Set theory Definition and 439 Threads

Hello, I know that the law of the excluded middle is implied in ZFC set theory, since it is implied by the axiom of choice. Taking away the axiom of choice, does ZF set theory (with axioms as stated in the Wikipedia article http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory)...

Hello! (Wave) What is the subject Set Theory about? What knowledge is required? (Thinking) That is the Course Content: Brief report on basic elements (algebra of sets, relations and functions, etc..). Construction of the set of natural numbers. Ordinal numbers and their arithmetic. The...

Could someone please explain how the image of a set A' ⊆ A is the set: f(A') = {b | b = f(a) for some a ∈ A'}. And how can the complement of A be a subset of A? Forgive my ignorance here, I'm a beginning student of set theory. Edit: Maybe I should rephrase my question: Could you explain what...

Could someone please explain how the image of a set A' ⊆ A is the set: f(A') = {b | b = f(a) for some a ∈ A'}. And how can the complement of A be a subset of A? Forgive my ignorance here, I'm a beginning student of set theory.

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...

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...

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.

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...

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 ##...

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...

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...

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...

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...

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 =...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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.

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...

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...

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...

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...

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...

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

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...

My goal: To show the dimension of space [SIZE="3"]L equals the length of any maximal flag of [SIZE="3"]L; Is the following valid? My attempt: Let [SIZE="3"]M \rightarrow {L_{i-1}, ... L_i} where [SIZE="3"]{e_i} \in L_i [SIZE="5"]| [SIZE="3"]e_i \not\in L_{i-1} Assuming...

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.

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...

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...

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...

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 =...

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...

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...

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 ≥...

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.)

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)...

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...

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