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. S

    Set theory: proofs regarding power sets

    Let X be an arbitrary set and P(X) the set of all its subsets, prove that if ∀ A,B ∈ P(X) the sets A∩B,A∪B are also ∈ P(X). I really don't know how to get started on this proof but I tried to start with something like this: ∀ m,n ∈ A,B ⇒ m,n ∈ X ⇒ Is this the right way to start on this proof...
  2. T

    Exercise from Naive Set Theory by Halmos

    For those who have read Halmos, in Section 6 Ordered Pairs (page 23 in my book), he gives a non-trivial exercise to find an intrinsic characterization of those sets of subsets of A that correspond to some order in A. I'm curious what that characterization is. A is suppose to be a quadruple...
  3. L

    MHB Need book recommendation for Set Theory word problems

    Hi - Glad to have found this forum. I am looking for a book which contains LOTS of Set Theory word problems with solutions. Anyone aware of a good resource? Thanks in advance.
  4. R

    Set Theory: Why use implication?

    Homework Statement I hope this does not violate copyright or anything but this problem originated from an assignment from Introduction to Mathematical Thinking in Coursera. I could not post there because the class ended and the discussion board there is dead. Let C be the set of all cars, let...
  5. K

    I'm sorry, I'm not sure I understand your question. Can you clarify?

    Set Theory -- Uncountable Sets Homework Statement Prove or disprove. There is no set A such that ##2^A## is denumberable. The Attempt at a Solution A set is denumerable if ##|A| = |N|## My book shows that the statement is true. If A is denumerable, then since ##|2^A| > |A|, 2^A ##...
  6. A

    MHB Proving the Inclusion Property for Sets in Axiomatic Set Theory

    For reference, my class is using The Joy of Sets by Keith Devlin. I've been asked to solve this as a practice problem, but this stuff is really confusing over the first read or two and I've yet to see any example proofs and I think I'll just mess it up. A link to the book can be found here if...
  7. vrble

    Can Basic Set Theory Explain Why an Element Belongs to a Set?

    1. Suppose A \ B\subseteqC\capD and x\inA. Prove that if x \notinD then x\inB 2. None 3. Proof: Suppose A \ B\subseteqC\capD, x\inA, and x\notinD. It follows that our first assumption is equivalent to A due to our third assumption. Thus, B\subseteqC\capD is disjoint and either x\notinB\subseteqC...
  8. Math Amateur

    Simple set theory problem - definition of a J-Tuple

    On page 113 Munkres (Topology: Second Edition) defines a J-tuple as follows: I was somewhat perplexed when I tried to completely understand the function \ x \ : \ J \to X . I tried to write down some specific and concrete examples but still could not see exactly how the function...
  9. Math Amateur

    MHB Introduction to J-tuples in set theory

    On page 113 Munkres (Topology: Second Edition) defines a J-tuple as follows: https://www.physicsforums.com/attachments/2153 I was somewhat perplexed when I tried to completely understand the function \ x \ : \ J \to X . I tried to write down some specific and concrete examples but still...
  10. J

    Set Theory and Binary Logic: Understanding XOR in Set Theory Operations

    First: relating some ideia of set theory and binary logic, like: U = 1 Ø = 0 thus, some identities appears: U ∪ U = U U ∪ Ø = U Ø ∪ U = U Ø ∪ Ø = Ø U ∩ U = U U ∩ Ø = Ø Ø ∩ U = Ø Ø ∩ Ø = Ø 1 + 1 = 1 1 + 0 = 1 0 + 1 = 1 0 + 0 = 0 1 × 1 = 1 1 × 0 = 0 0 × 1 = 0 0 × 0 =...
  11. R

    Looking for a Better Textbook on Set Theory?

    I don't like Jech's textbook on set theory because he gives these definitions written in this bizarre language and he doesn't restate the definition in colloquial English. That mathematicians feel its unnecessary to give colloquial examples of their definition or examples, in my opinion, is a...
  12. saybrook1

    Abstract algebra or set theory?

    I'm trying to round out my math skills in order to apply to graduate school for physics and I've already taken all of the calculus offered along with linear algebra, power series etc... I'm wondering which would be better should I choose to take a math course this term: abstract algebra or set...
  13. R

    MHB How to remember set theory properties?

    I'm an undergraduate studying math taking intermediate proof-writing courses, and there are certain basic identities of set theory and functions that still confuse me - i.e., I have to reprove them or think about them carefully every time. Examples: (A\times B)\cap (C\times D)=(A\cap C)\times...
  14. K

    Set Theory Logic: Finding True Statements in a Given Domain

    Homework Statement In each of the two following open sentences P(x) and Q(x) over a domain S are given. Determine all ##x \in S## for which P(x) → Q(x) is a true statement. ## P(x): x \in [-1, 2]; Q(x): x^{2} \leq 2; S=[-1,1] ## Homework Equations According to truth values for →: a...
  15. Fredrik

    Real numbers without set theory

    I understand the definition of real numbers in set theory. We define the term "Dedekind-complete ordered field" and prove that all Dedekind-complete ordered fields are isomorphic. Then it makes sense to say that any of them can be thought of as "the" set of real numbers. We can prove that a...
  16. P

    Set Theory: Is {a} a Subset of {S}?

    Homework Statement I am not sure if set theory is precalc or not but here is my question. Find a pair set such that {a} belongs to the set and {a} is not a subset of S. The Attempt at a Solution So I thought that a set like this would work S = {{a}, b} because {a} belongs to the set...
  17. C

    Topological indistinquisable points and set theory.

    In set theory a set is defined to be a collection of distinct objects (see http://en.wikipedia.org/wiki/Set_%28mathematics%29), i.e. we must have some way of distinguishing anyone element from a set, from any other element. Now a topological space is defined as a set X together with a...
  18. D

    Set Theory Proof: A∩B=Ø implies C∩D=Ø

    Homework Statement Hey guys! I am new to this forum but saw the helpful posts on set theory proofs and wondered if I could finally get some help with this problem: Suppose A, B, C, and D are sets with A⊆C and B⊆D. If A∩B=Ø then C∩D=Ø. This is a biconditional so I have to prove it...
  19. D

    Set Theory Proof Help: Proving C∩D=Ø When A⊆C and B⊆D

    Hey guys! I am new to this forum but saw the helpful posts on set theory proofs and wondered if I could finally get some help with this problem: Suppose A, B, C, and D are sets with A⊆C and B⊆D. If A∩B=Ø then C∩D=Ø. This is a biconditional so I have to prove it both ways correct...
  20. A

    Is P(E) U P(F) Equal to P(E U F)?

    Homework Statement Prove that(power set) P(E) U P(F) is a subset of P(E U F) Homework Equations P(E) U P(F) is a subset of P(E U F) The Attempt at a Solution P(E)U P(F)={x:xεP(E) or xεP(F)} but P(E)={X:X is a subset of E} or P(E)={x:xεX→xεE} so we get P(E)U P(F)={x:xεX→xεE or...
  21. M

    Set theory: find the intersection

    Homework Statement In a group of 30 people each person twice read a book from books A, B, C. 23 people read book A, 12 read book B and 23 read book C. (a) How many people read books A and B? (b) How many people read books A and C? (c) How many people read books B and C? Homework...
  22. K

    Set Theory, Functions. Injective/Surjective

    Homework Statement Give f:A→A and g:A→A where f is surjective, g is injective, but f*g is neither surjective nor injecive The Attempt at a Solution I don't know why I can't really think of two... I assume it's easiest to do one in ℝ, but when it comes to producing...
  23. I

    Proving the Truth of 3(b) in Basic Set Theory

    how do I go about doing 3(a) and 3(b)? I'm guessing that for 3(a), it is true, since we have for LHS: P((A or B) and C) we can consider the case P(A and C) by excluding B, and this is a subset of the RHS when we also exclude B: (P(A) and P(C)). We can consider excluding B because...
  24. R

    Southeastern Massachusetts Conference Math League: Set theory, gcf,lcm

    Homework Statement 2.) if jimmy piles his baseball cards in stacks of 4, then there is 1 left over. if he piles them in stacks of 7, there are 4 left over. If he piles them in stacks of 9, there are 6 lefty over. If he piles them in stacks of 10, there are 7 left over. compute the smallest...
  25. U

    Probability and set theory

    Homework Statement Let X be a set containing n elements. If two subsets A and B of X are picked at random, the probability that A and B have the same number of elements is Homework Equations The Attempt at a Solution Total number of subsets possible is 2^n. Now the subsets containing 1...
  26. Chris L T521

    MHB Saint's question from Yahoo Answers regarding set theory

    Here is the question: Here is a link to the question: Show that for any three sets A; B; C , we have: A - (B -C) = (A-B) U (A ? C)? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  27. B

    Set theory and baye's theorem problem

    A computer consulting firm presently has bids out on three projects. Let Ai = {awarded project i}, for i = 1, 2, 3, and suppose that P(A1) = 0.22, P(A2) = 0.25, P(A3) = 0.29, P(A1 ∩ A2) = 0.07, P(A1 ∩ A3) = 0.09, P(A2 ∩ A3) = 0.05, P(A1 ∩ A2 ∩ A3) = 0.02. The question is to find the...
  28. stripes

    Intro abstract algebra along with basic set theory

    Homework Statement An interesting example of a ring: Begin with a nonempty set X and form the power set of X, P(X), which is the set of all subsets of X. On P(X), define addition + and multiplication × as follows: For A, B in P(X): A × B = A ∩ B A + B = (A\B) ∪ (B\A), where as...
  29. D

    Confusing Axiomatic Set Theory Proof

    This proof makes no sense to me. The theorem to be proved is Theorem 44. {x,y} = {u,v} → (x = u & y = v) V (x = v & y = u) where {x,y} and {u,v} are sets with exactly two members, which can be either sets or individuals. The proof relies on: Theorem 43. z \in {x,y} z = x V z = y...
  30. E

    Russel's Paradox in Naive Set Theory

    I realize that Russell's Paradox in naive set theory is considered to be, well... a paradoxical fallacy. Despite the fact that it is paradoxical and goes against logical intuition, is it really illogical though? It seems to me that the method in which the paradox arises is perfectly sound and as...
  31. B

    Proving A×B ≠φ for A≠φ and B≠φ

    I am supposed to prove: If A \neq \phi and B \neq \phi then A\times B \neq \phi The HINT in the back of the book gives: A \neq \phi \wedge B \neq \phi \Rightarrow \existsa\subseteqA \wedgeb\subseteq B so that (a,b) \subseteq A\times B I have 2 questions 1.Is it enough for the...
  32. Fernando Revilla

    MHB Solve Set Theory Question: Prove Iy o f = f

    I quote a question from Yahoo! Answers I have given a link to the topic there so the OP can see my response.
  33. reenmachine

    Set Theory - Counting - Binomial Coefficient - Factorials

    Homework Statement A department consists of 5 men and 7 women.From this department you select a committee with 3 men and 2 women.In how many ways can you do this? Homework Equations Since the "overall set" (the entire department) is composed of both men and women and each has a specific...
  34. C

    Introduction to Set Theory (precursor to better evaluation of LA)

    My goal: To show the dimension of space L equals the length of any maximal flag of L; Is the following valid? My attempt: Let M \rightarrow {L_{i-1}, ... L_i} where {e_i} \in L_i | e_i \not\in L_{i-1} Assuming e_i \in M and e_i \not\in L_{i-1}, we can say: e_i \in L_i and L_i...
  35. D

    Set Theory Theorem: Existence of Natural Number Sequences & Large Cardinals

    Is there a theorem which says that if certain natural number sequences exist, then some large cardinals exist. Can anyone tell me if it's true and what it says? I vaguely remember my set theory professor mention this theorem years ago.
  36. G

    Prove an Associative Law (set theory)

    Homework Statement Prove that: A\cup(B\cupC) = (A\cupB)\cupCHomework Equations The Attempt at a Solution I never had to prove anything but I'll try. A\cup(B \cupC). Take: A = {1, 2, 3, 4, 5}, B = {5, 6, 7, 8, 9, 10}, C = {7, 8, 9, 10} (B\cupC) = P If A\cupP means A, united with the union of B...
  37. U

    Set Theory for Beginners - Exploring Ideas

    Gentlemen, I am writing to you in order to identify an idea I have and to see how this idea can be mathematically expressed. My understanding is that this idea pertains to Set Theory and I am going to do my best in expressing this idea for you- x y z are formed into two sets, x|y and y|z...
  38. D

    A couple of questions on set theory

    Hi. I'm studying calculus in high school right now, and I became really interested in how calculus could be developed from a mathematically rigorous point of view. One of my instructors suggested I learn some set theory, so for about a week I've been researching stuff on the internet - very...
  39. reenmachine

    Does A Subset B Equate to A Minus B Being Empty?

    Homework Statement Prove that if A and B are sets , then ##A \subseteq B \ \ \leftrightarrow \ \ A - B = \varnothing## The Attempt at a Solution Let ##A \subseteq B## be arbitrary.The definition of ##A \subseteq B## implies that ##\forall x \in A## , ##x \in B##.This implies that ##A - B =...
  40. reenmachine

    Set Notation , Set Theory

    Homework Statement Build a notation for the set: ##\{ ... , -8 , -3 , 2 , 7 , 12 , 17 ,...\}## Homework Equations ##2+5(0) = 2## ##2+5(1) = 7## ##2+5(-1) = -3## etc... The Attempt at a Solution ##\{ \ 2+5y \ | \ y \in \mathbb{Z} \ \}## Take note that you could replace ##2## by any...
  41. reenmachine

    Set Theory , building a set notation

    Homework Statement Build a set notation for $$\bigcup_{i \in N}R × [i , i + 1]$$ The Attempt at a Solution ##\{(x,y) \in R : x \in R \ \ \exists z \in N \ \ z ≤ y ≤ (z+1)\}## Last time I tried one of these kind of sets I struggled quite a bit , so I'm interested in knowing how much...
  42. K

    Quick simple set theory question

    Homework Statement Let X be a set and ≥ be a binary relation on X Provide a mathematical definition for ≥ is reflexive ≥ is symmetric ≥ is transistive ≥ is antisymmetric X is a lattice The attempt at a solution So I'm not really sure what this is asking... specifically if ≥...
  43. caffeinemachine

    MHB A Very Standard Theorem in Set Theory

    Let $A$ be any set. Show that there is no bijection between $A$ and the power set $\mathcal P(A)$ of $A$. (The power set of a set is the set of all its subsets including the empty-set.)
  44. S

    Does the Operation in Set Theory Imply a Singleton Set?

    Homework Statement Let S be a set with an operation * which assigns an element a*b of S for any a,b in S. Let us assume that the following two rules hold: 1. If a, b are any objects in S, then a*b = a 2. If a, b are any objects in S, then a*b = b*a (Herstein, Abstract Algebra, 2ed)...
  45. Fredrik

    Short summary of the essentials of set theory

    I've been talking to a guy who doesn't know anything about sets, and I couldn't think of anything good to recommend that he should read. I know that there are lots of good books about set theory, but don't they all cover too many details so that it takes too long to get an overview of the...
  46. micromass

    Foundations Is Naive Set Theory by Paul Halmos a Must-Read for Math Enthusiasts?

    Author: Paul Halmos Title: Naive Set Theory Amazon Link: https://www.amazon.com/dp/1614271313/?tag=pfamazon01-20
  47. L

    Amount of Set Theory Required to Study Logic

    Amount of Set Theory "Required" to Study Logic Hello, I have been self-studying set theory to try and get into logic. The book I bought is published by Dover (I love how cheap their books are) entitled Set Theory and Logic by Robert Stoll. I have gone over the basic set theory section...
  48. M

    MHB Set Theory Proofs: A, B, and C - Solving for Set Equality and Complements

    I have gotten to this point with a and b but do i am totally lost with c. Any help would be much appreciated Consider any three arbitrary sets A, B and C. (a) Show that if A ∩ B = A∩ C and A ∪ B = A ∪ C, then B = C. (b) Show that if A − B = B − A, then A = B. (c) Show that if A∩B = A∩C = B ∩C...
  49. 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.
  50. 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...
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