Set theory Definition and 438 Threads

Anyone heard of causal set theory before? Basically, it is a concept that our universe should be viewed solely as set of discrete events and the causal relations between them. I wrote a thesis where I described the Lagrangians of quantum fields. Please let me know what you think: arXiv:0905.2263

Homework Statement Suppose {Ai| i \in I} is an indexed family of sets and I does equal an empty set. Prove that \bigcap i \in I A[SIZE="1"]i \in \bigcap i\in I P(A[SIZE="1"]i ) and P(Ai) is the power set of Ai Homework Equations none The Attempt at a Solution Suppose x \in...

Homework Statement Prove that for all sets A, B, and C, (A-C) \cap (B-C) \cap (A-B) = ∅ Homework Equations The Attempt at a Solution Proof: Suppose A, B, and C are sets Let x \in (A-C) \cap (B-C) \cap (A-B) Since x \in (A-C), by definition of difference, x \in A and x \notin C...

Homework Statement For all sets A and B, if A \subseteq B then Bc \subseteq Ac. Homework Equations The Attempt at a Solution Proof: Suppose A and B are sets and A \subseteq B. Let x \in Bc By definition of complement, if x \in Bc then x \notin B Since x \notin B, x \notin A...

Ideally covers lots of content in depth with lots of exercises and doesn't skip anything in hardcover. The only bit of set theory I know is the most very basic that would occupy the first chapter in a book that would require it. Self study, very motivated. :) Thanks!

I found this question in a book. Q-Suppose you own a hotel with a countable number of rooms. One night a traveler wishes to stay in your hotel, but all the rooms are occupied. Can you give him a room without kicking anybody out of the hotel? What if a tour bus shows up with countably many...

Homework Statement A,B and C are sets. Prove (A∩B)C = AC∩BC is FALSE That is, I have to give a counterargument for this statement. Homework Equations I can't find a counterargument directly. My professor suggest trying to prove the statement to find a problem and come up with the...

Suppose that one card is to be selected from a deck of 20 cards taht cointains 10 red cards numbered from 1 to 10 and 10 blue cards numbered from 1 to 10. Let A be the event that a card with an even number is selected; let B be the event that the blue card is selected; and let C be the event...

Homework Statement Find the cardinality of the set of all equivalence relations on N Homework Equations by what we have learned yet we only have to determine if it's countable or not The Attempt at a Solution I know that the set of all relations on N is equivalent to P(NXN) thus is...

I am more precisely looking for a book on mathematical logic which presupposes only minimal exposure to set theory. Preferably something which includes an introductory chapter delineating relevant set theoretic principals. I am familiar with only basic set theory. More precisely this means...

Homework Statement Suppose X ⊂ R^n is a compact set, and U_1, U_2, U3, ... ⊂ R^n are open sets whose union contains X. Prove that for some n ∈ N (the natural numbers) we have X ⊂ U_1 ∪ ... ∪ U_n. Homework Equations A set is called compact if it is both closed and bounded. The Attempt at a...

Homework Statement Prove or find counterexamples. For any sets A, B, C in a universe U: if A union C contained B union C then A contained B Homework Equations none. The Attempt at a Solution im just not sure if i did it right. id appreciate if you can check my work and let me...

-------------------------------------------------------------------------------- I tried to do the questions but I am just not sure if i did it right. id appreciate if you can check my work and let me know what changes i have to make. thanks the symbol "n" means "intersect" U for Union...

I need help on how to get started with this question: Im stocked and i just don't have a clue on how to figure this out. Prove: If A intersect B = C intersect B and A intersect B' = C intersect B' then A = C

I'm looking for a book on Set Theory, currently. I've found one by Halmos which looks good, but I'd like some input on it.

Let A,B,C,X,Y be subsets of E,and A' MEAN the compliment of A in E i.e A'=E-A,and A^B = A \cap B Then prove the following: a) (A^B^X)U(A^B^C^X^Y)U(A^X^A') = A^B^X b) (A^B^C)U(A' ^ B^C)U B' U C' = E Thanks

They seem to be different fields but both try to underpin maths. There has been suggestions that set theory is problematic, where some paradoxes cannot be resolved. But how about Category theory? Any problems or paradoxes? Is it more promising then set theory?

since a lot of talking is going on with sets, will somebody write down the axioms in ZFC theory as a point of reference , when a discussion is opened up. thanx

Hi! I am having trouble constructing the sentences in this proof. Its very simple, proof that A \cup \left( B \cap C \right) = \left( A \cup B \right) \cap \left( A \cup B \right) So basically I need to show that if x \in A \cup \left( B \cap C \right) then x \in \left( A \cup B...

Philosophy of basic set theory proofs involving "or". Hey! I'm working through an Introduction to Analysis text, and I'm currently on the first chapter, which covers set theory. In one of the end-of-chapter problems, I'm asked to prove a basic theorem which leads to the following statement...

My analysis text mentions in passing that the real numbers can be constructed rigorously starting from set theory. I was wondering if there were a resource on the web that might go over this and show the proofs of how this is done?

I've been working on these problems and unfortunately i can't make heads or tails of these two. Any insight where to start the proof would be great. 1)Let A, B and C be sets. Show that if A~B⊆C, then A~C⊆B holds. What I got so far: Is it correct to state that A~B = A⋂B' and A~C = A⋂C'...

I've been trying to wrap my brain around sets lately. Please bear with me, as I've been trying to teach myself. So, from what I've read, you can construct most everything in modern mathematics from sets. You can form the natural numbers from the successor function, you can construct the...

Page 87 of "Introduction to Set Theory" Sorry, if this post doesn't fit into this forum, but i had no other choice. I got a electronic version of "Introduction to Set Theory" by "Karel Hrbacek" and "Thomas Jech". It is a magnificent book which opens up every window of understandings of...

Homework Statement Assume that S is a function with domain w such that S(n) is a subset of S(n^+) for each n in w. (Thus S is an increasing sequence of sets.) Assume that B is a subset of the union of S(n)'s for all n such that for every infinite subset B' of B there is some n for which B'...

I started reading a book on writing proofs (it is all about set theory), and I really enjoy it. If I do physics at uni, will I get to use things like set theory and to write proofs? And if so what specific applications does set theory have in physics?

Homework Statement Show that the equation f(m,n) = 2^m(2n+1)-1 defines a one-to-one correspondence between w x w to w. Where w (omega) is a symbol used to represent the 0,1,2,3,4,5,6... Question: The book defines a one to one correspondence as a one to one function from A onto B. Is...

I was wondering if someone could please look over my proof of this set theory problem and let me know if I am doing it right or not and give me some help. Provide a counterexample for the following: If (A-B)intersect(A-C)=empty set, then B intersect C = empty set. Proof: Assume...

Homework Statement This is the problem stated verbatim. xo is supped to be x with a subscript o. Suppose that A is a set and there exists xo ε A for which lx-xol ≤ r. Is it necessarily true true that for all x,y εA, we will have lx-yl ≤2r? Homework Equations Well, this problem is just...

Very simple question :smile: Are the Pairing Axiom and the Union axiom in the Zermelo–Fraenkel set theory the same? I have a book that states them as the following: Pairing Axiom: For any sets u and v, there is a set having as members just u and v. Union axiom: For any sets a and b there...

I'm looking for a book that can stand as an introduction to axiomatic set theory (if it contains basic logic even better). Only thing I need it to be in the public domain and freely available online, anyone know of anything? Thanks in advance!

Any books that really stand out? Currently, I'm reading "Set Theory and Logic" by Stoll. I'm not interested in the axiomatic type of set theory, like Godel's theory and all those unreadable symboic proofs. I'm more interested in stuff like the axiom of choice proofs and such. Also, is there...

I'm doing to come up with a subject in either of them to do either an "independent study" or "project" on, the former is a course which simply requires you to learn the subject and the latter is "independent study" + a x-page paper. Unfortunately I don't know either subject too well so I can't...

Just checking here. Propositional logic connectives like AND and OR have analogs or representations in set theory. For example, the logical connective AND is represented in set theory by intersection, an element of X AND Y is the element of the intersection of sets X and Y. And similarly, the...

Homework Statement Prove that, for all n, for all m with 0 <= m <= n, the number of subsets of {1, . . . , n} of size m is the same as the number of subsets of {1, . . . , n} of size n − m. Homework Equations n/a The Attempt at a Solution My problem is that I don't know where to...

Homework Statement I would like to show that if we have a non-negative real valued function f defined on f a set X, and that the set of points where f is non-vanishing is uncountable, then for any M > 0, I can find a sequence {x_n} of points in X such that \sum_n f(x_n)>MHomework Equations...

Homework Statement Show that if every total order of a set x is a well-order, then there is no bijection between x and x\cup\{ x\} = Sx.The Attempt at a Solution Suppose there was, then you can have a total order on x and an induced total order on Sx. But this induced order on Sx is a total...

In Munkres' Topology he defines a Cartesian product AxB to be all (a,b) such that a is in A and b is in B. He says that this is a primative way of looking at things. And then defines it to be {{a},{a,b}} He says that if a = b then {a,b} will just be {a,a} = {a} and therefore will only be...

Homework Statement Let X be any set, f a function. Let f:X->Y Does f(A) make sense for A in X? I know f^(-1)(B) makes sense for B in Y. The Attempt at a Solution I can't see why not

Homework Statement THIS PROBLEM IS DRIVING ME INSANE! HELP! Let M be a metric space in which the closure of every open set is open. Prove that M is discrete. Homework Equations The Attempt at a Solution

Homework Statement Trying to prove some of the basic laws in set theory, and would like any opinions on 1 of my proofs (eg hints on how can I improve it, is it even a valid proof). Thanks in advance. (A \subseteq B \wedge B \subseteq C) \rightarrow (A \subseteq C) Homework Equations...

Homework Statement I have to prove that if A blis a subset of B then B' is a subset of A'. Homework Equations The Attempt at a Solution I did: Let x belongs to B but x does not belong to A =>x does not belong to B' but x belongs to A' Hence proved. please tell me if I am...

Homework Statement Let X be a finite set with n elements. Prove that P(X) has 2^n elements. <This is an extra credit problem for a summer class I'm taking.> Homework Equations P(X) is the power set of X, the set of all possible subsets of X. The principle of induction. The...

Does anyone know how Cantor showed the existence of Transcendental numbers. How can he say that most numbers are transcendental? Is that why everyone critised it? Cheers Ash

Homework Statement A is some set. R is a relation (set of ordered pairs), and is transitive on A. S = {(x,y) | (x,y) is element of R, (y,x) is not element of R} Show that S is transitive and trichotomic on A. Homework Equations Transitivity: With xRy and yRz ==> xRz The...

I am interested in learning set theory. It is an independent study. I already have previous knowledge of logic and deduction. Does anyone know of any good resources for learning set theory? Also, the reason I plan on learning set theory is so I can learn topology afterward, so any learning...

Homework Statement Suppose B is a set and suppose \mathcal{F} is a family of sets. Prove that \cup {A\setminus B|A \in \mathcal{F}}\subseteq \cup(\mathcal{F}\setminus \mathcal{P}(B)) For want of a better way I'm denoting powerset of B as \mathcal{P}(B)) Homework Equations The...

Homework Statement Prove that if R is a symmetric relation on A, and Dom(R) = A, then R = the identity relation. 2. The attempt at a solution My problem is... I don't believe the claim. At all. If A = {1, 2, 3} and R = {(1, 2), (2, 1), (3, 1), (1, 3)}, that satisfies the antecedent, and...

So I'm taking a probability class right now. We are going over elementary set theory, and the professor brought something up which seems non-intuitive to me. He said that a set must have distinct objects, so... A = \{ 1, \,\, 1, \,\, 2, \,\, 3 \} is not properly defined, because the 1 is...

It has been known for some time that the Axiom of Choice (if you treat it as a proposition to be proved rather than an axiom) and the Continuum Hypothesis are independent of Zermelo-Fraenkel set theory (ZF). These and other statements (Suslin's Problem, Whitehead's Problem, the existence of...