Set theory Definition and 439 Threads

Homework Statement Let f: X ----> Y and g: Y ----> Z be functions and let h = g o f: X ----> Z Homework Equations a. If h is surjective then g is surjective b. If h is surjective then f is surjective. The Attempt at a Solution Here h: X ----> Z a. Suppose h: x ---> z is surjective...

Homework Statement For each structure, draw a directed graph representing the membership relation. Then determine which of the following axioms is satisfied by the structure: Extensionality, Foundation, Pairing, Union U= {a,b} a in b , and b in a The Attempt at a Solution The directed...

Homework Statement I'm reviewing my powerpoints from class and see the formula A ∩ B ⊆ A. Is this a correct formula? I interpret this as all elements of set A intersected with set B is a subset of set A. I don't think this is a true statement, is it? Sorry it's been a while since I have...

Homework Statement So I was doing this problem in Munkres's Topology book: Determine whether the statement is true or false, If a double implication fails, determine whether one or the other of the possible implications holds: A ⊂ B or A ⊂ C ⇔ A ⊂ ( B ∪ C ) Homework Equations - The Attempt...

Hi As I am venturing in graduate level mathematics, I am having a recurring problem; I keep getting stuck in the abstraction of it. Usually it involved set theory; I never get "fluent" in it. However, the main problem is abstraction. For instance, this semester I had topology, and the...

I am reading Micheal Searcoid's book: Elements of Abstract Analysis ( Springer Undergraduate Mathematics Series) ... I am currently focussed on Searcoid's treatment of ZFC in Chapter 1: Sets ... I am struggling to attain a full understanding of the Axiom of Replacement which reads as shown...

I am reading Micheal Searcoid's book: Elements of Abstract Nalysis ( Springer Undergraduate Mathematics Series) ... I am currently focussed on Searcoid's treatment of ZFC in Chapter 1: Sets ... I am struggling to attain a full understanding of the Subset Principle which reads as shown below...

Hello, I am trying to understand the intuition of the definition of the period of a state in a Markov chain. Say for example we can go from state i to state i in 3,5,7,9,11... and so on steps. The gcd here is one. So is this aperiodic state or one with periodicity of 2. Thanks

We can easily find out below rules in set theory: 1. Let consider set “A” as follows: A = {a1, a2, a3, a4… an} and also power set of A is set C: C = {{}, {a1}, {a2}, {a3}, {a4}, {a1, a2}, {a1, a3},….{an}} Rule 1: To find the number of subsets with precise members number, we can use Binomial...

I am looking for a book that starts at the standard ZFC axioms and progresses to the point where some recognisable non-trivial mathematical statement is proved. By recognisable I mean something that you may encounter in school/early university level and is not purely set-theoretical (e.g...

Hi Guys, I am self teaching myself analysis after a long period off. I have done the following proof but was hoping more experienced / adept mathematicians could look over it and see if it made sense. Homework Statement Question: Suppose D is a non empty set and that f: D → ℝ and g: D →ℝ. If...

Hi. I hope this is not too far into philosophy. Set Theory is commonly accepted as the foundations of Mathematics. Is it possible to develop a different type of Mathematics by using Fuzzy sets or Rough sets instead?

Homework Statement Problem Consider a random experiment with a sample space S={1,2,3,⋯}. Suppose that we know: P(k) = P({k}) = c/(3^k) , for k=1,2,⋯, where c is a constant number. Find c. Find P({2,4,6}). Find P({3,4,5,⋯}) I am primarily interested in part 1, finding C. The rest...

Homework Statement Let A and B be two events such that P(A) = 0.4, P(B) = 0.7, P(A∪B) = 0.9 Find P((A^c) - B) 2. Homework Equations I can't think of any relevant equations except maybe the Inclusion Exclusion property. P(A∪B) = P(A) + P(B) - P(A∩B) This leads us to another thing P(A∩B^c)...

Hello all, Can anyone give me a pointer on how to start this proof?: f:E\rightarrow F we consider f^{-1} as a function from P(F) to P(E). Show f^(-1) is injective iff f is surjective.

Homework Statement Hello, I need some help on the following. I am BRAND new to set theory and this was in my first HW assignment and I am not sure where to start. The question is as follows: Let A and B be parts of a set E Let P(E)\rightarrow P(A) X P(B) be defined by f(X)=(A\cap X,B\cap X)...

Hi guys. So I've been wondering, what's so controversial about the axiom of choice? I heard it allows the Banach-Tarski Paradox to work. A little insight would be much appreciated, thanks.

Homework Statement $$ A \subset B \Rightarrow A \cap C \subset B \cap C $$ 2. Homework Equations [/B] $$ A \subset B \Leftrightarrow A \cup B \subset B$$ $$ A \cap C \Leftrightarrow A \cap C \subset A \wedge A \cap C \subset C$$ The Attempt at a Solution For sets A and C $$A \cap C...

I read somewhere that mathematical functions can be implemented as sets by making a set of ordered tuples <a,b> where a is a member of A and b is a member of B. That should create a function that goes from the domain A to the range B. But set theory has functions too, could they be sets too...

Hi! I would like to know if my assumptions are right: Topology is the merging domain of analysis and algebra; Relational algebra use topological operators; Relational algebra is a specification of topology ?

Homework Statement [/B] I am going through Apostol's Calculus volume 1 and am working through I 2.5 #3. I'm not very familiar with doing proofs so I just wanted to make sure that I got the right idea here. Here's the question: Let A = {1}, B = {1,2} Prove: 1. ## A \subset B ## 2. ## A...

Homework Statement 1. Prove that if A \cap B = A and A \cup B = A , then A = B 2. Show that in general (A-B) \cup B \neq A 3. Prove that (A-B) \cap C = (A \cap C) - (B \cap C) 4. Prove that \cup_{\alpha} A_{\alpha} - \cup_{\alpha} B_{\alpha} \subset \cup_{\alpha} (A_{\alpha} -...

Homework Statement We have the set D which consists of x, where x is a prime number. We also have the set F, which consists of x, belongs to the natural numbers (positive numbers 1, 2, 3, 4, 5..) that is congruent with 1 (modulo 8). What numbers are in the intersection of these two sets...

I would appreciate any and all feedback regarding this document currently housed in Google docs. Basically, I generalize induction among natural numbers to an extreme in an environment regarding what I call grammatical systems. Then an induction principle is derived from that which holds in...

Homework Statement If I am given ##n(A)## and ##n(B)## for two sets A and B, and also provided with ##n(A\cap B)^2##. We are supposed to find ##n((AXB) \cap (BXA))##. Homework Equations My teacher said that the formula for ##n((AXB) \cap (BXA)) = n(A \cap B)^2##. I am not sure how do you get to...

I am planning to study the following pure mathematics areas (on my own) and wanted to know if this is the best sequence: 1- Formal Logic 2 -Philosophical Logic 3- Sentential Logic 4- Predicate Logic 5- Symbolic Logic 6 -Set Theory 7 -Pure Mathematics (Intro, Pure Math I and II and Hardy) -...

So I just finished "Book of Proof" and I'm looking for a more rigourous book on axiomatic set theory, including Gödel's theorems.Any recommendations?

The book I am using for my Introduction to Topology course is Principles of Topology by Fred H. Croom. Problem: Prove that if ##X=X_1\times X_2## is a product space, then the first coordinate projection is a quotient map. What I understand: Let ##X## be a finite product space and ##...

I'm currently an undergrad in math who's going to graduate next year. I'm interested in doing research in set theory (not now of course, perhaps in grad school). Unfortunately, I'm at a liberal arts school and there are no set theorists in the math department. All they offer is a naiive set...

Lately I have been attempting (and failing miserably at) whatever sample Putnam questions I can find on the internet. Here is my latest endeavor. I found this question on the Kansas State University website, so I think I am allowed to post it. I must warn you that I know almost nothing about...

So I'm reading Naive Set Theory by Paul Halmos. He asks: His response is that no ##x## fails to meet the requirements, thus, all ##x##es do. He reasons that if it is not true for a given ##x## that ##x \in X~ \mathrm{for ~ every} ~X~ \mathrm{in} ~ \phi##, then there must exist an ##X## in...

So I'm reading up on some set theory, and I came to the axiom of pairing. The book uses that axiom to prove/define a set which contains the elements of two sets and only the elements of those two sets. ##~~B## is the set which contains the elements and only the elements of sets ##a## and...

I am considering the following question and I want you to agree (but perhaps you don’t):Rutherford wrote a letter to Bohr, as an answer to a previous letter from Bohr containing one of the first of Bohr’s descriptions of the atomic model, saying that he understood the atom model Bohr advocated...

I'm coming from a physics background, but find pure mathematics extremely interesting, so have decided to try and gain a more fundamental understanding of the subject. I've recently been reading up on relations and how one can define them as sets of ordered pairs. I am particularly interested in...

The class is called Math for EE and CE. The professor teaches from his own notes and doesn't give many examples. Any help checking my work would be appreciated and/or if you could point me in the direction of more examples like these. I've looked trough Set Theory and discrete math books but...

Apologies if this is in the wrong forum, but I chose to post here as the question pertains to equivalence relations and classes. Sorry if it's such a trivial question, but what is the mathematical difference between equivalence and equality? My understanding is the following, but I'm a little...

Homework Statement Show that the set S of all (x,y) ∈ ℝ2such that 2x2+xy+y2 is closed but not compact. Homework Equations set S of all (x,y) ∈ ℝ2such that 2x2+xy+y2 The Attempt at a Solution I set x = 0 and then y = 0 giving me [0,±√3] and [±√3,0] which means it is closed However, for it to...

Homework Statement Ok I created this question to check my thinking. Are the following Sets: Open, Closed, Compact, Connected Note: Apologies for bad notation. S: [0,1)∪(1,2] V: [0,1)∩(1,2] Homework Equations S: [0,1)∪(1,2] V: [0,1)∩(1,2] The Attempt at a Solution S: [0,1)∪(1,2] Closed -...

Homework Statement I am trying to understand why ℕ the set of natural numbers is considered a Closed Set. 2. Relevant definition A Set S in Rm is closed iff its complement, Sc = Rm - S is open. The Attempt at a Solution I believe I understand why it is not an Open Set: Given that it...

Homework Statement Problem: Given a regular deck of 52 cards, let A be the event {king is drawn} or simply {king} and B the event {club is drawn} or simply {club}. Describe the event A ∪ B Solution: A ∪ B = {either king or club or both (where "both" means "king of clubs")} Homework Equations...

Homework Statement One way of modeling tuples in set theory is through nested ordered pairs. A notation I'm not familiar with (I'm assuming it means that the following elements are nested into the last one) is used. (a1, a2, a2,... an) = (a1(a2, a3,..., an)). I have never seen the second "(" in...

I have two quick questions: With P being the power set, P(~A) = P(U) - P(A) and P(A-B) = P(A) - P(B) I'm told if it's true to prove it, and if false to give a counterexample. To be they're both false, since the null set is part of any power set, the subtraction of two power sets would get...

Are these three sets equivalent? $$A=\left\{(x,y):x,y\in\Bbb{R},y\ge x^2-1\right\}$$ $$B=\left\{x,y\in\Bbb{R}:y\ge x^2-1\right\}$$ $$C=\left\{(x,y)\in\Bbb{R}^2:y\ge x^2-1\right\}$$ I am thinking that $A$ and $C$ are, but not $B$ as it might be ambigious as to which dimension it is in, i.e it...

I'm introducing myself to set theory. My reference doesn't seem to address the fact that 1/1 = 2/2 = 1. If we make a correspondence between natural numbers and rational numbers using sequential fractions, should we just skip equivalent fractions so as to make it a bijection? In other words, does...

Homework Statement Prove if an ordered set A has the least upper bound property, then it has the greatest lower bound property. Homework Equations Definition of the least upper bound property and greatest lower bound property, set theory. The Attempt at a Solution Ok, I think that my main...

Could anyone recommend some good books on set theory?

Hi folks, 1. Homework Statement I don't fully understand the question statement, how is it supposed to be read? Question: Give a formula for the minimizer x* (to be read as x-star) of the function ƒ:ℝn → ℝ, x → ƒ(x) = ||Ax-b||22, where A∈ℝm×n and b∈ℝm are given. You can assume that A has rank...

Mod note: Moved from a technical section, so missing the homework template. Here is what I'm trying to prove. Let f:A->B. If there are two functions g:B->A and h:B->A such that g(f(a))=a for every a in A and f(h(b))=b for every b in B, then f is bijective and g=h=f^(-1). I think I have most...

My question is on how to answer if two statements are equal in set theory. Like De'Morgans laws for example. I'm currently reading James Munkres' book "Topology" and am working through the set theory chapters now, and this isn't the first time I've seen the material, but every time I see this...

Homework Statement Show that, ∪n=2∞[0,1 - 1/n] = [0,1) Homework EquationsThe Attempt at a Solution