈ℝm×n and b∈ℝm are given. You can assume that A has rank...
  46. S

    Proving a function is bijective

    Mod note: Moved from a technical section, so missing the homework template. Here is what I'm trying to prove. Let f:A->B. If there are two functions g:B->A and h:B->A such that g(f(a))=a for every a in A and f(h(b))=b for every b in B, then f is bijective and g=h=f^(-1). I think I have most...
  47. S

    Question on testing logical truths for set operations

    My question is on how to answer if two statements are equal in set theory. Like De'Morgans laws for example. I'm currently reading James Munkres' book "Topology" and am working through the set theory chapters now, and this isn't the first time I've seen the material, but every time I see this...
  48. B

    Solving Discrete Math Question: Proving ∪n=2∞[0,1 - 1/n] = [0,1)

    Homework Statement Show that, ∪n=2∞[0,1 - 1/n] = [0,1) Homework EquationsThe Attempt at a Solution
  49. H

    ZF Set Theory and Law of the Excluded Middle

    Hello, I know that the law of the excluded middle is implied in ZFC set theory, since it is implied by the axiom of choice. Taking away the axiom of choice, does ZF set theory (with axioms as stated in the Wikipedia article http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory)...
  50. evinda

    MHB Introduction to Set Theory: Fundamentals, Construction, and Arithmetic

    Hello! (Wave) What is the subject Set Theory about? What knowledge is required? (Thinking) That is the Course Content: Brief report on basic elements (algebra of sets, relations and functions, etc..). Construction of the set of natural numbers. Ordinal numbers and their arithmetic. The...
Back
Top