Questions tagged [elementary-set-theory]
For elementary questions on set theory. Topics include intersections and unions, differences and complements, De Morgan's laws, Venn diagrams, relations and countability.
29,088 questions
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I.N.Herstein "Topics in algebra" Sec. 1.1 The Set theory
I was recently reading I.N.Herstein's "Topics in algebra" and stumbled across interesting proposition and it's proof:
For any three sets, $A, B, C$ we have:
$$A \cap (B \cup C) = (A \cap B) ...
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A question about the formal definition of a function graph.
With some friends I am currently reading and trying to understand Category Theory by Steve Awodey. As I am no trained mathematician, even simple issues can halt my progress. One occurred when I tried ...
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Why is $\mathbb N^{\mathbb N}$ uncountable
Before answering, I do know that $\mathbb N^{\mathbb N}$ is uncountable because there is a one to one correspondence between it to the irrational numbers. Via the unique representation of simple ...
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Closure operator over simplicial complex
A simplicial complex (sometimes refereed to as abstract complex or abstract simplicial complex) is a set system $(S, \Delta)$ where $S$ is a set and $\Delta\subseteq \mathcal P(S)$ is a family of ...
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Is it true that $X \times Y$ Hausdorff $\Rightarrow$ both $X$ and $Y$ are Hausdorff?
I've been trying to get my head across this problem. I think I found a proof, however I'm not sure if it is valid.
Let's assume $X \times Y$ is Hausdorff and prove $X$ is Hausdorff (then the same ...
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Why is $V$ not in the product sigma algebra? [duplicate]
I am reading G. De Barra's Measure and Integration. The author proves the following theorem in Page 179:
Let $(X, \mathcal{S}, \mu)$ and $(Y, \mathcal{T}, \nu)$ be $\sigma$-finite measure spaces. For $...
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A few clarifications about multiplication in subgradient calculus
In the subgradient calculus linearity properties, the appropriate side of the addition rule utilizes Minkowski addition of sets. Ordinarily in linearity, a scaling rule agrees with, and is basically ...
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Ordered Pair of Sets, Ordered Pair and Cartesian Product.
I'm confused by some definition from a book. The book gives a defeninition of an ordered pair of sets:
(This is a translation so it could be inaccurate.) The unordered pair of sets $X$ and $Y$ is $\{...
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6
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Example that the union of an infinite number of closed sets is not necessarily closed.
Example. Let $I_n = [1/n, 1]$, which is clearly closed, and consider
$$S=\bigcup_{n=2}^{\infty}I_n=[1/2,1]\cup[1/3,1]\cup[1/4,1]\cdots\tag{1}$$
This is the set
$$S=\bigg\{x\bigg\lvert x\in \mathbb{R},\...
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5
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How many elements are present in the subset of a null set?
Consider :
How many elements are present in the subset of a null set?
This is one of the question that appeared in my math exam.
Definition $1.1$ - Subset:
A set $A$ is a subset of set $B$ if all ...
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Is Jaccard distance a metric function?
We know that a function is a metric if it satisfies the following properties:
$d(p, q) > 0 \; \text{if} \; p \ne q; \; d(p, p) = 0$
$d(p, q) = d(q, p)$
$d(p, q) \le d(p, r) + d(r, q)$ (triangle - ...
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Relative to what set theory do we do model theory? [duplicate]
In regards to category theory, it's clear to me what the start point is, we collect up all possible spaces have some sort of structure which can be created w.r.t to some fundamentals of mathematics, ...
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Bijection between infinite sequences of real numbers (i.e. all functions $f \colon \mathbb{N} \to \mathbb{R}$ ) and real numbers set. [closed]
Please tell me if my reasoning is correct. We construct a bijection between these sets. We know that $\mathbb{R}$ has the same cardinality as the set of all infinite sequences of zeros and ones. Let's ...
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Prove or disprove that $\bigcup A=A$, for any set $A$
The union of a set $A$ is defined as $\bigcup A=\{x\in F | F\in A\}$. I'm going to prove the opposite.
Lemma For any set $Y, \bigcup P(Y)=Y$.
Proof: Let $x\in Y$. We have $Y\subseteq Y\Rightarrow Y\...
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Complement map on $k$-subsets: standard notation?
Let $[n]=\{1,\dots,n\}$, and write $\binom{X}{k}$ for the family of all $k$-element subsets of a set $X$.
Taking complements in $[n]$ gives the bijection
$$
\binom{[n]}{k}\xrightarrow{\;S\mapsto [n]\...
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2
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257
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$11$ committees, each has $5$ members, every $2$ have exactly one member in common
Singapore Mathematical Olympiad 2008 Round 2
There are $11$ committees in a club. Each committee has $5$ members and every two committees have a member in common. Show that there is a member who ...
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120
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How to express logical phrase "Take union of two sets if their intersection is not empty" in set algeba
This is not homework, I need it to simplify a certain kind of filtration in my research but I am stuck. I have a set of subsets $\mathcal{S}$ of a finite set $\Omega$ and I want to express "take ...
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Justifying universal generalization after existential elimination using dependence between quantifiers
In an attempt to transform following equivalent definitions of the cartesian product from the first into the second, trying to be rather formal about it:
$$
\forall A : A \in X \times Y \...
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Definition of a family of sets
Actually I'm confused about what is a family of sets. I learned that a family of elements of a set $X$ indexed by a set $I$ is a function from the $I$ to $X$. So when we talk about $\left(X_i\right)_{\...
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Is $\mathbb{Z_2}$ a subset of $\mathbb{Z_4}$ [closed]
Let $\mathbb{Z_2}=\lbrace0,1\rbrace$ where $0$ is equivalence class of all integers who gives remainder $0$ when divided by $2$ means $0$ is nothing but set of even integers and $1$ is set of odd ...
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Recommended book or note for methods of mathematical proof [duplicate]
I asking about good book for undergraduate student to learn methods of mathematical proofs in more details and has lot of examples. I found "book of proof by richard hammack" but I want more....
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Different definitions of a saturated set
I saw the definition of a saturated set on Wikipedia as follows:
Def $1$: Let $f: X \rightarrow Y$ be an arbitrary mapping. A subset $C$ of $X$ is called saturated if $C = f^{-1}(f(C))$.
...
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Is there a word for saying that for two members $x$ and $y$ of a set, any true statement will be true if $x$ and $y$ are swapped?
Is there a word for saying that for two members $x$ and $y$ of a set, any true statement will be true if $x$ and $y$ are swapped?
For example, consider the group that you get when you take from the ...
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If $f$ is a surjection from $\mathbb{R}$ to $\mathbb{N}$, why is finding an injection $g$ of which $f$ is a right inverse independent of $ZF$?
I am working through Classic Set Theory for Guided Independent Study by Derek Goldrei. At the beginning of Chapter 5, titled The Axiom of Choice, the author propositions the student to solve the ...
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set, class and collection in Halmos books "Naive set theory" (1960) and "Measure theory" (1950) [duplicate]
By looking at the definition of "set" and "collection" in the book Naive Set Theory, Halmos says that they are synonyms. The word "class" is to be left to mean a ...
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Similar result of "every non-empty open set in $\mathbb{R}$ is the union of a countable, disjoint collection of open interval"
I've been reading Royden's Real Analysis - 5th edition, and it stated a well-known result in Proposition 1.4.9 that in $\mathbb{R}$, every non-empty open set is a union of a countable, disjoint ...
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Prove that the ordered set of numerical sequences has uncountable cofinality
Prove that there does not exist a countable set $P$ of numerical sequences such that $\forall(x_n)_{n∈\mathbb{N}}\in\Bbb R^{\Bbb N}\ \exists(p_n)_{n∈\mathbb{N}}\in P\ \forall n \in \mathbb{N}\ x_n\le ...
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why is it impossible to find onto map S to its power set?
I want to verify if my proof of this problem is correct or not.
If $S$ is any set, prove that it is impossible to find a mapping from $S$ onto $S^*$, the power set of $S$.
My proof: Let $f: S- S^*$ be ...
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What is wrong with this proof that the set of all polynomials with rational coefficients is not countable?
I constructed this proof that the set of all polynomials with rational coefficients is uncountably infinite, but I see here that this result is not true. I am not sure where my proof is incorrect and ...
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Proving that there is a bijection between any two Peano systems
I want to prove that if two given 'models' satisfy the Peano axioms, there must be a bijection between them. In other words, there is only one version of the natural numbers in set theory. The ...
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What is a set that can have infinite elements, the order matters and can have repeated elements?
I has been learning about Set Theory, and trying to figure out how to write some concepts we use using it, but there is some... issues I'm having to do this.
If I have something that can store things, ...
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Is the following set empty and can its infimum be equivalent to positive infinity?
(See the motivation for more info.)
Suppose $\dim_{\text{H}}(\cdot)$ is the Hausdorff dimension, and $\mathcal{H}^{\dim_{\text{H}}(\cdot)}(\cdot)$ is the Hausdorff measure in its dimension on the ...
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Can any non-injective function be restricted to an injective one?
Conjecture: For any arbitrary non-injective function $f:X \to Y$, there exists $C \subseteq X$ ($C \neq \emptyset$), such that $g:C \to Y$ is an injective function, and such that $g(C) = f(X)$. This $...
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Show that $\mathcal{P}(\emptyset) \subseteq \{\emptyset\}$
This is my (to some extent) self-educated proof that $\mathcal{P}(\emptyset) \subseteq \{\emptyset\}$. What do you think about the proof? I am asking because it is mostly independent self-learning, so ...
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2
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Proving that a recursive function is unique - Tao's Analysis
In Exercise $3.5.12$ in Tao's Analysis $1$, he asks the following:
Let $X$ be a set, and $f:\mathbb{N} \times X \to X$ and let $c \in X$. Prove that there exists an $a:\mathbb{N} \to X$, such that (i)...
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Can the words: set, collection, list and family be synonyms, representing the same concept in mathematics?
Looking at my favorite programming languages, they have all their definitions of what they are calling being a set, a collection ...
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What is the preferred way to prove that $A\setminus A = \varnothing?$
As a highschool student learning elementary set theory, the way we were taught to prove $A\setminus A=\varnothing$ at least from where I am was to show that $A\setminus A \subseteq \varnothing$ and $\...
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If $f:X→Y$ is injective and compatible with equivalence relations $E_X$ and $E_Y$ is $\widetilde f: X/E_X→Y/E_Y$ injective?
Let $f$ be a map from a set $X$ into a set $Y$; let $\mathcal E_X$ be an equivalence relation on $X$ and let $\mathcal E_Y$ be an equivalence relation on $Y$. We say that $f$ is compatible with $\...
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VC dimension of monotone Boolean conjunctions
The following is an exercise from High-dimensional statistics by Wainwright.
Consider the class of functions $\mathcal{B}_d = \{h_S \colon \{0,1\}^d \to \{0,1\} \mid S \subseteq \{1, \ldots, d\}\} \...
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Tao's Analysis 1 - Cardinality Lemma, proving that a function is bijective
I have a question related to Lemma 3.6.9 in Tao's Analysis $1$ book. The Lemma states the following:
Suppose that $n \geq 1$, and $X$ has cardinality $n$. Then $X$ is non-empty, and if $x$ is any ...
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Representation Theory of Groups on Sets
Is there any body of literature on representing groups on pure sets, instead of necessarily vector spaces? It seems like a lot of the theory holds and has meaning. For example, if the action of a ...
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$S$ is a bounded set in $\mathbb{R}^n$ and $B$ is the set of isolated points of $S$. Then $B$ is countable.
I am reading "Analysis on Manifolds" by James R. Munkres.
Let $S$ be a bounded set in $\mathbb{R}^n$;
Let $B$ be the set of isolated points of $S$;
(c) Show that $B$ is countable.
I proved ...
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Given $A∩B ⊆ C\setminus D \text{ and } x ∈ A$ prove that if $x ∈ D \text{ then } x ∉ B.$
I have just started learning about proofs. I'm using Velleman's book 'How to Prove It', and I would really appreciate it if you could say what you think about my proof for this theorem:
Theorem. ...
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How to identify fake coins
Given 16 identical coins in a row. There is exactly one counterfeit coin in the first 8 coins (Left group) and exactly one counterfeit coin in the last 8 coins (Right group). Both counterfeit coins ...
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What does it mean to be interpretable in $PA$? [closed]
Let $PA$ to be the First Order Axiomatization of Peano Arithmetic.
Upto my understanding, ALL Members of any potential Models of this Theory would have to be Natural Numbers.
Then how are fragments of ...
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0
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71
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$V_\omega \subset Z$ ($Z$ being The Minimal Zermelo Universe)?
Let $Z$ be The Minimal Zermelo Universe, meaning:
$$ Z = U\{P^n(\omega) : n \in \omega\} $$
where $\omega$ is the Natural Numbers (including $0 = \emptyset$), and $P^n$ is the Iterated Power Set ...
- 1,573
1
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1
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111
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Constructing $V_\omega$ without the Axiom of Replacement
How to actually construct $V_\omega$ without the Axiom of Replacement, but rather with just the Axiom of Infinity (Obviously also the other Axioms)?
I have been thinking about this problem for almost ...
- 1,573
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1
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87
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Proof the properties of image and reverse image of a mapping
Hello im learning linear algebra. Today, my teacher gives me a problem
Proof the properties of the image and reverse image of a mapping $f:X\rightarrow Y$
$f^{-1}(A \cup B) = f^{-1}(A) \cup f^{-1}(B)...
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1
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Idea of proving that a countable union of countable sets is countable without the axiom of choice
There were some previous discussions and the consensus was that AC (or ACC, axiom of countable choice) is required to prove the fact that a countable union of countable sets is countable. For quite a ...
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3
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The use of the word "precisely" in mathematical statements
I am having trouble with the word precisely in this sentence and generally, probably, in any mathematical context.
A statement reads
The points within the parallelpiped determined by $\boldsymbol{a}$,...
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