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. M

    Proof: A Subset of B implies A/D Subset of B/D

    Homework Statement Prove that if A is a subset of B then A/D is a subset of B/D. Homework Equations The Attempt at a Solution Consider element x of A. Since A is a subset of B then for all x element of A, x is an element of B. Consider element x of A/D. If x is an element of D...
  2. M

    Proof in Set Theory: Subset of ]0,2]

    Homework Statement {1/x+1/y / (x,y) in (IN*)^2} subset of ]0,2] Homework Equations The Attempt at a Solution When x=y=1 u get a sum of 2 which is in ]0,2] and for any x and y greater than 1 u get a sum between 0<sum≤2. It's a simple problem but i just don't know how to show the...
  3. Z

    Conditional probability question with set theory things

    Homework Statement Basically, I'm given the probability of 4 independent events: P(A) = 0.04 P(B) = 0.03 P(C) = 0.02 P(D) = 0.01 If anyone of these occur, a failure will happen. More than one can happen at the same time. I need to find the probability that more than one of them...
  4. D

    Ordinal Numbers: Bridging English & Set Theory

    How are ordinal numbers in set theory/order theory related to ordinal numbers in English? There should somehow be a bit of relationship for them to share the same name.
  5. C

    Proving Non-Empty Set Theory Equations: A Solution Using Induction Method

    Homework Statement Let A = A1 x ... x An B = B1 x ... x Bn C = C1 x ... x Cn Such that A,B,C are non empty, A=B\cup C and B\cap C = \emptyset prove that there exists a k in {1,...,n} such that B_k\cap C_k = \emptyset and for i\neq k, A_i = B_i = C_i Homework Equations...
  6. L

    Proving Set Inclusion: P(A) ⊆ P(B) Implies A ⊆ B

    Homework Statement Prove that if P(A) \subseteq P(B) then A \subseteq B, where A and B are two sets and P symbolizes the power set (set of all subsets) of a particular set. Homework Equations The Attempt at a Solution Okay, so here goes. Because it's a conditional, we suppose...
  7. B

    Set Theory| Proof if A subset B then f(A) subset f(B)

    Homework Statement f: X -> Y is a map from X to Y. And A, B subset X are random subsets of X. Proof the following: a) if A subset B then f(A) subset f(B) The Attempt at a Solution (1)Take an arbitrary element x in f(A). (2)For every x there has to be a y in A so that f(y)=x (3)From A...
  8. A

    Venn Diagram problem (Set Theory)

    Homework Statement The universal set u=40 Set A = 20 Set B=17 n(A∩B) = 1/2n(A'∩B') What is the value of n(A∩B)? Homework Equations none^^The Attempt at a Solution The first thing I thought was that because Set A and Set B add up to 37 there must be 3 remaining outside the two sets, and since...
  9. A

    Consistency of ZF Set Theory

    I've never been exposed to this axiom schemata of replacement before, so here's my understanding of it: the axiom includes an arbitrary formula, and that formula may have arbitrarily many free variables. Therefore, a separate axiom is needed for formulas with one free variable, with two free...
  10. A

    Solve Set Theory Problem: 30 People, 7 Black Hair, 24 Right Handed

    Homework Statement There are 30 people in a class of which 7 people have black hair and 24 people are right handed. 2 people are neither right handed nor have black hair. 1. How many have black hair and are right handed? 2. How many have black hair and are not right handed? I'm sure this...
  11. B

    Simple Set Theory - Union, Intersect and Complement

    Homework Statement If A, B, and C are subsets of the set S, show that A^C \cup B^C = \left(A \cap B\right)^C Homework Equations A^C = \{x \in S: x \not \in A\} A\cup B = \{x \in S:\; x \in A\; or\; x\in B\} A\cap B = \{x \in S:\; x \in A\; and\; x\in B\} The Attempt at a Solution...
  12. T

    Set Theory: 70 Students Visiting 3 Universities

    in a coaching centre of 70 but 4 students went on a university visit.31 went to unilag,35 went to lasu and 36 went to u.i. 10 went to all the three universities, 12 went to unilag only, 13 also went to lasu only, 15 went to u.i only. how many students visited 1 . . unilag and lasu 2 . .at...
  13. W

    Elementary Set Theory Proof

    Hello, I am teaching myself Set Theory, and in doing some exercises I came across the problem: Given sets A and B, prove that A \subseteq B if and only if A \cap B = A. My proof, in natural language, is in two parts: 1) Prove that if A \subseteq B, A \cap B = A. By the definition...
  14. L

    Set theory homework - Theoretic reasoning

    Homework Statement Prove where X and Y are both sets use theoretic reasoning i) Z \ (X \cap Y) = (Z \ X) \cup (Z \ Y) ii)(Y \ X) \cup Z = (Y \cup Z) \ (X \ Z) iii) Z \ (Y \ X) = (X \capZ) \cup(Z \ Y) Homework Equations \ = set difference The Attempt at a Solution i know you don't do other...
  15. U

    Set Theory Homework find ∪i=0Ai and ∩ i=0Ai

    Homework Statement Find ∪i=0Ai (with infinite symbol) and ∩ i=0Ai (with infinite symbol) in each of the cases when for each natural number i, Ai is defined as: 1. Ai = {i,−i, i + 1,−(i + 1), i + 2,−(i + 2), . . .} 2. Ai = {0, i, 2i} 3. Ai = {x : x is a real number such that i < x...
  16. BloodyFrozen

    Set Theory Symbols: Is A ∩ C ⊆ B Equal?

    Homework Statement Is A ∩ C ⊆ B equal (A ∩ C) ⊆ B or A ∩ (C ⊆ B)? Homework Equations N/A The Attempt at a Solution I think it's the first one due to it being in order, but I'm not sure...
  17. G

    Books on NBG Axiomatic Set Theory

    Hi! I am looking for a text concerned with NBG axiomatic set theory( Neumann-Bernays-Godel). Could you recommend some books related with it?
  18. O

    Set theory. Is the converse true?

    Homework Statement Prove that \cup_{x \in C} \{ 2^{x} \} \subseteq 2^{\cup C} Homework Equations \cup_{x \in C} \{ 2^{x} \} = \{ A | \exists x \in C, A \subseteq 2^{x} \} 2^{x} is the powerset of x. i.e. 2^{x} = \{ y | y \subseteq x \} The Attempt at a Solution Suppose A \in...
  19. C

    Set Theory Question(inclusion-exclusion principle related)

    Homework Statement An auto insurance has 10,000 policyholders. Each policyholder is classified as: (i) young or old; (ii) male or female; (iii) married or single. Of these policyholders, 3000 are young, 4600 are male, and 7000 are married. The policyholders can also be classi ed as 1320...
  20. O

    Set theory. What does this mean?

    If \mathbb{Z} is the set of integers, what does \mathbb{Z/2Z} mean?
  21. S

    Solving Set Theory Question Homework: Assume A Closed Algebra

    Homework Statement Assume I have the property that for any {Ei} (i=1 to infinity) contained in some algebra A, if E1 contained in E2 contained in E3... (infinite nesting), then Union Ei (i=1 to infinity) is also contained in A. I simply want to show that for any {Ei} (i=1 to infinity) in A, I...
  22. I

    Prove the irrationality of ar+s or ar-s using proof by contradiction

    Use proof by contradiction to prove the following: Let a be an irrational number and r a nonzero rational number. Prove that if s is a real number, then either ar+s or ar-s is irrational. I am stuck with this proof. Here's what I have so far, Proof Suppose, by way of contradiction, that...
  23. A

    Set theory proof - counter examples

    I'm having a problem with providing counter examples when disproving a statement. For example A - (B U C) = (A - B) U (A - C). The solution given was A = {a}, B = {a} and C = empty set. My question is how can you work this out - i was told it's possible from the Venn diagrams but I'm not...
  24. O

    Set Theory Proof(Using Identities)

    Hi, I've been trying for 3 hours to solve this proof using identities. I can't seem to get it. Can i get a little help please? Prove: A U B = (A ∩ B') U (A' ∩ B) U (A ∩ B) thanks
  25. I

    Disproving the Statement: A Contradiction in Set Theory | Proof Help

    I am having issues with a proof, as follows. *U = universal set , P(U) = power set of a universal set For all sets A, B, C ∈ P(U), if A ⊆ C and B ⊆ C, then A ⊆ B or B ⊆ A. I am pretty sure the statement is false and so I have to disprove it, i.e. prove the negation. I am stuck on how to...
  26. I

    Set Theory Proof Help | Prove (A-C) U (B-C) = (A U B) - C

    I need to prove the following: (A-C) U (B-C) = (A U B) - C I know that the union means that I have to do a proof by cases to show that these two sets are equal. But where do I start?! thanks
  27. B

    Set theory, Foundations, Physics and Causal Structure

    Hey, I would like to start a discussion about the use of set theory in mathematical physics. I myself have done research in categorical physics and have seen the debates on how it can be an alternate foundation for mathematics. We can discuss here a few things, but try to stick to these...
  28. S

    Is the statement on the UNION and INTERSECTION of Indexed Sets always true?

    Homework Statement Can't quite figure out the LaTeX for Indexed Sets, so bear with me: From "Book of Proof" Section 1.8 #11 http://www.people.vcu.edu/~rhammack/BookOfProof/index.html Is the UNION of Aa, where a is in I, a subset of the INTERSECTION of Aa always true for any collection of sets...
  29. R

    How to Approach Solving Set Theory Equalities

    Hi, I'm struggling to understand how to approach set theory equality questions for example: True or false? (A n B) is a subset of (A u B) Is quite simple as its obvious the intersection will contain everything that is in the union But what about a more complex question like ...
  30. B

    Cardinality Proof (Set Theory)

    I'm trying to prove the following: if E is infinite set and F is finite set. prove that E and E U F have the same cardinality ? So what I did: I'm going to use Schroeder-Bernstein Thm. 1st, it's easy to show that |E| is less of equal to |E U F| since it is a subset of this latter. Now...
  31. Q

    Understanding Set Theory: Order and Intersection of Sets A and B

    All elements in \mathcal{A} match all elements of \mathcal{B}. The order of the information in \mathcal{B} is important to understand the dynamics of the information. \mathcal{A} does not have a logical order of information. However \mathcal{B} does have a logical order. Because there is no...
  32. Q

    LaTeX How Can You Express Complex Numbers in LaTeX?

    I know all the numbers of a real part of an equation is given as \Re How do you express your complex part in the same form?
  33. E

    [Set Theory] Proving Linear Order on a Subset of a Partially Ordered Set

    Homework Statement Let L be a partially ordered set. Every countable chain in L has an upper bound. Let S be a countable subset of L such that for arbitrary a,b in S there exists a c in S such that a (less-than-or-equal) c and b (less-than-or-equal) c. Prove S has an upper bound in L...
  34. N

    Set Theory Proof Homework: Can You Help?

    Homework Statement Homework Statement are located in the pdf below. I also upload the file onto an online viewer for pdf for those people who are afraid to download attachments. Here is the link: http://view.samurajdata.se/psview.php?id=c1f5a372&page=1 Homework Equations None.The Attempt...
  35. S

    What Are the Best Books to Learn Set Theory from Beginner to Advanced?

    HI friend , I am confused about how to start set theory , i want to learn it fully, so please help me in choosing books on this topic which covers from very basics to full advance. thanks Sid
  36. A

    Anyone have read Naive Set Theory by Halmos?

    Hi, I just found cheap Naive Set Theory by Halmos. I am wondering if it is worth buying. I have read somewhere it helps reading more advanced books. Plus it's small, it's easy to carry around. I do not like to bring heavy bag to the library. I am a Physics major. I have taken Calc...
  37. D

    How Do You Prove Sets Have Cardinality Aleph-Nought?

    URGENT HELP PLEASEEEE, a question on set theory Homework Statement the question is: Prove that these sets have cardinality aleph-nought:(there is two 2 prove) (a) {1/(2^k) : k∈ℕ} (b) {x∈ℤ : x >= -5} im not sure how to work this out, please help on this, i did ask on a previous...
  38. D

    Set theory question it , thank you

    set theory question please help it urgent, thank you Homework Statement 1. Homework Statement Let A, B and C be any sets inside our universal set U. Decide whether each of the following statements is true or false. Justify your answers by giving a proof or a counterexample as...
  39. G

    Proof of Fraction Position in Countable Set of Positive Rational Numbers

    Homework Statement Prove that the fraction m/n occurs in position \frac{m^2 +2mn + n^2 - m -3n}{2} of the enumeration {1/1, 1/2, 2/1, 1/3, 2/2, 3/1,...} of the set Q+ of positive rational numbers. (Hint: Count how many terms precede m/n in the enumeration.) Homework Equations The Attempt...
  40. H

    Set Theory Problem Involving Partitions

    This problem is from Hrbacek and Jech, Introduction to Set Theory, Third Edition, right at the end of chapter 2. Homework Statement Let A \neq {}; let Pt(A) be the set of all partitions of A. Define a relation \leq in Pt(A) by S_{1} \leq S_{2} if and only if for every C \in S_{1}...
  41. G

    Set Theory (Not too difficult)

    Homework Statement describe exactly when x intersecting (y union z) = (x intersecting y) union z Homework Equations The Attempt at a Solution I just for some reason cannot see this solution and need a shove in the right direction
  42. E

    Solve the paradox of set theory V7.4.1

    Solve the paradox of set theory V7.5 by LiJunYu 2010.12.25 email: myvbvc@tom.com or 165442523@qq.com Brief:All power sets of real number set R: P(R),P(P(R)),P(P(P(R))),...,Pn(R),...Because all Pn(R) does not contain its own,in Russell's paradox,"all sets which does not...
  43. V

    Best Books on Set Theory: Axioms & Theory

    What are the best books on Set Theory?? I mean a book with all axioms and theory and dispenses other books.
  44. D

    Proving 1-1 and Onto Functions in Set Theory

    Hello, So I've been running into problems with rigorously proving that a function I've defined in ZFC is a bijection (1-1 and onto). For example, if I know that a function between two numbers "n" and "m" (defined in the standard von neumann way) is a bijection (call the function "f"), how...
  45. R

    Set Theory - Proving Contrapositive

    Homework Statement using set theroetic notation, write down and prove the contra-positive of: GOD WHAT IS WRONG WITH LATEX? It is completely ruining my set notation! And i can't fix it! If B \cap C \subseteq A Then (C-A) u (B-A) is empty. The Attempt at a Solution I'm awful with set...
  46. T

    Optimizing Set Theory Hash Tables for Efficient Data Representation

    1. Homework Statement I posted this on Calculus & Beyond because all of this came from a math idea, but I realize now that it belongs in this section. Given that I have a set W, with a multitude of subsets w1...wn, with arbitrary intersections, worst-case-scenario-unordered, I want to know...
  47. T

    Comp sci. and set theory

    Homework Statement Given that I have a set W, with a multitude of subsets w1...wn, with arbitrary intersections, worst-case-scenario-unordered, I want to know what would be a good representation in a hash table. Basically I want to have things like A\cupB, A\capB, A - B, etc (the basic set...
  48. C

    Set Theory: Proving A-(BUC)=(A\cup B)-C

    Homework Statement [A-(BUC)]U[(A\cup B)-C]U[(A\cap C)-B]U[A\capB\capC]; The Attempt at a Solution Sorry about the crappy formatting (btw). Anyway, I'm trying to "prove" that this is is equal to A. So basically cancelling out the Bs and Cs? I'm not sure how to go about this. de...
  49. S

    Building a Discussion Group Using Set Theory

    Hi, I was wondering if anyone had any ideas how to approach this model building exercise. I've got to use set theory to design a discussion group, I'm given two initial sets (Persons and Strings), from this I have to make the following sets: Members - set of members, each member is a...
  50. X

    Two set theory questions (isomorphism and countable sets)

    First Homework Statement Let X be a finite non-empty set and Y a countable set. Prove that XxY (X cross Y) is countable Homework Equations X isomorphic {1, 2, ..., n} for some n\inN, N is natural numbers Y isomorphic to natural numbers The Attempt at a Solution I wasn't able to...
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