Suppose I have the following ( arbitrary ) statement:
$$ \forall x\in{S} \ ( P(x) ) $$
Which means: For all x that belongs to S such that P(x).
Can I write it as the following so that they are equivalent? ( although it is not conventional ):
$$ \forall x\in{S} \land ( P(x) ) $$
Can I write...
Hello there,
I had another similar post, where asking for proof for Hilbert’s Hotel.
After rethinking this topic, I want to show you a new example. It tries to show why that the sentence, every guest moves into the next room, hides the fact, that we don’t understand what will happen in this...
"The fact that the above eleven properties are
satisfied is often expressed by
saying that the real numbers form a
field with respect to the usual addition and
multiplication operations."
-what do these lines mean?
in particular the line "form a field with
respect to"?
is it something like...
Homework Statement: See attached image.
Homework Equations: ZFC set theory.
Consider the text in the attached image. What is meant with "We require of an axiom system that it be possible to decide whether or not any given formula is an axiom."? Is consistency synonymous with soundness? Is...
The following assertion quoted from the paper below seems as though it couldn’t be true. It is the issue that I would like some help addressing please:
“The restriction of ##g(A)## to ##A \cap \omega_1## ensures that ##B## remains countable for this particular ##T## sequence.”
...
Define...
<Moderator's note: Moved from a technical forum.>
Hi PF,
I am learning how to prove things (I have minimal background in math). Would the following proof be considered valid and rigorous? If not any pointers or tips would be much appreciated!
Problem:
Prove that the notion of number of...
Is it possible to calculate this :
Suppose the iterative root of ##2^x## :
##\phi(\phi(x))=2^x## (I suppose the Kneser calculation should work, it affirms that there is a real analytic solution)
Then how to compute ##\phi(\aleph_0)## ? (We know that ##2^{\aleph_0}=\aleph_1##).
Could this be...
First, I want to be pedantic here and underline the distinction between a set (in the model, or interpretation) and a sentence (in the theory) which is fulfilled by that set, and also constant symbols (in the theory) versus constants (in the universe of the model)
Given that, I would like to...
Hello, I am stuck on deciding if given sets are recursive or recursively enumerable and why. Those sets are:
set ƒ(A) = {y, ∃ x ∈ A ƒ(x) = y}
and the second is
set ƒ-1(A) = {x, ƒ(x) ∈ A}
where A is a recursive set and ƒ : ℕ → ℕ is a computable function.
I am new to computability theory and any...
Hello Everyone,
I am trying to write the intersection of a physical problem in the most compact way.
I am not really familiar with Set Theory notation, but I think it has the answer.
It is about the intersection of two circular areas:
- Area 1: A
- Area 2: B
If I want to write this in Set...
Homework Statement
Let ##f:\Bbb{R} \to \Bbb{R}## be a function such that ##f## has a local minimum for all ##x \in \Bbb{R}## (This means that for each ##x \in \Bbb{R}## there is an ##\epsilon \gt 0## where if ##\vert x-t\vert \lt \epsilon## then ##f(x) \leq f(t)##.). Then the image of ##f## is...
Let a function ##f:X \to X## be defined.
Let A and B be sets such that ##A \subseteq X## and ##B \subseteq X##.
Then which of the following are correct ?
a) ##f(A \cup B) = f(A) \cup f(B)##
b) ##f(A \cap B) = f(A) \cap f(B)##
c) ##f^{-1}(A \cup B) = f^{-1}(A) \cup f^{-1}(B)##
d) ##f^{-1}(A \cap...
<Moderator's note: Moved from a technical forum and thus no template. Also re-edited: Please use ## instead of $$.>
If ##R_{1}## and ##R_{2}## are relations on a set S with ##R_{1};R_{2}=I=R_{2};R_{1}##. Then ##R_{1}## and ##R_{2}## are bijective maps
##R_{1};R_{2}## is a composition of two...
1. The problem statement, all variables and given
Prove that ##\sqrt{2}\in\Bbb{R}## by showing ##x\cdot x=2## where ##x=A\vert B## is the cut in ##\Bbb{Q}## such that ##A=\{r\in\Bbb{Q}\quad \vert \quad r\leq 0 \quad\lor\quad r^2\lt 2\}##.
I believe that I have to show ##A^2=L## however, it...
The theorem: Let ##X##, ##Y## be sets. If there exist injections ##X \to Y## and ##Y \to X##, then ##X## and ##Y## are equivalent sets.
Proof: Let ##f : X \rightarrow Y## and ##g : Y \rightarrow X## be injections. Each point ##x \in g(Y)⊆X## has a unique preimage ##y\in Y## under g; no ##x \in...
Homework Statement
Wherein α, β are strings, λ = ∅ = empty string, βr is the shortest suffix of the string β, βl is the longest prefix of the string β, and T* is the set of all strings in the Alphabet T, |α| denotes the length of a string α, and the operator ⋅ (dot) denotes concatenation of...
Homework Statement
Wherein α is a string, λ = ∅ = the empty string, and T* is the set of all strings in the Alphabet T.
Homework Equations
(exp-Recursive-Clause 1) : α0 = λ
(exp-Recursive-Clause 2) : αn+1 = (αn) ⋅ α
The Attempt at a Solution
[/B]
This one is proving difficult for me. I...
Hello experts,
Full disclosure: I am a total layman at math, nothing in my training aside from high school courses and one college calculus class. I'm sure a week doesn't pass without someone posting a question about or challenge to Cantor. I am not here to challenge anything but rather to...
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...
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...
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]...
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...
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...
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...
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...
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...
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 Equations
The Attempt at a Solution
Since ##\bf{A}...