What is Set theory: Definition and 442 Discussions

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. H

    Set Theory Problem: equality of functions

    Homework Statement Prove: Let F and G be functions. F = G if and only if domF = domG and F(x) = G(x) for all x \in domF. Homework Equations ? The Attempt at a Solution If F = G, then (xFy if and only if xGy) (Substitutivity of identicals) If (xFy if and only if xGy), then...
  2. G

    Struggling with Set Theory Proofs? Any Tips?

    Hello All, I am taking a Discrete math course and am having trouble with set theory proofs. I can do basic ones, like prove <a,b,c>=<u,v,w> if a=u, b=v, and c=w. But as soon as it changes even a little bit, I cannot prove it. I was wondering what tips some of you might suggest to me? I do...
  3. S

    Basic Set Theory (Indexed Collection of Sets)

    Homework Statement Give an example of an indexed collection of sets {A_{\alpha} : \alpha\in\Delta} such that each A_{\alpha}\subseteq(0,1) , and for all \alpha and \beta\in\Delta, A_{\alpha}\cap A_{\beta}\neq \emptyset but \bigcap_{\alpha\in\Delta}A_{\alpha} = \emptyset.Homework Equations...
  4. F

    Set Theory By Naylor and Sell Homework Problems

    Homework Statement I am trying understand the approach to the problems given in the following textbook. Linear Operator Theory in Engineering and Science (Applied Mathematical Sciences) (Volume 0) [Paperback] Arch W. Naylor (Author), George R. Sell (Author). Pages 12-29. There are excercise...
  5. C

    Can Proofs Involving the Empty Set Be Solved by Contradiction?

    I am doing some non-homework exercises in preparation for my midterm, and am struggling with the following proofs: First Prove {} is a subset of {}, where {} refers to an empty set My professor told me to do this by contradiction. So I assume that {} is not a subset of {}. That would imply...
  6. R

    Set Theory: Basic Language

    I'm reading the book Basic Set Theory by: Azriel Levy as I thought it might help me better understand Group Theory and Matrix Math. I have read the first chapter a number of times but I keep getting hung up on some of the syntax of the basic language or language of first-order predicate...
  7. C

    Set Theory: Need help understanding the question

    Homework Statement Let T be a family of finite subsets of the natural numbers N = {1, 2, 3,...} such that if A and B are any members of T, then the intersection of A and B is nonempty. (a) Must N contain a finite subset F such that the intersection of A, B and F is nonempty for any sets A...
  8. R

    Question about set theory and ordered pairs

    Hi I was reading through a textbook and I came across the set theoretic definition of an ordered pair (Kuratowski), where (x,y)={{x},{x,y}}, which apparently can be shortened to {x,{x,y}}. This seems to be the standard definition for an ordered pair in set theory so that we can determine...
  9. R

    Is the Definition of an Ordered Pair Set Theoretic?

    Homework Statement Determine whether or not the following definition of an ordered pair is set theoretic (i.e. you can distinguish between the "first" element and the "second" element using only set theory). (x,y)={x,{y}} Homework Equations The Attempt at a Solution I am inclined to think...
  10. I

    Equivalence Relations in Set Theory: Homework Statement and Solutions

    Homework Statement Homework Equations An equivalence relation on a set A, is for a,b,c in A if: a~a a~b => b~a a~b and b~c => a~cThe Attempt at a Solution It seems uncomplicated, but I don't know how I would write down a proof. The book I'm using is Topics in Algebra, 1st Edition Herstein
  11. N

    Set theory - set builder notation

    Homework Statement X={8^n-7n-1/n belongs to N} Y={49(x-1)/x belongs to N} Homework Equations Then, x is subset of y ] or y is subset of x or x=y,none of these
  12. S

    Set Theory and Circumference of a Circle

    Hey friends! How can the set of all points on the circumference of a circle be an Infinite Set? Anyone could explain? Thanks in advance! -Saphira :)
  13. ╔(σ_σ)╝

    Please help me understand a set theory proof.

    Homework Statement Let A be a set and let P(A) denote the set of all subsets of A. Prove that A and P(A) do not have the same cardinality Homework Equations NOne The authors proof is as follows: Suppose there is a bijection f:A -> P(A). Let B= {x\in A| x \notin f(x)}. There is a y \in A...
  14. S

    Proving Basic Set Theory: Trichotomy, Union, Intersection, and Multiplication

    At first glance these things seem so intuitive and familiar from other maths (like distribution) that I don't see how/where to start in proving them; while I know its probably quite simple. I understand what union and intersection are, but I'm unsure if multiplying two sets means a new set with...
  15. B

    What is the purpose of studying set theory?

    What is the great mathematical utility of set theory? When you study set theory or when you studied do(did ) you know for what ecxacly you are(were) studying?And what do you know now?
  16. S

    Set Theory Identities: A = B if A, B, and C satisfy key set relations

    Homework Statement Can you conclude that A = B if A, B, and C are sets such that A \cup C = B \cup C and A \cap C = B \cap C Homework Equations The above is part c of a problem. The problems a and b are as follows A) A \cup C = B \cup C My answer: I gave a counter example...
  17. R

    Proving Set Theory Union in Cartesian Products

    Homework Statement Suppose A,B,C are sets. Prove that A× (B U C)= (AxB) U (C x A)
  18. J

    Recursion theorem (set theory)

    There are probably a million theorems called "the recursion theorem" and I'm not sure if this is actually one of them, but there's a remark saying that it defines recursion. It's proven by piecing together 'attempts' (functions defined on subsets of a domain) and states: For X a well-ordered...
  19. J

    Proving Set Theory Equality: How to Use Sentential Calculus Rules

    Hi Everyone! I would really appreciate some help with this set proof. Homework Statement Show that, for any sets A, B, ((A ∪ B) ∩ A) = (A ∩ A') ∪ (A ∩ B). (Hint: Remember that a complement of a complement is just the original set.) Homework Equations I can use any sentential...
  20. B

    Help with some set theory questions

    Greetings all. I was wondering if someone could give me some asistance on a few simple set theory proofs. 1) An(B\C)=(AnB)\(AnC) 2) A+(B+C)=(A+B)+C where + = symmetric difference 3) If A+B = A+C then B=C and 4) An(B+C) = (AnB)+(AnC)
  21. I

    Check my proof for cartesian product (set theory)

    Homework Statement Prove that \forall sets A, B, C , if B\subseteq C, then A \times B \subseteq A \times CHomework Equations The Attempt at a Solution Haven't done set theory proofs in a while. Does this suffice in proving the statement?: Let x \in B be arbitrary. Assuming B \subseteq C...
  22. C

    Set Theory: Prove the set of complex numbers is uncountable

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  23. D

    Prove property of classes (introductory set theory)

    this problem is from Apostol's Calculus Vol.1, i just started doing proofs so I'm still getting used to it. B - U (A) = ∩ (B-A) the U is the union of sets in a class F and ∩ is the intersection of sets in a class F. written under both the U and the ∩ are A∈F. so i let an element x ∈ B...
  24. C

    Solving a Set Theory Dilemma

    Homework Statement Let A,B,C be sets where A n C = B n C and A n Cc = B n Cc. Then A=B. Homework Equations The Attempt at a Solution To prove two sets equal, i think we want to let x be in A, and then show as a result that x in B also. However, i don't see how this is possible...
  25. S

    A proof in Set theory about the number of different subset in a Set

    hi guys, I am a physics major, recently doing a few self-reading on analysis, however, I got stuck at some excises of proofs. Homework Statement If a set A contains n elements, prove that the number of different subsets of A is equal to 2^n. The Attempt at a Solution First, I teared...
  26. C

    Set Theory (Proof): Show E is Open/Closed

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  27. T

    Set theory - expression describing subset

    Homework Statement I have two sets E and L. I have to show that E is a subset of L, using only complement, union and the empty set. I.e. all members of E also have to be members or L, but L may contain members not included in E.Homework Equations The Attempt at a Solution Am I right in...
  28. D

    Simple set theory question

    Homework Statement We're working more or less with the standard ZF axioms. Prove that A \subseteq \mathcal{P}(\bigcup A) for any set A, whose elements are all sets. When are they equal? Homework Equations Just the axioms I) Extensionality II) Emptyset and Pairset III) Separation IV)...
  29. H

    Proving Pigeonhole Principle in Set Theory

    Been working on this for 2 days and have gotten no where. Maybe someone out there can show me the light U = {1, 2, 3, ... , n, ..., 2n} for some natural number n Let P be a subset of U such that |P| = n + 1 show that there exists x, y in P where x not equal y such that x divides y or y...
  30. O

    Set theory: Axioms of Construction

    Homework Statement We're asked to prove that a few constructions of the sets a,b are themselves sets, stating which axioms we use to do so. a) a\b b)the function f:a->b c)the image of f Homework Equations The following standard definitions of axioms of construction...
  31. heff001

    Collection of Formulas of Set Theory (Symbols)

    In 'AN INTRODUCTION TO SET THEORY' by Professor William A. R. Weiss, the following symbols are applied to the snippet below: ... The collection of formulas of set theory is defined as follows: 1. An atomic formula is a formula. 2. If ϕ and Ψ are formulas, then (ϕ ^ Ψ) is also a formula. ... I...
  32. A

    Prerequisite Before Learning Set Theory

    I am in calculus, I was just wondering what math classes (or prerequisites) before trying to learn set theory. For example Should I learn multivariable calculus, linear algebra, etc. (or what)?
  33. O

    Set Theory proof on well ordered sets

    Homework Statement Without using the Axiom of Choice, show that if A is a well-ordered set and f : A -> B is a surjection to any set B then there exists an injection B -> A. Homework Equations The Attempt at a Solution I was wondering if the existence of the surjection from a well...
  34. phoenixthoth

    Conversion of Set Theory Problems to other fields

    Is there a way to prove axioms are consistent and/or independent by converting the problem of consistency/independence to another field, such as algebra?
  35. C

    What is a proof of set theory problems?

    Problem: (i)A\subseteqB \Leftrightarrow A\cupB = B (ii) A\subseteqB \Leftrightarrow A\capB = A and For subsets of a universal set U prove that B\subseteqA^{c} \Leftrightarrow A\capB = empty set. By taking complements deduce that A^{c}\subseteqB \Leftrightarrow A\cupB = U. Deduce that...
  36. C

    Proff of half infinite intervals through set theory

    Problem: We define half infinite intervals as follows: (a, \infty) = {x\in R | x>a}; [a, \infty) = {x\in R | x\geqa}; Prove that: (i) (a, \infty) \subseteq [b, \infty) \Leftrightarrow a\geqb, (ii) [a, \infty) \subseteq (b, \infty) \Leftrightarrow a>b. I've got pretty much no idea how...
  37. M

    What are subsets and proper subsets in set theory?

    Why does "Set theory" exist? Is there actually a use for Set theory? I understand why Algebra, Calculus, Trig, Physics, etc.. exist. Does anyone ever use it at their job (besides at some type of school).
  38. F

    Explore Set Theory and Logic with Stoll: A Beginner's Guide"

    Hi, Right now I am currently going through Set Theory and Logic by Stoll. This is my first time going through Set Theory, Logic, and everything else in the book. I feel like I'm not getting all I can out of this book because I don't feel like I'm on the same level, I struggle to answer the...
  39. T

    Set Theory and Predicate Calculus?

    Set Theory and Predicate Calculus (12 points) Given: P ⊆ Q Q ⊆ (S ∩ T) S ⊆ (R ∪ T^c) x(sub)1 ∈ P Use predicate calculus to prove x(sub)1 ∈ R. Studying for a test but I don't have this worked out for me. I honestly don't even know where to start. I know what union, intersect, etc and...
  40. T

    Power set (set theory) question

    I'm reading a book about set theory and it introduced the concept of power set. Ok, I understand what is a power set and the entire concept but I have a question about the number of elements of a power set. There's written in the book that the number of elements of a power set is 2n where n...
  41. M

    Set theory and analysis: Cardinality of continuous functions from R to R

    Homework Statement Prove the set of continuous functions from R to R has the same cardinality as RHomework Equations We haven't done anything with cardinal numbers (and we won't), so my only tools are the definition of cardinality and the Schroeder-Bernstein theorem and its consequences. I...
  42. E

    Definition of a function in NBG set theory

    Hi, I have a situation where I want to define a function over all possible rings. For example, I would like to define a function that accepts a ring as an input and returns its additive identity. However, this seems impossible to do in ZF set theory since we can not define the domain of...
  43. K

    Set theory cardinality question

    Can anyone please give a really explicit proof (omitting no steps) and with as simple words as possible that any infinite set can be writtern as the union of disjoint countable sets? Thank you.
  44. D

    Set theory, functions, bijectivity

    f:X\rightarrowY, A\subsetX f(Ac)=[f(A)]c implies f bijective. Just trying to apply the definitions of injective and bijective. The equivalence makes sense but I am having a hard time showing it. f(x)=f(y) implies x=y and for every y in Y there exists a x in X s.t. f(x)=y. I mean all I have is...
  45. D

    Set Theory Proof: Proving Accumulation Point is Interior/Boundary

    Homework Statement I have been asked to prove the following statement: An accumulation point of a set S is either an interior point or a boundary point of S. Here is my attempt at a solution: I started from the definition of accumulation points: A point is an accumulation point (x_0) when...
  46. J

    Set theory & equiv classes

    How do you write in proper set theoretic notation that a set A = (x,x) where x is a non-negative real number? Also, (x_1, y_1) R (x_2, y_2) if x_1 ^2 + y_1 ^2 = x_2 ^2 + y_2 ^2 The equiv. classes are circles at (0,0), right? How do you write this formally (using set theoretic notation)?
  47. J

    Set Theory Q: Show A∩⋃ⁿᵢ=1Bᵢ = ⋃ⁿᵢ=1(A∩Bᵢ)

    Letting A, B_1, B_2, ..., B_n subsets of X, then show A\cap\bigcup_{n}^{i=1}B_{i} = \bigcup_{n}^{i=1}\left(A \cap\right B_{i}) ---- Is it sufficient to say... By De Morgan law, we have \left(A\cap \right B_{i})\cup\left(A \right \cap\ B_{2})\cup\ --- \cup\left(A\cap \right\ B_{n}) =...
  48. R

    Proving Set Theory: A⊆B Equivalence

    Hi, Im only starting to learn about naive set theory from a book , so pardon me if my answer to the question is really obvious.. Prove that .. A\subseteqB , if and only if A\capB =A,if and only if A\cupB=B, if and only if A-B=empty set.. I was thinking of using venn diagrams to...
  49. 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...
  50. 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...
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