Set theory Definition and 439 Threads

Homework Statement Show Hausdorff's Maximality Principle is true by the Well-Ordering Principle. 2. Relevant propositions/axioms The Attempt at a Solution Case 1: ##\forall x,y\in X## neither ##x\prec y## or ##y\prec x## is true. Hence any singleton subset of ##X## is a maximal linear order...

why is not always true that if ##\vert A\vert\leq\vert B\vert## then there exist an injection from ##A## to ##B##?

Hi. Usually, Computer Programmers use Flow Charts, Algorithms, or UML diagrams to build a great software or system. In the same manner, in Mathematics, what do Mathematicians use to build a great system that they want to build. Category Theory is at the highest level of abstraction; then...

Prove in any axiomatic set theory that:2ε{a,2,b} , where a,b are letters

This post is to set forth a little game that attempts to demonstrate something that I find to be intriguing about the real numbers. The game is one that takes place in a theoretical sense only. It starts by assuming we have two pieces of paper. On each is a line segment of length two: [0,2]...

Could someone please check if these are right?

I know there are many proofs of this I can google, but I am interested in a particular one my book proposed. Also, by countable, I mean that there is a bijection from A to ℕ (*), since this is the definition my book decided to stick to. The reasoning is as follows: ℤ is countable, and so iz ℤxℤ...

The Godel theorem shows that the standard Peano axiomatization or arithmetic is undecidable. However, there is an alternative Presburger's axiomatization of arithmetic, which is decidable. Similarly, the standard ZFC axiomatization of set theory is undecidable. For instance, the continuum...

So, here's my question. I read somewhere that all universal truths on empty domains are vacuously true, whereas all existential are false. However, if all statements of the form (∀x∈A)(P(x)) , where A is an empty set, are vacuously true, then the statement (∃x∈A)(P(x)) should also be true...

I am reading "Introduction to Set Theory" (Third Edition, Revised and Expanded) by Karel Hrbacek and Thomas Jech (H&J) ... ... I am currently focused on Chapter 1: Sets and, in particular on Section 3: The Axioms where Hrbacek and Jech set up an axiomatic systems (which they do NOT call ZFC ...

What do MHB members think are the best books at an undergraduate or senior undergraduate level on set theory ... Further what are the best books on set theory at a graduate level ... ... Peter

My main question is regarding whether the membership relation is taken as an undefined concept (as is usually hinted in set theory books) or if the membership relation can be defined within the language of first order predicate theory. Let me describe a method to define the membership relation...

Homework Statement Prove that if ##A \subseteq B##, then ##\bigcup A \subseteq \bigcup B##. Homework EquationsThe Attempt at a Solution This is a simple problem, but I just want to make sure I am writing out the proof correctly: Suppose that ##A \subseteq B##. We want to show that ##\bigcup A...

Homework Statement Assume that ##x## and ##y## are members of a set ##B##. Show that ##\{ \{x\}, \{x,y\} \} \in \mathcal{P} \mathcal{P} B## Homework EquationsThe Attempt at a Solution I know that ##\{ \{x\}, \{x,y\} \} \in \mathcal{P} \mathcal{P} B## iff ##\{ \{x\}, \{x,y\} \} \subseteq...

Homework Statement Show that ##\bigcup \{\mathcal{P} X : X \in A \} \subseteq \mathcal{P} \bigcup A## Homework EquationsThe Attempt at a Solution Suppose that ##c \in \bigcup \{\mathcal{P} X : X \in A \}##. Then by definition this means that ##\exists a \in A## such that ##c \in \mathcal{P}...

Homework Statement Suppose that the sample space is the set of all real numbers and that every interval of the form (-infinity, b] for any real number b is an event. Show that for any real number b (-infinity, b) must also be an event. The Attempt at a Solution use the 3 conditions required...

We can prove that When A and B are two sets(A≠B) (A-B) = (A∩B') = (A-(A∩B)) {We can also confirm them using venn diagram} From first and third relation A-B = A - (A∩B) By cancelling A from both side I get B = (A∩B) Which is only possible when A and B are same set. What is wrong in my proof , is...

Homework Statement From Introduction to Set Theory Chapter 8.1 exercise 1.4 Prove that Zorn's Lemma is equivalent to the following statement: For all ##(A,\leq)##, the set of all chains of ##(A,\leq)## has an ##\subseteq##-maximal element.[/B]Homework Equations N/A The Attempt at a Solution...

This is no homework for me. I am working as a teaching assistant in a lecture about logic and discrete structures for Informatics students. This should be a piece of cake, but I am not exactly sure of the logic behind. 1. Homework Statement Translate into words ∃c . ∀a ∈ A . ∀b ∈ B . ¬(a =...

Let Q denote the theory of Robinson Arithmetic. A theory T is nice iff T is consistent, is p.r. adequate and extends Q. The fixed-point lemma states that for all nice theories T, for any formula φ, there is a sentence σ such that T ⊢σ↔φ("σ")...

Homework Statement Suppose R1 and R2 are relations on A and R1 ⊆ R2. Let S1 and S2 be the transitive closures of R1 and R2 respectively. Prove that S1 ⊆ S2. Please check my proof and please explain my mistakes. thank you for taking the time to help. Homework Equations N/A The Attempt at a...

Homework Statement Let ##X## and ## Y## be non-empty sets, ##i## be the identity mapping, and ##f## a mapping of ##X## into ##Y##. Show the following a) ##f## is one-to-one ##~\Leftrightarrow~## there exists a mapping ##g## of ##Y## into ##X## such that ##gf=i_X## b) ##f## is onto...

Find a function g from {0,1} to B\A such that f^-1(g(x)) = x +2 for x∈{0,1}. Present it in the 2-row form. A = {{1},2,3} and B = {∅,1,{2},3} I know that B\A = {∅,1,{2}} and f is a bijection from A to B\A how do I find such function g? It obviously can't be bijection, how do I match one value to...

Homework Statement If ##\bf{A}## ##= \{A_i\}## and ##\bf{B}## ##= \{B_j\}## are two classes of sets such that ##\bf{A} \subseteq \bf{B}##, show that ##\cap_j B_j \subseteq \cap_i A_i## and ##\cup_i A_i \subseteq \cup_j B_j## Homework EquationsThe Attempt at a Solution Since ##\bf{A} \subseteq...

Given a set, there are subsets and possible relations between those arbitrary subsets. For a given example set, the possible relation between the subsets of the example set will narrow down to the "true" possible relations between those subsets. a) {1} Number of Subsets: ##2^1 = 2## (∅, {1})...

Sets which doesn't contain themselves are called normal sets while sets that contain themselves are called abnormal. Let ##N## be a set of all normal sets. Prove that ##N## is normal if and only if ##N## is abnormal. Proof. ##~~\rightarrow ~~ ## Suppose ##N## is normal such that ##N \not\in N##...

I bought a maths book and have discovered it's somewhat above my level. In particular I'm confused about one bit of notation. I understand the "is a member of" operator when it takes a set as argument (e.g. n ∈ ℝ) but not when the book uses it with functions (e.g. n ∈ f) Does n ∈ f mean that...

In the paper John Corcoran & Hassan Masoud (2014): Existential Import Today: New Metatheorems; Historical, Philosophical, and Pedagogical Misconceptions, History and Philosophy of Logic, http://dx.doi.org/10.1080/01445340.2014.952947, already in the introduction it says, as self-evident, that...

Hi guys, Here is a wacky question for you: Suppose you have a simple recursive function f(x)=x. Given the fact that a function f(x)=y can be rewritten as a set of ordered pairs (x, y) with x from the domain of f and y from the range of f, it would seem that the function f(x)=x can be written...

I'd really like some help in answering the next question...anything that might help will save my life: F is defined this way: F:A→B where A,B⊂P(N) and P(N) is the power set of the naturals. Let S,R∈A such that S is a proper subset of R if and only if F(S) is a proper subset of F(R) My question...

Homework Statement Homework Equations I don't think there are any in this case The Attempt at a Solution I know that in order to prove R is an equivalence relation, I'd have to show that it is Reflexive, Symmetric, and Transitive. I'm not sure why, but I'm finding this a bit difficult in...

I am reading D. J. H. Garling: "A Course in Mathematical Analysis: Volume I Foundations and Elementary Real Analysis ... ...At present I am focused on Chapter 1: The Axioms of Set Theory and need some help with Theorem 1.2.2 and its relationship to the Separation Axiom ... ... The...

I am reading D. J. H. Garling: "A Course in Mathematical Analysis: Volume I Foundations and Elementary Real Analysis" ... ... At present I am focused on Chapter 1: The Axioms of Set Theory and need some help with Theorem 1.2.2 and its relationship to the Separation Axiom ... ... The...

Hey guys! I have heard of this concept in various places and sort of understands what it attempts to do. Can anybody please explain it to me in more detail like how it works, how to notate it, and how to expand it to infinities and infinitesimals. Thanks in advance! Aakash Lakshmanan xphysx.com...

Homework Statement ##C \subseteq A \cap B \implies A \cap B \cap C = C## Homework Equations How do I get rid of the "belongs to" term on the right hand side? I know I need to prove either the left hand or the right hand side of the "or" term is correct, I'm just not sure how to get there. The...

Homework Statement Let ##A_n = (n − 1, n + 1)##, for all natural numbers n. Find, with proof, ##∪_{n≥1}A_n## Homework Equations What does that last statement mean? Union for n greater than or equal to one times the interval? The Attempt at a Solution I can't understand the question.

Homework Statement Prove that ##A \cup (A \cap B) = A## Homework Equations In the previous exercise, we proved: Let A, B be sets. Then, the following statements are equivalent: 1) ##A \subseteq B## 2) ##A \cup B = B## 3) ##A \cap B = A## The Attempt at a Solution The proof of ##A \cup (A...

It has been stated that in axiom of regularity , a set cannot be an element of itself and there is a proof for which S={S} . I can understand his proof since S is the only element and hence its method of proof is viable here . But , what if I change the question to S= {S,b} ( it is a set which...

Homework Statement [/B] P(A | \overline{B}) = ? Homework Equations Multiplicative rule: \begin{equation} P(A | B) = \frac{P(A \cap B)}{P(B)} \end{equation} Additive rule: \begin{equation} P(A \cup B) = P(A) + P(B) - P(A \cap B) \end{equation} Difference: \begin{equation} A \backslash B = A...

I was wondering if anyone could please check my work and reasoning for this problem. Thank-you! (Also, would this be considered a direct proof? How might a contradiction and IFF proof look like and compare?) Problem: Suppose F, G1 and G2 are nonempty families of sets. Prove that if F ⊆ G1 ∩ G2...

Hi, Is the following notation correct? X = {xt> max(xt-k,...,xt-1,xt+1,...,xt+k)|(t-k,...,t+k) ∈ T2k+1∧ k ∈ ℕ\{0}} where T = [1,n]∩ℕ denotes the time periods over which x runs. I basically want to say that X is the set of points that are local maxima in a neighbourhood of k points to the right...

Homework Statement Suppose A⊂B⊂C. What is A/B, A/C, and A∪B Homework EquationsThe Attempt at a Solution This isn't really a homework question, I am just trying to get some exposure to discrete math before I take it in the fall. The set differences A/B and A/C are both empty sets and the 'or'...

Could someone help me in this by simplifying/Proving the equation using Theorems/ Rules on Operation of Sets (i.e Commutative property, idempotent, assoc, dist. , definition of union , def. intersection , DeMorgan ,etc.). Letting A, B and C be three sets .. Prove/Disprove : Any solution...

Hi All, Sorry, to begin with, if the question seems to be out of place. My question has to do with the notation (#) for the number of elements in a set: $$ N = \# \{ v_i \in V \vert v_i v_j \in E \} $$ Is it well known? If written in a scientific communication, must one offer a description of...

From the text it says (P -> Q) or (P -> R) is equivalent to P -> (Q or R) I tried to see if this is true so I tried (P \to Q) \lor (P \to R) \\ (P \lor \neg Q) \lor (P \lor \neg R) \\ P \lor \neg Q \lor \neg R \\ P \lor \neg(Q \land R) \\ P \to (Q \land R) and P \to (Q \lor R) \\ P \lor...

Im just reading this one example and i am stumped at this one step. (R\to C) \land (S \to C) \\ (\neg R\lor C) \land (\neg S \lor C) \ \ \ \ \ \textrm{by conditional law}\\ (\neg R\land \neg S) \lor C \ \ \ \ \textrm{by distributive law} I don't understand how it went from the second step to...

Homework Statement Assume {B, C, D} is a partition of the universal set U, A is a subset of U and A is not a subset of B complement, A is not a subset of C complement, A is not a subset of D complement. Prove that {A ∩ B, A ∩ C, A ∩ D} is a partition of A. Homework EquationsThe Attempt at a...

Homework Statement . Let A be a set and {B1, B2, B3} a partition of A. Assume {C11, C12} is a partition of B1, {C21, C22} is a partition of B2 and {C31, C32} is a partition of B3. Prove that {C11, C12, C21, C22, C31, C32} is a partition of A. Homework EquationsThe Attempt at a Solution I know...

Homework Statement Attached is the problem Homework EquationsThe Attempt at a Solution So I have to show that each side is a subset of the other side Assume x∈ A ∪ (∩Bi) so x∈A or x∈∩Bi case 1 x∈ ∩ Bi so x∈ (B1∩B2∩B3...∩Bn) which implies x∈B1 and x∈B2 ... and x∈Bn so x∈B1∪A and x∈B2∪A...

I am trying to prove that two definitions of a finite set are equivalent. 1.) A set ##A## is finite if and only if it is equipollent to a natural number ##n##. ( natural number as the set containing all the previous natural numbers including ##0## ) 2.) A set ##A## is finite if and only if...