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. S

    MHB Prove: 2ε{a,2,b} in Set Theory

    Prove in any axiomatic set theory that:2ε{a,2,b} , where a,b are letters
  2. A

    B A Rational Game: Exploring the Paradox of Aligning Irrational Numbers

    This post is to set forth a little game that attempts to demonstrate something that I find to be intriguing about the real numbers. The game is one that takes place in a theoretical sense only. It starts by assuming we have two pieces of paper. On each is a line segment of length two: [0,2]...
  3. ertagon2

    MHB Set Theory Questions (check if right)

    Could someone please check if these are right?
  4. D

    I Proof of Countability of ℚ: Bijection from A to ℕ

    I know there are many proofs of this I can google, but I am interested in a particular one my book proposed. Also, by countable, I mean that there is a bijection from A to ℕ (*), since this is the definition my book decided to stick to. The reasoning is as follows: ℤ is countable, and so iz ℤxℤ...
  5. Demystifier

    A Is there a decidable set theory?

    The Godel theorem shows that the standard Peano axiomatization or arithmetic is undecidable. However, there is an alternative Presburger's axiomatization of arithmetic, which is decidable. Similarly, the standard ZFC axiomatization of set theory is undecidable. For instance, the continuum...
  6. D

    B Empty domains and the vacuous truth

    So, here's my question. I read somewhere that all universal truths on empty domains are vacuously true, whereas all existential are false. However, if all statements of the form (∀x∈A)(P(x)) , where A is an empty set, are vacuously true, then the statement (∃x∈A)(P(x)) should also be true...
  7. Math Amateur

    MHB Axioms of Set Theory .... and the Union of Two Sets ....

    I am reading "Introduction to Set Theory" (Third Edition, Revised and Expanded) by Karel Hrbacek and Thomas Jech (H&J) ... ... I am currently focused on Chapter 1: Sets and, in particular on Section 3: The Axioms where Hrbacek and Jech set up an axiomatic systems (which they do NOT call ZFC ...
  8. Math Amateur

    MHB Best Books on Set Theory: Recommendations from MHB Members

    What do MHB members think are the best books at an undergraduate or senior undergraduate level on set theory ... Further what are the best books on set theory at a graduate level ... ... Peter
  9. J

    A Defining the membership relation in set theory?

    My main question is regarding whether the membership relation is taken as an undefined concept (as is usually hinted in set theory books) or if the membership relation can be defined within the language of first order predicate theory. Let me describe a method to define the membership relation...
  10. Mr Davis 97

    Is My Proof That the Union of Subsets is a Subset Correct?

    Homework Statement Prove that if ##A \subseteq B##, then ##\bigcup A \subseteq \bigcup B##. Homework EquationsThe Attempt at a Solution This is a simple problem, but I just want to make sure I am writing out the proof correctly: Suppose that ##A \subseteq B##. We want to show that ##\bigcup A...
  11. Mr Davis 97

    Show Set Theory Subset Relationship: x, y $\in$ B

    Homework Statement Assume that ##x## and ##y## are members of a set ##B##. Show that ##\{ \{x\}, \{x,y\} \} \in \mathcal{P} \mathcal{P} B## Homework EquationsThe Attempt at a Solution I know that ##\{ \{x\}, \{x,y\} \} \in \mathcal{P} \mathcal{P} B## iff ##\{ \{x\}, \{x,y\} \} \subseteq...
  12. Mr Davis 97

    Is the Number of Elements in a Set Equal to Its Power Set?

    Homework Statement Show that ##\bigcup \{\mathcal{P} X : X \in A \} \subseteq \mathcal{P} \bigcup A## Homework EquationsThe Attempt at a Solution Suppose that ##c \in \bigcup \{\mathcal{P} X : X \in A \}##. Then by definition this means that ##\exists a \in A## such that ##c \in \mathcal{P}...
  13. K

    Is (-infinity, b) an event for any real number b?

    Homework Statement Suppose that the sample space is the set of all real numbers and that every interval of the form (-infinity, b] for any real number b is an event. Show that for any real number b (-infinity, b) must also be an event. The Attempt at a Solution use the 3 conditions required...
  14. parshyaa

    I Is It Valid to Cancel Sets in Set Theory?

    We can prove that When A and B are two sets(A≠B) (A-B) = (A∩B') = (A-(A∩B)) {We can also confirm them using venn diagram} From first and third relation A-B = A - (A∩B) By cancelling A from both side I get B = (A∩B) Which is only possible when A and B are same set. What is wrong in my proof , is...
  15. VrhoZna

    Prove Zorn's Lemma is equivalent to the following statement

    Homework Statement From Introduction to Set Theory Chapter 8.1 exercise 1.4 Prove that Zorn's Lemma is equivalent to the following statement: For all ##(A,\leq)##, the set of all chains of ##(A,\leq)## has an ##\subseteq##-maximal element.[/B]Homework Equations N/A The Attempt at a Solution...
  16. Smalde

    Basic set theory with quantors question

    This is no homework for me. I am working as a teaching assistant in a lecture about logic and discrete structures for Informatics students. This should be a piece of cake, but I am not exactly sure of the logic behind. 1. Homework Statement Translate into words ∃c . ∀a ∈ A . ∀b ∈ B . ¬(a =...
  17. N

    Help me Prove that such a set does NOT exist

    Let Q denote the theory of Robinson Arithmetic. A theory T is nice iff T is consistent, is p.r. adequate and extends Q. The fixed-point lemma states that for all nice theories T, for any formula φ, there is a sentence σ such that T ⊢σ↔φ("σ")...
  18. T

    Is S1 Always a Subset of S2 If R1 Is a Subset of R2?

    Homework Statement Suppose R1 and R2 are relations on A and R1 ⊆ R2. Let S1 and S2 be the transitive closures of R1 and R2 respectively. Prove that S1 ⊆ S2. Please check my proof and please explain my mistakes. thank you for taking the time to help. Homework Equations N/A The Attempt at a...
  19. F

    MHB How to Prove a Set Theory Ordinal Relationship?

    Let $\beta$ be an ordinal. Prove that $A\cap \bigcup\beta=\bigcup\{A\cap X\mid X \in \beta\}$ I'm not sure on this. It looks a bit like union distributing over intersection. Please help.
  20. F

    Topology by Simmons Problem 1.3.3

    Homework Statement Let ##X## and ## Y## be non-empty sets, ##i## be the identity mapping, and ##f## a mapping of ##X## into ##Y##. Show the following a) ##f## is one-to-one ##~\Leftrightarrow~## there exists a mapping ##g## of ##Y## into ##X## such that ##gf=i_X## b) ##f## is onto...
  21. FallArk

    MHB Find g: Solving a Set Theory Problem

    Find a function g from {0,1} to B\A such that f^-1(g(x)) = x +2 for x∈{0,1}. Present it in the 2-row form. A = {{1},2,3} and B = {∅,1,{2},3} I know that B\A = {∅,1,{2}} and f is a bijection from A to B\A how do I find such function g? It obviously can't be bijection, how do I match one value to...
  22. F

    Topology by Simmons Problem 1.2.1

    Homework Statement If ##\bf{A}## ##= \{A_i\}## and ##\bf{B}## ##= \{B_j\}## are two classes of sets such that ##\bf{A} \subseteq \bf{B}##, show that ##\cap_j B_j \subseteq \cap_i A_i## and ##\cup_i A_i \subseteq \cup_j B_j## Homework EquationsThe Attempt at a Solution Since ##\bf{A} \subseteq...
  23. F

    I Sets, Subsets, Possible Relations

    Given a set, there are subsets and possible relations between those arbitrary subsets. For a given example set, the possible relation between the subsets of the example set will narrow down to the "true" possible relations between those subsets. a) {1} Number of Subsets: ##2^1 = 2## (∅, {1})...
  24. F

    I Russell's Paradox: Proving N is Normal if Abnormal

    Sets which doesn't contain themselves are called normal sets while sets that contain themselves are called abnormal. Let ##N## be a set of all normal sets. Prove that ##N## is normal if and only if ##N## is abnormal. Proof. ##~~\rightarrow ~~ ## Suppose ##N## is normal such that ##N \not\in N##...
  25. Jehannum

    I What does n ∈ f mean in set theory notation?

    I bought a maths book and have discovered it's somewhat above my level. In particular I'm confused about one bit of notation. I understand the "is a member of" operator when it takes a set as argument (e.g. n ∈ ℝ) but not when the book uses it with functions (e.g. n ∈ f) Does n ∈ f mean that...
  26. nomadreid

    I Existential Import paper by Corcoran and Massoud

    In the paper John Corcoran & Hassan Masoud (2014): Existential Import Today: New Metatheorems; Historical, Philosophical, and Pedagogical Misconceptions, History and Philosophy of Logic, http://dx.doi.org/10.1080/01445340.2014.952947, already in the introduction it says, as self-evident, that...
  27. Stoney Pete

    I Can an ordered pair have identical elements?

    Hi guys, Here is a wacky question for you: Suppose you have a simple recursive function f(x)=x. Given the fact that a function f(x)=y can be rewritten as a set of ordered pairs (x, y) with x from the domain of f and y from the range of f, it would seem that the function f(x)=x can be written...
  28. E

    MHB Proving Existence of Surjective Function F from P(N)\N to P(N)

    I'd really like some help in answering the next question...anything that might help will save my life: F is defined this way: F:A→B where A,B⊂P(N) and P(N) is the power set of the naturals. Let S,R∈A such that S is a proper subset of R if and only if F(S) is a proper subset of F(R) My question...
  29. E

    Difficulty proving a relation is an equivalence relation

    Homework Statement Homework Equations I don't think there are any in this case The Attempt at a Solution I know that in order to prove R is an equivalence relation, I'd have to show that it is Reflexive, Symmetric, and Transitive. I'm not sure why, but I'm finding this a bit difficult in...
  30. Math Amateur

    MHB Axioms of Set Theory: Separation Axiom and Garling Theorem 1.2.2 .... ....

    I am reading D. J. H. Garling: "A Course in Mathematical Analysis: Volume I Foundations and Elementary Real Analysis ... ...At present I am focused on Chapter 1: The Axioms of Set Theory and need some help with Theorem 1.2.2 and its relationship to the Separation Axiom ... ... The...
  31. Math Amateur

    I Set Theory: Separation Axiom and Garling's Theorem 1.2.2

    I am reading D. J. H. Garling: "A Course in Mathematical Analysis: Volume I Foundations and Elementary Real Analysis" ... ... At present I am focused on Chapter 1: The Axioms of Set Theory and need some help with Theorem 1.2.2 and its relationship to the Separation Axiom ... ... The...
  32. A

    B What are surreal numbers and how do they work?

    Hey guys! I have heard of this concept in various places and sort of understands what it attempts to do. Can anybody please explain it to me in more detail like how it works, how to notate it, and how to expand it to infinities and infinitesimals. Thanks in advance! Aakash Lakshmanan xphysx.com...
  33. J

    Solve Set Theory Homework: Right Hand Side of "Or

    Homework Statement ##C \subseteq A \cap B \implies A \cap B \cap C = C## Homework Equations How do I get rid of the "belongs to" term on the right hand side? I know I need to prove either the left hand or the right hand side of the "or" term is correct, I'm just not sure how to get there. The...
  34. J

    Find the Union of Intervals: A_n

    Homework Statement Let ##A_n = (n − 1, n + 1)##, for all natural numbers n. Find, with proof, ##∪_{n≥1}A_n## Homework Equations What does that last statement mean? Union for n greater than or equal to one times the interval? The Attempt at a Solution I can't understand the question.
  35. M

    Proof of A Union of A Intersection B Equals A

    Homework Statement Prove that ##A \cup (A \cap B) = A## Homework Equations In the previous exercise, we proved: Let A, B be sets. Then, the following statements are equivalent: 1) ##A \subseteq B## 2) ##A \cup B = B## 3) ##A \cap B = A## The Attempt at a Solution The proof of ##A \cup (A...
  36. Ling Min Hao

    A Question about axiom of regularity

    It has been stated that in axiom of regularity , a set cannot be an element of itself and there is a proof for which S={S} . I can understand his proof since S is the only element and hence its method of proof is viable here . But , what if I change the question to S= {S,b} ( it is a set which...
  37. TheSodesa

    How Do You Calculate Conditional Probability with Complements?

    Homework Statement [/B] P(A | \overline{B}) = ? Homework Equations Multiplicative rule: \begin{equation} P(A | B) = \frac{P(A \cap B)}{P(B)} \end{equation} Additive rule: \begin{equation} P(A \cup B) = P(A) + P(B) - P(A \cap B) \end{equation} Difference: \begin{equation} A \backslash B = A...
  38. L

    MHB Proving a Set Theory Statement Regarding Families of Sets

    I was wondering if anyone could please check my work and reasoning for this problem. Thank-you! (Also, would this be considered a direct proof? How might a contradiction and IFF proof look like and compare?) Problem: Suppose F, G1 and G2 are nonempty families of sets. Prove that if F ⊆ G1 ∩ G2...
  39. rattma

    I Is This Set Theory Notation Correct for Describing Local Maxima?

    Hi, Is the following notation correct? X = {xt> max(xt-k,...,xt-1,xt+1,...,xt+k)|(t-k,...,t+k) ∈ T2k+1∧ k ∈ ℕ\{0}} where T = [1,n]∩ℕ denotes the time periods over which x runs. I basically want to say that X is the set of points that are local maxima in a neighbourhood of k points to the right...
  40. R

    Elementary Set Theory (Discrete)

    Homework Statement Suppose A⊂B⊂C. What is A/B, A/C, and A∪B Homework EquationsThe Attempt at a Solution This isn't really a homework question, I am just trying to get some exposure to discrete math before I take it in the fall. The set differences A/B and A/C are both empty sets and the 'or'...
  41. T

    MHB Proving Set Operations [Set Theory]

    Could someone help me in this by simplifying/Proving the equation using Theorems/ Rules on Operation of Sets (i.e Commutative property, idempotent, assoc, dist. , definition of union , def. intersection , DeMorgan ,etc.). Letting A, B and C be three sets .. Prove/Disprove : Any solution...
  42. DaTario

    I Question about notation in set theory

    Hi All, Sorry, to begin with, if the question seems to be out of place. My question has to do with the notation (#) for the number of elements in a set: $$ N = \# \{ v_i \in V \vert v_i v_j \in E \} $$ Is it well known? If written in a scientific communication, must one offer a description of...
  43. T

    I Prove Logical Equivalence of P->(Q or R)

    From the text it says (P -> Q) or (P -> R) is equivalent to P -> (Q or R) I tried to see if this is true so I tried (P \to Q) \lor (P \to R) \\ (P \lor \neg Q) \lor (P \lor \neg R) \\ P \lor \neg Q \lor \neg R \\ P \lor \neg(Q \land R) \\ P \to (Q \land R) and P \to (Q \lor R) \\ P \lor...
  44. T

    I How Does the Distributive Law Simplify Logical Expressions?

    Im just reading this one example and i am stumped at this one step. (R\to C) \land (S \to C) \\ (\neg R\lor C) \land (\neg S \lor C) \ \ \ \ \ \textrm{by conditional law}\\ (\neg R\land \neg S) \lor C \ \ \ \ \textrm{by distributive law} I don't understand how it went from the second step to...
  45. TheMathNoob

    Proof of Partition Property for Subset A in Universal Set U

    Homework Statement Assume {B, C, D} is a partition of the universal set U, A is a subset of U and A is not a subset of B complement, A is not a subset of C complement, A is not a subset of D complement. Prove that {A ∩ B, A ∩ C, A ∩ D} is a partition of A. Homework EquationsThe Attempt at a...
  46. J

    Proof of Partition of a Set with Nested Partitions | Set Theory Proof 2

    Homework Statement . Let A be a set and {B1, B2, B3} a partition of A. Assume {C11, C12} is a partition of B1, {C21, C22} is a partition of B2 and {C31, C32} is a partition of B3. Prove that {C11, C12, C21, C22, C31, C32} is a partition of A. Homework EquationsThe Attempt at a Solution I know...
  47. J

    Proving Set Equality: A Simple and Effective Method

    Homework Statement Attached is the problem Homework EquationsThe Attempt at a Solution So I have to show that each side is a subset of the other side Assume x∈ A ∪ (∩Bi) so x∈A or x∈∩Bi case 1 x∈ ∩ Bi so x∈ (B1∩B2∩B3...∩Bn) which implies x∈B1 and x∈B2 ... and x∈Bn so x∈B1∪A and x∈B2∪A...
  48. M

    A Somewhat difficult set theory proof

    I am trying to prove that two definitions of a finite set are equivalent. 1.) A set ##A## is finite if and only if it is equipollent to a natural number ##n##. ( natural number as the set containing all the previous natural numbers including ##0## ) 2.) A set ##A## is finite if and only if...
  49. BubblesAreUs

    Set Theory: Proving h is Surjective Implies f & g

    Homework Statement Let f: X ----> Y and g: Y ----> Z be functions and let h = g o f: X ----> Z Homework Equations a. If h is surjective then g is surjective b. If h is surjective then f is surjective. The Attempt at a Solution Here h: X ----> Z a. Suppose h: x ---> z is surjective...
  50. C

    Question about the axioms of set theory

    Homework Statement For each structure, draw a directed graph representing the membership relation. Then determine which of the following axioms is satisfied by the structure: Extensionality, Foundation, Pairing, Union U= {a,b} a in b , and b in a The Attempt at a Solution The directed...
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