Set theory Definition and 439 Threads

Homework Statement Prove: Let F and G be functions. F = G if and only if domF = domG and F(x) = G(x) for all x \in domF. Homework Equations ? The Attempt at a Solution If F = G, then (xFy if and only if xGy) (Substitutivity of identicals) If (xFy if and only if xGy), then...

Hello All, I am taking a Discrete math course and am having trouble with set theory proofs. I can do basic ones, like prove <a,b,c>=<u,v,w> if a=u, b=v, and c=w. But as soon as it changes even a little bit, I cannot prove it. I was wondering what tips some of you might suggest to me? I do...

Homework Statement Give an example of an indexed collection of sets {A_{\alpha} : \alpha\in\Delta} such that each A_{\alpha}\subseteq(0,1) , and for all \alpha and \beta\in\Delta, A_{\alpha}\cap A_{\beta}\neq \emptyset but \bigcap_{\alpha\in\Delta}A_{\alpha} = \emptyset.Homework Equations...

Homework Statement I am trying understand the approach to the problems given in the following textbook. Linear Operator Theory in Engineering and Science (Applied Mathematical Sciences) (Volume 0) [Paperback] Arch W. Naylor (Author), George R. Sell (Author). Pages 12-29. There are excercise...

I am doing some non-homework exercises in preparation for my midterm, and am struggling with the following proofs: First Prove {} is a subset of {}, where {} refers to an empty set My professor told me to do this by contradiction. So I assume that {} is not a subset of {}. That would imply...

I'm reading the book Basic Set Theory by: Azriel Levy as I thought it might help me better understand Group Theory and Matrix Math. I have read the first chapter a number of times but I keep getting hung up on some of the syntax of the basic language or language of first-order predicate...

Homework Statement Let T be a family of finite subsets of the natural numbers N = {1, 2, 3,...} such that if A and B are any members of T, then the intersection of A and B is nonempty. (a) Must N contain a finite subset F such that the intersection of A, B and F is nonempty for any sets A...

Hi I was reading through a textbook and I came across the set theoretic definition of an ordered pair (Kuratowski), where (x,y)={{x},{x,y}}, which apparently can be shortened to {x,{x,y}}. This seems to be the standard definition for an ordered pair in set theory so that we can determine...

Homework Statement Determine whether or not the following definition of an ordered pair is set theoretic (i.e. you can distinguish between the "first" element and the "second" element using only set theory). (x,y)={x,{y}} Homework Equations The Attempt at a Solution I am inclined to think...

Homework Statement Homework Equations An equivalence relation on a set A, is for a,b,c in A if: a~a a~b => b~a a~b and b~c => a~cThe Attempt at a Solution It seems uncomplicated, but I don't know how I would write down a proof. The book I'm using is Topics in Algebra, 1st Edition Herstein

Homework Statement X={8^n-7n-1/n belongs to N} Y={49(x-1)/x belongs to N} Homework Equations Then, x is subset of y ] or y is subset of x or x=y,none of these

Hey friends! How can the set of all points on the circumference of a circle be an Infinite Set? Anyone could explain? Thanks in advance! -Saphira :)

Homework Statement Let A be a set and let P(A) denote the set of all subsets of A. Prove that A and P(A) do not have the same cardinality Homework Equations NOne The authors proof is as follows: Suppose there is a bijection f:A -> P(A). Let B= {x\in A| x \notin f(x)}. There is a y \in A...

At first glance these things seem so intuitive and familiar from other maths (like distribution) that I don't see how/where to start in proving them; while I know its probably quite simple. I understand what union and intersection are, but I'm unsure if multiplying two sets means a new set with...

What is the great mathematical utility of set theory? When you study set theory or when you studied do(did ) you know for what ecxacly you are(were) studying?And what do you know now?

Homework Statement Can you conclude that A = B if A, B, and C are sets such that A \cup C = B \cup C and A \cap C = B \cap C Homework Equations The above is part c of a problem. The problems a and b are as follows A) A \cup C = B \cup C My answer: I gave a counter example...

Homework Statement Suppose A,B,C are sets. Prove that A× (B U C)= (AxB) U (C x A)

There are probably a million theorems called "the recursion theorem" and I'm not sure if this is actually one of them, but there's a remark saying that it defines recursion. It's proven by piecing together 'attempts' (functions defined on subsets of a domain) and states: For X a well-ordered...

Hi Everyone! I would really appreciate some help with this set proof. Homework Statement Show that, for any sets A, B, ((A ∪ B) ∩ A) = (A ∩ A') ∪ (A ∩ B). (Hint: Remember that a complement of a complement is just the original set.) Homework Equations I can use any sentential...

Greetings all. I was wondering if someone could give me some asistance on a few simple set theory proofs. 1) An(B\C)=(AnB)\(AnC) 2) A+(B+C)=(A+B)+C where + = symmetric difference 3) If A+B = A+C then B=C and 4) An(B+C) = (AnB)+(AnC)

Homework Statement Prove that \forall sets A, B, C , if B\subseteq C, then A \times B \subseteq A \times CHomework Equations The Attempt at a Solution Haven't done set theory proofs in a while. Does this suffice in proving the statement?: Let x \in B be arbitrary. Assuming B \subseteq C...

How to prove the set of complex numbers is uncountable? Let C be the set of all complex numbers, So C={a+bi: a,b belongs to N; i=sqrt(-1)} -------------------------------------------------- set of all real numbers is uncountable open intervals are uncountable...

this problem is from Apostol's Calculus Vol.1, i just started doing proofs so I'm still getting used to it. B - U (A) = ∩ (B-A) the U is the union of sets in a class F and ∩ is the intersection of sets in a class F. written under both the U and the ∩ are A∈F. so i let an element x ∈ B...

Homework Statement Let A,B,C be sets where A n C = B n C and A n Cc = B n Cc. Then A=B. Homework Equations The Attempt at a Solution To prove two sets equal, i think we want to let x be in A, and then show as a result that x in B also. However, i don't see how this is possible...

hi guys, I am a physics major, recently doing a few self-reading on analysis, however, I got stuck at some excises of proofs. Homework Statement If a set A contains n elements, prove that the number of different subsets of A is equal to 2^n. The Attempt at a Solution First, I teared...

Homework Statement The boundary \partial E of a set E if defined to be the set f points adherent to both E and the complement of E, \partial E=\overline{E}\bigcap \overline{(X\backslash E)} Show that E is open if and only if E \bigcap \partial E is empty. Show that E is closed if and only...

Homework Statement I have two sets E and L. I have to show that E is a subset of L, using only complement, union and the empty set. I.e. all members of E also have to be members or L, but L may contain members not included in E.Homework Equations The Attempt at a Solution Am I right in...

Homework Statement We're working more or less with the standard ZF axioms. Prove that A \subseteq \mathcal{P}(\bigcup A) for any set A, whose elements are all sets. When are they equal? Homework Equations Just the axioms I) Extensionality II) Emptyset and Pairset III) Separation IV)...

Been working on this for 2 days and have gotten no where. Maybe someone out there can show me the light U = {1, 2, 3, ... , n, ..., 2n} for some natural number n Let P be a subset of U such that |P| = n + 1 show that there exists x, y in P where x not equal y such that x divides y or y...

Homework Statement We're asked to prove that a few constructions of the sets a,b are themselves sets, stating which axioms we use to do so. a) a\b b)the function f:a->b c)the image of f Homework Equations The following standard definitions of axioms of construction...

In 'AN INTRODUCTION TO SET THEORY' by Professor William A. R. Weiss, the following symbols are applied to the snippet below: ... The collection of formulas of set theory is defined as follows: 1. An atomic formula is a formula. 2. If ϕ and Ψ are formulas, then (ϕ ^ Ψ) is also a formula. ... I...

I am in calculus, I was just wondering what math classes (or prerequisites) before trying to learn set theory. For example Should I learn multivariable calculus, linear algebra, etc. (or what)?

Homework Statement Without using the Axiom of Choice, show that if A is a well-ordered set and f : A -> B is a surjection to any set B then there exists an injection B -> A. Homework Equations The Attempt at a Solution I was wondering if the existence of the surjection from a well...

Is there a way to prove axioms are consistent and/or independent by converting the problem of consistency/independence to another field, such as algebra?

Problem: (i)A\subseteqB \Leftrightarrow A\cupB = B (ii) A\subseteqB \Leftrightarrow A\capB = A and For subsets of a universal set U prove that B\subseteqA^{c} \Leftrightarrow A\capB = empty set. By taking complements deduce that A^{c}\subseteqB \Leftrightarrow A\cupB = U. Deduce that...

Problem: We define half infinite intervals as follows: (a, \infty) = {x\in R | x>a}; [a, \infty) = {x\in R | x\geqa}; Prove that: (i) (a, \infty) \subseteq [b, \infty) \Leftrightarrow a\geqb, (ii) [a, \infty) \subseteq (b, \infty) \Leftrightarrow a>b. I've got pretty much no idea how...

Why does "Set theory" exist? Is there actually a use for Set theory? I understand why Algebra, Calculus, Trig, Physics, etc.. exist. Does anyone ever use it at their job (besides at some type of school).

Hi, Right now I am currently going through Set Theory and Logic by Stoll. This is my first time going through Set Theory, Logic, and everything else in the book. I feel like I'm not getting all I can out of this book because I don't feel like I'm on the same level, I struggle to answer the...

Set Theory and Predicate Calculus (12 points) Given: P ⊆ Q Q ⊆ (S ∩ T) S ⊆ (R ∪ T^c) x(sub)1 ∈ P Use predicate calculus to prove x(sub)1 ∈ R. Studying for a test but I don't have this worked out for me. I honestly don't even know where to start. I know what union, intersect, etc and...

I'm reading a book about set theory and it introduced the concept of power set. Ok, I understand what is a power set and the entire concept but I have a question about the number of elements of a power set. There's written in the book that the number of elements of a power set is 2n where n...

Homework Statement Prove the set of continuous functions from R to R has the same cardinality as RHomework Equations We haven't done anything with cardinal numbers (and we won't), so my only tools are the definition of cardinality and the Schroeder-Bernstein theorem and its consequences. I...

Hi, I have a situation where I want to define a function over all possible rings. For example, I would like to define a function that accepts a ring as an input and returns its additive identity. However, this seems impossible to do in ZF set theory since we can not define the domain of...

Can anyone please give a really explicit proof (omitting no steps) and with as simple words as possible that any infinite set can be writtern as the union of disjoint countable sets? Thank you.

f:X\rightarrowY, A\subsetX f(Ac)=[f(A)]c implies f bijective. Just trying to apply the definitions of injective and bijective. The equivalence makes sense but I am having a hard time showing it. f(x)=f(y) implies x=y and for every y in Y there exists a x in X s.t. f(x)=y. I mean all I have is...

Homework Statement I have been asked to prove the following statement: An accumulation point of a set S is either an interior point or a boundary point of S. Here is my attempt at a solution: I started from the definition of accumulation points: A point is an accumulation point (x_0) when...

How do you write in proper set theoretic notation that a set A = (x,x) where x is a non-negative real number? Also, (x_1, y_1) R (x_2, y_2) if x_1 ^2 + y_1 ^2 = x_2 ^2 + y_2 ^2 The equiv. classes are circles at (0,0), right? How do you write this formally (using set theoretic notation)?

Letting A, B_1, B_2, ..., B_n subsets of X, then show A\cap\bigcup_{n}^{i=1}B_{i} = \bigcup_{n}^{i=1}\left(A \cap\right B_{i}) ---- Is it sufficient to say... By De Morgan law, we have \left(A\cap \right B_{i})\cup\left(A \right \cap\ B_{2})\cup\ --- \cup\left(A\cap \right\ B_{n}) =...

Hi, Im only starting to learn about naive set theory from a book , so pardon me if my answer to the question is really obvious.. Prove that .. A\subseteqB , if and only if A\capB =A,if and only if A\cupB=B, if and only if A-B=empty set.. I was thinking of using venn diagrams to...

Hey guys I signed up for a "set theory and topology" class for the fall and was planning on taking real analysis in the spring. Set theory got canceled and so i am taking real analysis instead and pushing set theory to the spring. Is this a wise idea? Taking real analysis before a formal set...

I'm supposed to write the following intervals as sets in descriptive form: a. (t, infinity), t a fixed real number b. (0, 1/n), n a fixed natural number --- I think it is: a. (t, infinity) = {x: t < x < infinity} b. (0,1/n) = {x: 0 < x < 1/n} Is this correct? Also, how do you...