This tag is for elementary questions on set theory, spanning topics usually found in introductory courses in set theory, in addition to review sections of graduate textbooks in the same field. Topics include intersections and unions, de Morgan's laws, Venn diagrams, relations, functions, (un)...

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1
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1answer
56 views

What does multiply mean in a set

I have seen this question today. $[A \cdot(B-C)]\cap[(A \cup C)\cdot(C-B)]$ 1.$(A-B) \cdot (A-C)$ 2.$(A \cdot B)\Delta (A\cdot C)$ 3.$(A \cup B)\cdot (A \cup C)$ 4.$\emptyset$ ...
1
vote
1answer
43 views

Is “closedness” a proper word?

In one of my papers I had to prove a list of properties of a set, say, $S=\{a,b,c\}$. Among them we have a fact that $S$ is downward closed with respect to a binary relation $R$. I found it awkward to ...
0
votes
1answer
21 views

Is $I \cap (\bigcup_{j=0}^{\infty} I_j)$ a half open interval?

A half open interval is a set of form $\emptyset$ or $[a,b[$ where $a < b$ If $I$ is a half open interval and $I_j, j=0,1,...$ is a sequence of half open intervals, is $I \cap (\bigcup_{j=0}^{\...
0
votes
0answers
32 views

How can I represent this set?

I have a set of variables: $X := \{x_1, \ldots, x_n\}$ Let the range for each variable $x_i$ be $R_i$. How can I represent the set $\{(x_i, v_i), \forall x_i \in X, \forall v_i \in R_i \}$ in terms ...
1
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6answers
64 views

Definition of Relation of a Set

The standard definition of a relation of an arbitrary set A is a subset of the set product of A, AxA. Is it okay to define relation R to be a subset of the set product AxA such that R has at least ...
0
votes
1answer
21 views

Intersecting a set with an arbitrary union

I want to verify that my proof is correct for the following fact $\bigcup_{\alpha \in \mathbb{J}} (U_\alpha \cap Y) = (\bigcup_{\alpha \in \mathbb{J}} U_\alpha)\cap Y$. let $x \in \bigcup_{\alpha \...
2
votes
1answer
26 views

Transitive Property of Proper Inclusion

$ Theorem:$ If $ A \subset B $ and $B \subset C$, then $A \subset C$. Here $X \subset Y$ is defined as $ X\subseteq Y$ and $X\neq Y$. I can prove the case of improper inclusion using the ...
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votes
5answers
102 views

About Cantor's proof of uncountability or real numbers

The proof is using reductio ad absurdum , i.e. contradiction. Start with that there is a sequence of all real numbers (in some interval) and then it is shown that there exists number that is different ...
0
votes
1answer
59 views

Function over non-numerical sets

Considering a finite lexicographically ordered set, for example, $\{a, b, c, d\}$ called $A$ with $A$ as domain and codomain of a function which returns the element with right shift of 1 over A, how ...
0
votes
0answers
13 views

Explaining the proof that the set of integer combinations of a,b is equal to the set of integer multiples of the GCD of a,b

I'm trying to understand how this proof works, but the website is using a different notation than my book and it's not making any sense to me. here is the proof on ProofWiki I know that the ...
0
votes
1answer
20 views

Isn't $\bigcap_{i=1}^5 A_i=A_5$ where $A_i=\{x \mid x \in \mathbb{N},0 \le x \le \frac{1}{n}\}$

If $A_i=\{x \mid x \in \mathbb{N},0 \le x \le \frac{1}{n}\}$ where $n \in \mathbb{N}$. which one is correct?(Note that $0 \notin \mathbb{N}$) 1.$\bigcup_{i=1}^n A_i=A_1$ 2.$\bigcap_{i=1}^5 ...
0
votes
1answer
27 views

Cartesian product of a family of sets by halmos [duplicate]

can someone please give me an example for this sentence, (i don't want a formal definition) thank's.
3
votes
1answer
30 views

Is a relation between A and B the same as a mapping from elements of A to subsets of B?

The way I always saw it was that a relation is a subset of $A \times B$, or a collection of ordered pairs $(a,b)$, where $a \in A$ and $b \in B$. Is there any meaningful distinction between the two ...
1
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4answers
58 views

prove $\bigcup_{n=1}^\infty A_n=\bigcup_{n=1}^\infty B_n$

prove$\bigcup_{n=1}^\infty A_n=\bigcup_{n=1}^\infty B_n$ if $A_i$ is a arbitrary set and $B_1=A_1$ and $B_n=A_n-\bigcup_{i=1}^{n-1} A_i$ for $n\ge 2$. I have no ideas for solving it there was another ...
0
votes
1answer
39 views

Proof of Ultrafilter lemma with two propositions and Zorn lemma

I would like to prove the following: Let $X$ be any set, then every filter $\mathcal{F}$ on $X$ is contained in an ultrafilter $F$ Using two propositions and Zorn Lemma. I am required to come ...
0
votes
3answers
33 views

Set theory confusion

Given a universal set $\{ 1,2,3,4,5,6,7,8,9 \} $, $A$ is defined as ... $A = \{ x : (x-1)(x-6) \lt 0 \}$ So what are the elements in $A$ , I'm a little confused here . The $x$ values is $= 1$ or ...
0
votes
1answer
26 views

formula for defining terms in a finite set

Suppose there's a finite set, $S$ of terms in $\mathbb{R}$ which have the property $P(x)$. Suppose we know how to define the maximum value of the set by the relation, $max(x)$. We also have the ...
0
votes
3answers
35 views

How to write half-open intervals as disjoint ones

I have a collection of half open intervals $(I_j, j \in \mathbb N)$ and I want to get a new collection $(J_j, j \in \mathbb N)$ out of the $I_j$ such that for $i \neq j$ $J_j \cap J_i = \emptyset$ and ...
0
votes
2answers
34 views

Proving the Inclusion-Exclusion Formula?

I've been given the following problem: ...
0
votes
1answer
18 views

Prove there is a $g:B\rightarrow\mathbb{R}$ s.t. $f(g(b))=b$ for each $b\in B$.

Suppose $f:\mathbb{R}\rightarrow B$ is surjective, where $B$ is finite. Prove there is a $g:B\rightarrow\mathbb{R}$ s.t. $f(g(b))=b$ for each $b\in B$. I'm given a hint to use induction on the ...
5
votes
1answer
119 views

Relation between open sentences and sets (conceptual question)

Hi I'm a college student getting into the more proof oriented side of math. I was reviewing Mathematical Proofs, A Transition to Advanced Mathematics 2nd edition and after thinking about chapters 1 ...
0
votes
0answers
31 views

Exercise from David Williams' “Probability with Martingales” page 25

I was stuck at the exercise page 25 and then I found the answer here : Noob Question : Need help to understand : Probability with Martingales : page 25 But there is still one point I don't get, how ...
2
votes
1answer
43 views

Understand a part of the proof the Schroder Bernstein theorem

This is the Lemma in a book: Let X be a set and p:℘X→℘X a function which is monotonic, in the sense that if A⊆B⊆X, then p(A)⊆p(B). Then there is a set Z⊆X such that p(Z)=Z This is its Proof in the ...
3
votes
1answer
49 views

In set theory, what does it mean for a variable to have a bar symbol above it?

Please see the image below. What does the bar symbol mean in this context?
1
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1answer
29 views

Modified cantor set can be identified with binary sequences

I'm reading this paper on sets of uniqueness, and on page 15 the author constructs the following Cantor-like set: The Cantor set in the interval $[0, 2π]$ is constructed by removing the middle $...
2
votes
4answers
65 views

My proof that if $P(A) \subseteq P(B)$, then $A \subseteq B$

I'm not sure if my proof is sound. Here it is: Assume that $P(A) \subseteq P(B)$, so any subset C of A is also a subset of B. Therefore, any element in C is also an element of A, and by the same ...
2
votes
0answers
78 views

Unable to prove that a statement is an equivalent form of the Axiom of Choice [duplicate]

The exercise 22 page 158 of Elements of Set Theory by B. Enderton is the following: Show that the following statement is another equivalent version of the axiom of choice: For any set $A$ there ...
1
vote
1answer
30 views

Projective limit of system sets compatible maps $\phi : \{x_1, \dots, x_n \} \hookrightarrow \mathbb{Z}$

This question is about the projective limit of the following system: Let $I$ be the poset of finite subsets of $\mathbb{R}$ partially ordered by inclusion. Let $S_U$ be the set of injective maps $\...
3
votes
1answer
33 views

Not understanding the proof that there is no surjection from a set to its powerset

Here is the question: If a set, $A$, is finite, then $|A| < 2^{|A|} = |P(A)|$, and so there is no surjection from set $A$ to its powerset. Show that this is still true if $A$ is infinite. ...
0
votes
1answer
33 views

What's the union of a set whose members are sets? like here?

The definition of union of a set in my book is the set formed by all the members of the members of the set. But if Z is a set formed by several sets and Z = {{{1,2},3,5}, {3,4}, {6},6, 7} what's the ...
3
votes
1answer
55 views

Is the set of hyperreal numbers a quotient ring?

It is easy to see that the set of real sequences $\mathbb{R}^{\mathbb{N}}$ is a ring. It suffices to define, for all $r,s\in\mathbb{R}^{\mathbb{N}}$, the operations $r\oplus s =(r_n+s_n)_{n\in\mathbb{...
1
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2answers
18 views

Checking reflexive, symmetric and transitive properties of $\neq$ on $\mathbb{N}$

QS: Indicate if the relation on the given set are reflexive on a given set, which are symmetric, and which are transitive. $\not = \text{on } \Bbb N$ So for this problem I am trying to ...
1
vote
1answer
50 views

prove sets cardinality inequality

I need to prove that if $$ A , B $$ are infinite sets and it holds that : $$ |A| > |B| $$ then: $$ |A \backslash B| = |A| $$ I guess I just don't what can I say about the cardinality of |A\B| ...
2
votes
1answer
113 views

What branch/field of mathematics is this? [closed]

I do not want solutions, I just want the field/branch of mathematics that these problems deal with, and possibly a good online source or two to learn it. Problems :- 1:- 2:- 3:- 4:- ...
2
votes
1answer
44 views

$A \subseteq B \subseteq C ; A' \subseteq C' ; |A|=|A'| , |C|=|C'|$ ; then $\exists B' $ s.t. $A' \subseteq B' \subseteq C' $ , $|B|=|B'|$?

Let $X$ be a non-empty set and $A,B,C,A',C' \in \mathcal P(X)$ be such that $A \subseteq B \subseteq C ; A' \subseteq C'$ and $|A|=|A'| , |C|=|C'|$ ; then is it true that $\exists B' \in \mathcal P(...
0
votes
2answers
26 views

Family of Sets - Please explain

Let $X$ be a set and $\mathfrak{M}\subset \mathcal{P}(x)$ be a family of sets over(?) $X$. Does this simply mean, that if for example $X$ contains all squares of a chess board, any set $A\subset\...
0
votes
1answer
24 views

What is the complement of a product of two sets?

I am given this information: Suppose $A=\{1,2,3\}$, $B=\{3,5\}$, $C=\{1,2,4,6,9\}$ and $U = \{0, 1, 2, 3, 4, 5, 6,7,8,9\}$. Enter "T" for each true, and "F" for each false statements. There ...
-2
votes
2answers
61 views

What are example of minimal but not least element? [duplicate]

I know that every least element are minimal. But, What are example of minimal but not least element?
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2answers
25 views

How do you prove that the greatest element of a partially ordered set (A,<=) is maximal? [duplicate]

1) How do you prove that the greatest element of a partially ordered set (A,<=) is maximal? 2) What will be some example that the maximal is not necessarily a greatest element?
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votes
4answers
92 views

How do you prove $X\times Y = \varnothing \iff X=\varnothing$ or $Y=\varnothing$? [closed]

How can I prove that these two are equivalent? $X\times Y = \{(x,y)\mid (x\in X)\land(y\in Y)\} = \varnothing$ $X=\varnothing$ or $Y=\varnothing$.
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votes
1answer
32 views

Cardinality question on set of symbols [closed]

Few moments ago I asked myself a question, that I not positive if, in fact, is well defined. Let $\mathbb{R}$ be the set of real numbers. Define $S$ to be a set of symbols, as follows: Let $x$ be ...
2
votes
2answers
48 views

uncountability proof [closed]

In the proof that the set of all sequences whose elements are 0 and 1 if we replace the set of reals with the set of natural numbers wouldn't that lead to the same contradiction that N is a proper ...
0
votes
0answers
32 views

Calculate maximum value limit in set partition

I have all the subset of 4 element as follows with value associated with each subset ...
0
votes
0answers
18 views

how to write a set of local extrema formally

Is the following notation correct? $X = \left\{x_t > \max\left(x_{t-z},\ldots,x_{t-1},x_{t+1},\ldots,x_{t+z}\right) \mid (t-z, \ldots, t+z)\in \mathbb{T}^{2z+1} ∧ k ∈ ℕ\setminus \{0\}\right\} $ ...
0
votes
1answer
41 views

How do I prove αβ = 0 ⇐⇒ α = 0 or β = 0?

Let α and β be cardinal numbers. Prove that αβ = 0 ⇐⇒ α = 0 or β = 0. Below is what I have done. Could you please review and point out illogical parts, and things I have missed out? I want to be ...
3
votes
2answers
96 views

Prove that $\bigcup\mathbb{N}=\mathbb{N}$

Prove that $\bigcup\mathbb{N}=\mathbb{N}$. Showing that $\mathbb{N}\subseteq \bigcup\mathbb{N}$ is simple. However, I'm not seeing how to handle showing $\bigcup\mathbb{N}\subseteq\mathbb{N}$. I ...
0
votes
1answer
65 views

Is there a bijection from $A = ]0,1[$ to $B = A \cup \{1,2,3,4\}$?

Is there a bijection from $A = ]0,1[$ to $B = A \cup \{1,2,3,4\}$? If there is, give example. I had this question on exam few days ago, and I have been googling for days, but I can't find a ...
0
votes
0answers
44 views

Exercise from David Williams' “Probability with Martingales” page 16

I am reading David Williams' "Probability with Martingales" and I'm completely stuck at the exercise page 16: Let $\mathcal{C}$ be the class of subsets $C$ of $\mathbb{N}$ for which $$\lim_{m \...
1
vote
2answers
38 views

How do you prove that If X is an infinite set, then there is a denumerable subset Y of X such that X and X-Y are equipotent? [closed]

How do you prove that If X is an infinite set, then there is a denumerable subset Y of X such that X and X-Y are equipotent?
1
vote
0answers
26 views

Constructing a Collection of Sets Satisfying Certain Intersecting Properties

I am trying to solve the following problem. We would like to construct $\{A_1, \ldots, A_n\}$, where $n$ is even, and each $A_i \subseteq [m]$, with $|A_i| = k$ and $m = \text{poly}(n)$. Now, I would ...