Set theory Definition and 439 Threads

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...

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...

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.

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...

Hello my name is Andy I'm in high school, and I have a bit of a confusion or lack of information. Ok so I have been reading a book on set theory, and I keep encountering … ,but up more in the use of a set. Like A,B … but again up more to the middle of the sentence. I feel dumb, I googled it...

Homework Statement Hi I could use some help getting an explanation that kinda twists my head a bit. I want to know what I'm missunderstanding so I can get this right from the beginning. ^c = complement U = union I want to simplify this set A U (A U B^c)^c intersect (A U C ) to...

I'm a physics undergraduate and I'll starting learning topology from Munkres next semester. But first I want to learn set theory to feel more comfortable. Do you know any good textbook? A friend of mne from the math department said I should go with Kaplansky's "Set Theory and Metric Spaces".

Set Theory "+" symbol 1. X, Y and Z are sets. Does X × (Y + Z) = X × Y + X × Z? The solution starts like so: X × (Y + Z) = {(x,(y,0)) | x \in X, y \in Y}\cup{(x,(z,1)) | x \in X, z \in Z} I don't understand how the "+" symbol works. Why does it equate to this (x,(y,0)) (x,(z,1))...

Hope this is the right forum for my question. I'm into statistics and quite often see assumptions involving set theory. I know some set theory but am having trouble understanding it for any application. I would like to narrow this gap, maybe because this type of mathematics seems most...

we have E=(a+b)/ab (a,b)EN* 1/ prove that E C ]0,2] (i already duid that ) 2/ is ]0,2] E E? shelp me in this one! Homework Equations The Attempt at a Solution x \in ]0,2] \Rightarrow x \in E

Homework Statement A = { pi + 2k pi / k \in Z } B = {(- pi / 3) + (2k pi / 3 ) / k \in A } Prove that A C B Homework Equations A C B = \forallXE E : x \ni A \Rightarrow X \ni B The Attempt at a Solution \ni[k E Z ]: x = pi + 2k pi \ni[k E Z ]: x = pi ( 1 + 2k) I'm sure i need to...

Sorry if the title of my question is wrong, lim sup_n A_n = ∩^∞ U^∞ A_k does it mean that first we are taking the union of A_n and then we are taking intersection? actually I am confused, what is it actually?

Let A={1,...,n}. Show that there is a bijection of P(A) with the cartesian product Xn, where X is the two element set X={0,1} and P(A) is the power set of A. Below is the start of my proof. I just want to make sure that my function "makes sense." Proof: Let A={1,...n}, and X={0,1}. Define f...

Claim: If A is an inductive set of postive integers, then A is Z+. I tried to prove this two different ways for the fun of it. I would like to get some feedback concerning the correctness of both. Thank you. :-p Proof: By definition, Z+ is the intersection of all inductive subsets of ℝ. Since...

Homework Statement Let A and B be sets. Prove that A U B=(A-B) U (B-A)U (A∩B)Homework Equations The Attempt at a Solution If I want to prove the foward direction: A U B⊆(A-B) U (B-A)U (A∩B) then from my understanding I know that xεA or xεB. And I can assume wolog that xεA. But since I assume...

Homework Statement Hi everyone, i have the following problem, that for the moment i couldn't find the solution or even how to search about... This is it, we have a family of subsets F ={F_1,...,F_p} of the set {1,2,...,k}. We know that, for every i != j, F_i !=F_j and for every pair of...

Let A, B, and C be sets. Show that a) (A-B) - C \subseteq A - C b) (B-A) \cup (C-A) = (B \cup C) - A I am using variable x to represent an element. Part A) I rewrote (A-B) - C as (x\inA ^ x\notinB) - C I think this could be rewritten as (x\inA ^ x\notinB) ^ x\notin C A-C can...

Hi there, here's the question I am given, i will provide the answer that I think is correct, do you mind checking it and possibly pointing out where I am wrong if I am? Give an example of a set S such that: a) S is a subset P(N) b) S belongs to P(N) c) S belongs to P(N) and |S|=5 here...

The notation has me a bit confused... Heres my logic for the P({1}) on the inside {EmptySet, {{1}}} reason being, you always include the empty set, {1} is a part of the set. The cardinality is two You have the set: {EmptySet, {{1}}}, and now you have to consider the outer "P" the...

Homework Statement show S1 U S2 = (S1' ∩ S2')' The Attempt at a Solution I'm pretty sure I have this right or I'm close Let x ∈ S1 U S2 x ∈ S1 or x ∈ S2 Since x ∈ S1 or S2, then x ∉ S1' and S2' If x ∉ S1' and S2', then x ∈ (S1' and S2')' Therefore, S1 U S2 = (S1' ∩ S2')' Homework Statement...

Homework Statement Determine whether the relations on three sets are Reflexive, Irrelfexive, Symmetric,, Asymmetric, Antisymmetric, Transitive, and Intransitive. The relation \subseteq on a set of sets. Homework Equations The Attempt at a Solution I am having trouble figuring out...

I am taking a philosophy course that covers basic set theory as part of the introduction. I’m not sure in which section of the forum set theory should be, but I think this is the right place. Homework Statement For each of the following relations, indicate whether it is Reflexive...

Hi, I have been trying for a very long time to prove the following set theory identity (A union B) intersect (B union C) intersect (C union A) = (A intersect B) union (B intersect C) union (C intersect A). I thought that I could simplify (A U B) intersect (B U C) intersect (C U A) as [B U...

Hello, I am trying to teach myself set theory...main problem is, as an engineer, mathematical proofs were never exactly stressed in my curriculum. (Scary, right?) The problem is stated as follows: "Prove the following, {x\inZ|for an integer y, x=6y}={x\inZ|for integers u and v, x=2u...

What, if anything, does set theory have to do with integrating x^2 or finding the center of symmetry for a polygon?

Question 2: -------------------- Homework Statement Consider the following sets, where U represents a universal set : U = {1, 2, 3, 4, ∅, {1}} A = {1, 3} B = {{1}, 1} C = {2 , 4} D = { ∅ , 1, 2 } Homework Equations A+D is the set : (Choose only one ) 1. {1, 3}...

Question 1 : -------------------- Homework Statement Consider the following sets, where U represents a universal set : U = {1, 2, 3, 4, ∅, {1}} A = {1, 3} B = {{1}, 1} C = {2 , 4} D = { ∅ , 1, 2 } Homework Equations Choose the correct option : D - B is the set ...

I am interested in a little fun... It has been some time since I have done any set theory, and I am forgetting the symbols and ideas... So, I wanted to construct a strong inductive proof, and get a bit of help with how to concisely write the proof, and how to get TEX here at the forums to...

Homework Statement The Question data is as follows : Consider the following sets, where U represents a universal set : U = {1, 2, 3, 4, ∅, {1}} A = {1, 3} B = {{1}, 1} C = {2 , 4} D = { ∅ , 1, 2 ) Homework Equations Which one of the following statements is true ? 1. The...

Hey everyone, I'm currently taking a mathematics course in set theory at my university as a general elective. I was wondering does set theory have any practical applications in engineering?

Is it true that for every standard formulation T of ZFC, T ⊢ the power set of {naturals}? After all, the empty set axiom and the pairing axiom are in T, and so we get N. Then by the power set axiom we get P(N).

Homework Statement I'm attempting some self-study in set theory using the text mentioned above. The exercises here are quite different from those in previous texts which I've used, so I was hoping I could present some of my attempts (so far, only from the first problem set) and receive some...

Let D be a set that has cardinality d WTS that D has 2^{d} subsets of cardinal number d. So I was thinking about slitting D into two sets C_{1} and C_{2} both of cardinality d. From there I think that there are d^{d} subsets that contain C_{1}. Since d is an infinite cardinal d^{d}=2^{d}...

HI there, Just getting into set theory just had a few questions/clarifications I guess you could call it. if X = {[FONT=sans-serif]Ø, a} Y={{[FONT=sans-serif]Ø}, a} and Z = {a, {a}, {[FONT=sans-serif]Ø}} So i understand X has 2 elements along with Y and Z has 3. I know what the cardinalities...

So the book asks me to sketch out these graphs, and of course there are no examples. I was wondering how this is done. (a) [0,1] X [1, 2] // The X here stands for the Cartesian product. (b) ([0,1] U {2}) X [1,2] // How can I graph this? The U stands for Union and the X here stands for...

Homework Statement Prove that if k>1 then, 5/(k-1)-3/k-2/(k+2) = (9k+6)/(k-1)k(k+2) Hence simplify Ʃ of k=2 to n {(3k+2)/(k-1)k(k+2)} Homework Equations The Attempt at a Solution Ok so the first part is ok I just multiplied the denominators with the numerators and...

One thing I've wondered for quite some time, and looked for, but not found anything I consider adequate is whether there is a theory of probability based on Zadeh's fuzzy set theory. The closest I've found is Zadeh's Perception Probability Theory, but this doesn't quite seem to cut it...

Homework Statement How to write "the largest number in a set of real numbers A" using the appropriate set theory notation? Homework Equations - The Attempt at a Solution Tried Googling and searching on Wikipedia with no relevant results.

Let A, B and C be sets. Prove that if A\subseteqB\cupC and A\capB=∅, then A\subseteqC. My attempted solution: Assume A\subseteqB\cupC and A\capB=∅. Then \veex (x\inA\rightarrowx\inB\cupx\inc). I'm not sure where to start and how to prove this. Any help would be greatly appreciated. Thank you.

Homework Statement Give an example of an indexed family of sets such that the intersection of any finite subfamily is not empty, but the intersection when the index=infinity, is empty. The Attempt at a Solution The family I came up with is the exclusive interval (-1/k , 1/k) where...

Homework Statement Is this the right power set for x? Homework Equations X={S,{S}} The Attempt at a Solution P(x)={ø, {S}, {{S}}, {S,{S}}}

How would I prove A \subseteq B \Leftrightarrow A \cap B^{c} = \emptyset ?

Homework Statement if A is a subset of B, then Bc \ C is a subset of Ac \ C for any set C Homework Equations A is a subset of B = for all elements in A are also elements in B A\B = the complement of a set B in a set A A\B= A and Bc The Attempt at a Solution I tried...

I'm attempting to teach myself topology from a textbook. I'm on the first chapter and came into some trouble with some of the set theory. Here is what the textbook says. We make a distinction between the object a, which is an elemant of a set A, and the one-element set {a}, which is a...

Hi guys, I someone help me to set my logic straight and help me understand the following situation. For three I was given the following definition: P(A ∪ B ∪ C) = P(A) + P(B) + P(C) − P(A ∩ B) − P(A ∩ C) − P(B ∩ C) + P(A ∩ B ∩ C). I don't understand why we need to + P(A ∩ B ∩ C)...

Homework Statement I'm working on some set theory stuff to prepare for Topology next semester. I'm actually working out of a Topology book from Dover Publications. I could really use some direction/correction. 1. If S ⊂ T, then T - (T - S) = S. 2. If S is any set, then ∅ ⊂ S. The...

I can't think of a counterexample to disprove this set theory "theorem" Assume F and G are families of sets. IF \cupF \bigcap \cupG = ∅ (disjoint), THEN F \bigcap G are disjoint as well.

Hi, I am a high school student, and I am trying to learn set theory (it's not for school - independent study; I love it). I have a book (Takeuti/Zaring Introduction to Axiomatic Set) Theory and I am going through it, but I feel like I'm going way to slowly (after trying to go through it for...

Hello all, I'm trying to decide my courses for next semester and I'm all set except for a choice between EM2 and set theory. I've had both teachers before, and the set theory teacher is a lot better. I need EM2 for my major, but I'm afraid I'll be wasting the opportunity to take a fun class...

Homework Statement Given the following four statements concerning the student body at CU: ... b) There are no women engineering students at CU ... Homework Equations n/a The Attempt at a Solution Let W be the set of all women Let E be the set of all engineering students...