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, ...

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1answer
26 views

Equivalence classes of $\Bbb Z$ with the operation $\mod n$

So I came across this phrase in my abstract algebra textbook: The integers $\mod n$ also partition $\Bbb Z$ into $n$ different equivalence classes; we will denote the set of these equivalence ...
1
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2answers
66 views

The length of a point and the interval

I think the length of a point is $0$, and since biunique corespondence between the points of [0, 1] and [0, 10], therefore I came to the conclusion that there is a same number of points between [0, 1] ...
3
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3answers
48 views

Let $(X, \mathfrak T)$ be a topological space and suppose that $A$ and $B$ are subsets of X. If $A\subseteq B$ then $A' \subseteq B'$

Let $(X, \mathfrak T)$ be a topological space and suppose that $A$ and $B$ are subsets of X. If $A\subseteq B$ then $Bd(A) \subseteq Bd(B)$ If $A\subseteq B$ then $A' \subseteq B'$ ($A'$ is the set ...
1
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5answers
61 views

Suppose that $A$ and $B$ are sets and that $f: A \rightarrow B$ is onto. Does being onto guarantee the sets are finite?

Suppose that $A$ and $B$ are sets and that $f: A \rightarrow B$ is onto. Determine which of the following statements are true: If $A$ is finite then $B$ is finite. If $B$ is finite, ...
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0answers
15 views

Pairwise crossing lines meaning

I'm having a hard time being certain of the meaning "pairwise crossing" in the context of Graph Theory... namely, if say 4 lines are pairwise crossing, may any be parallel. the question states: "A ...
4
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0answers
37 views

finding a bijective function from the real plane to the real line

As part of a HW assignment in the course elementary set theory, I was given the following question: Prove explicitly (don't use any theorems or known facts, but find a bijective function) that ...
1
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2answers
29 views

Problem involving finite sets condition

I stumbled upon this innocent looking problem in my old high school algebra textbook and I just can't figure it out . It goes like this : How many finite, non-empty sets satisfy the following ...
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0answers
22 views

finding an injective function to prove cardinality equality

As part of a HW assignment in the course elementary set theory, I was given the following question: Prove that the set of all binary sequences (sequences of $0$ and $1$) except for the binary ...
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4answers
76 views

Proving that $A\subset B$ if given $A=A\cap B$

Let $A = A \cap B$. Prove $A \subseteq B$ I go about like this : Let $x \in (A \cup B)$ $\implies x\in A ~~\text{and} ~~ x\in B$ Question 1 : Is this true? Will and come here? Ideally or ...
0
votes
1answer
31 views

Proving that there are as many infinite binary sequences and infinite binary sequences not containing 11

I need to prove that all the infinite binary sequence are equal in cardinality to the infinite binary sequences which don't include 1 twice in a row. And I'm supposed to use ...
8
votes
5answers
455 views

Precise meaning of “extension”?

Halmos's Naive Set Theory explains the "extensionality" in "axiom of extensionality" as: Every set is determined by its extension. and that's it. What is a set's extension, then? Intuitively it ...
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2answers
22 views

Name of an element of an element of a set

Is there some way of notating that an object can be related to a set through element relations, even if it is not an element of the set? e.g. $a\not\in\{\{a,b\},\{c,d\}\}$, but ...
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0answers
21 views

Show that $(0,1]$ and $(0,1)$ are equinumerous. [duplicate]

I know that one approach is to find a bijective function and another to find a set which is equinumerous to both of them, but I dont have ideas for any of those approaches. Any suggestions?
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3answers
43 views

Show that $A \cap B = B$ iff $A \cup B = A$, where $A \subseteq B$.

Show that $A \cap B = B$ iff $A \cup B = A$, where $A \subseteq B$. I have tried to do this by element-chasing, but I just end up saying that $x \in A$ and $x \in B$. I am really stuck for ...
1
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3answers
71 views

Show that ~ creates a partition of $M_2(\mathbb{R})$

Let $M_2 (\mathbb{R})$ be the set of 2x2 matrices over $\mathbb{R}$: $$ M_2 (\mathbb{R}) = \biggl\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \; \biggm| \; \text{with } a,b,c,d \in ...
1
vote
1answer
19 views

What is the notation for a sequence of elements (non number elements)?

I am new to math and am exploring how to formally represent a sequence of events. I want to be able to say "an event sequence $E_s$ is a sequence of events $\langle e_1, e_2,\ldots, e_n\rangle$". Just ...
1
vote
1answer
25 views

Let $A = (0,1) \cup [2,3)$ be a subset of $(\mathbb R, \mathfrak T_C)$. Find the following sets. How am I doing?

Let $A = (0,1) \cup [2,3)$ be a subset of $(\mathbb R, \mathfrak T_C)$ $\mathfrak T_C = \{(a,\infty) :a \in \mathbb R\} \cup \{\mathbb R, \emptyset\}$ I need to find the following sets: Int(A) ...
1
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3answers
36 views

Give a family of sets $\{F_a\}$ with each $F_a\subseteq (0,1)$ and $F_a \cap F_b \neq \emptyset$ but $\bigcap_aF_a = \emptyset$.

Give a family of sets $\{F_a\}$ with each $F_a\subseteq (0,1)$ and $F_a \cap F_b \neq \emptyset$ but $\bigcap_aF_a = \emptyset$. What is an example of such a family of sets?
3
votes
1answer
44 views

Is this proof that $\kappa^{<\kappa}=\kappa$, when $2^{<\kappa}=\kappa$, correct?

Let $\kappa$ be a cardinal number. I want to show that if $\kappa$ is regular and if $2^{<\kappa} = \kappa$ then $\kappa^{<\kappa}= \kappa$. Here is what I got so far: $$ ...
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2answers
35 views

Proof of surjectivity of a function [closed]

Let $g:[0,\infty) \rightarrow [0,1)$ defined by $g(x)= \frac{x}{1+x}$. How i can prove that's onto, with the definition?
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2answers
41 views

Is there a sample of a $f(x)=y$ multivalued function whose inverse $f(y)=x$ is also multivalued?

Trying to learn about the properties of the multivalued functions, I found the definition at the Wikipedia as "a left-total relation (that is, every input is associated with at least one output) in ...
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2answers
79 views

When can variables simply be variables?

This may seem a somewhat strange question, but I've been tying myself in knots about it recently. When constructing a polynomial ring, you must formally define a polynomial as an ordered ω-tuple, ...
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2answers
65 views

Prove every subset of $\Bbb N$ is countable.

This isn't a homework problem. I've seen a proof of the following statement online, and I think the proof is suspect, or at least incomplete. Theorem. Every subset of $\Bbb N$ is countable. ...
0
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2answers
36 views

no. of elements in $(B \times A) \cap (A \times B)$

Can someone prove that $n((A \times B) \cap (B \times A)) = n(A \cap B)^2$ I am unable to derive this. (Frankly, I don't know how to even start). I just let $(A \times B) = M, and (B \times A) = N$, ...
1
vote
1answer
23 views

intersection closure for boolean functions

This seems a basic thing yet I'm having hard time understanding it. Let $X=\{x_1,x_2,\dots,x_n\}$ be a set of $n$ elements and $S$ be a set of boolean functions from $X$ to $\{0,1\}$. So every ...
2
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1answer
65 views

Can we prove AC from the statement “There is no $\aleph$ cardinal strictly between $\operatorname{CARD}(X)$ and $\operatorname{CARD}(2^X)$”?

If $X$ is a set, let $\operatorname{CARD}(X)$ denote the Cardinal number of $X$. Let GCH(1) be the statement "If $K$ is an infinite initial ordinal number, then there exists no initial ordinal number ...
0
votes
1answer
32 views

Show $f^{-1}(\bigcup_{B_i\in B}B_i) = \bigcup f^{-1}(B_i)$ proof done correct?

Show $f^{-1}(\bigcup_{B_i\in B}B_i) = \bigcup f^{-1}(B_i)$ Attmept: I used induction: We know it is true for $i = 1$ where $i\in \mathbb N$ Confirm for $i = 2$: $f^{-1}(B_1\cup B_2) = f^{-1}(B_1) ...
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0answers
21 views

Let $(X, \mathfrak T)$ be a topological space and supposed that A is a subset of X. Then $Bd(A) = Cl(A) \cap Cl(X-A)$.

Let $(X, \mathfrak T)$ be a topological space and supposed that A is a subset of X. Then $Bd(A) = Cl(A) \cap Cl(X-A)$. I know this is a true statement. I am trying to prove if because I would also ...
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5answers
68 views

what does function from a set to its power set mean?

I am having some confusion in understanding, what exactly does a function from a set to a power set means. I don't want a proof to the cantor's theorem. Consider a set A = {1,2,3} , P(A) = ...
0
votes
1answer
27 views

De Morgan's laws [duplicate]

one can prove de Morgan's laws in set theory by induction. Are these laws true for an uncountable set of indices ( sets ) ? Can somebody give a proof for one of the laws in the case of uncountable ...
1
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5answers
189 views

Is this element a subset of set A? [duplicate]

Let $A$ be the set $\{\propto,\{1,\propto\},\{3\},\{\{1,3\}\},3\}$. The statement "$\{1,\propto\}\subseteq A$" is false. (Taken from A Concise Introduction to Pure Mathematics) But I think ...
0
votes
1answer
28 views

Eventually constant variable assignments

One proof of the Downward Löwenheim Skolem Theorem is via consideration of elementary substructures. In a discussion of this theorem, Christopher Leary writes the following: "Suppose that $ ...
1
vote
1answer
39 views

Isn't reflexivity and symmetry implied in equivalence relations?

It looks like for all "nice" sets, the set $S\times S$ will have symmetry and reflexivity by default. The tough part is usually showing transitivity. However, are there any non-empty sets such that ...
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4answers
95 views

Is the logarithm of $\aleph_0$ infinite?

In classical mathematics $2^{\aleph_0}=\aleph_1$, right? So if $2^x=\aleph_0$, what does $x$ equal? In other words, can we define a logarithm for $\aleph_0$, and what should it be. Is it infinite? ...
2
votes
3answers
102 views

Is $\mathbb{N} = \mathbb{Z}^+$?

Because $\{1,2,3,4,\ldots\}$ contains all natural numbers, which are also all positive integers.
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2answers
27 views

Cardinality of $E=\left\{\left(x,y\right):x,y>0\text{ and }x+y,xy\in\mathbb{Q}\right\}\subseteq\mathbb{R}^2$

I want to find the cardinality of $$E=\left\{\left(x,y\right):x,y>0\text{ and }x+y,xy\in\mathbb{Q}\right\}\subseteq\mathbb{R}^2.$$ This problem came from a recent real analysis comprehensive exam ...
2
votes
1answer
40 views

Is it consistent without the axiom of choice that every permutation of some infinite set have fixed points?

A "permutation" of a non-empty set means an injective mapping of the set onto itself. Let $S(1)$ be the statement "There exists a permutation of every set containing at least two elements, which has ...
2
votes
2answers
17 views

Show that all intervals contained in $[0, 1]$ is a semi-algebra

Suppose $J = \{\text{All intervals contained in }[0,1]\}$ I am having trouble showing that the complement of any element of $J$ is equal to a finite disjoint union of elements of $J$. It seems ...
1
vote
2answers
38 views

Show $f(f^{-1}(B_0))\subset B_0$ and that equality holds if $f$ is surjective

Let $f: A\to B$. Let $A_0\subset A$ and $B_0\subset B$. Show $f(f^{-1}(B_0))\subset B_0$ and that equality holds if $f$ is surjective. Attempt: I already did the first part. It is showing that ...
1
vote
1answer
53 views

Let $\mathfrak T_X = \{f^{-1} (U) : U \in \mathfrak T_Y\}$ then $\mathfrak T_X$ is a topology on X. False?

Let $f :X \rightarrow Y$ be a function and suppose that $\mathfrak T_Y$ is a topology on $Y$. Let $\mathfrak T_X = \{f^{-1} (U) : U \in \mathfrak T_Y\}$ then $\mathfrak T_X$ is a topology on X. ...
0
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0answers
25 views

Elementary substructures and eventually constant variable assignments [duplicate]

One proof of the Downward Löwenheim Skolem Theorem is via consideration of elementary substructures. In a discussion of this theorem, Christopher Leary writes the following: "Suppose that $ ...
1
vote
0answers
19 views

Uniting sequentially a countable set with a “dense barrage”

consider following setup: $$ D = \{1-\frac{1}{2^n} |\space n \in {\mathbb N_0} \} $$ $$ f(x) \begin{cases} x, & \text{if $x \in D$} \\ 1, & \text{if $x \in \mathbb N_+$} \\ 0, & else ...
0
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2answers
140 views

How to use the element method to prove the following sets are equal?

I have been asked to describe the following sets, and then prove my answers using the element method, but i am not sure how to do this. I am trying to prove that (b) is equal to $0$ as $i$ ...
1
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3answers
68 views

The cantor set is uncountable

I am reading a proof that the cantor set is uncountable and I don't understand it. Hopefully someone can help me. Let $C$ be the Cantor set and $x\in C$. Then there exists unique $x_k\in \{0,2\}$ ...
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2answers
45 views

Proving Set using the laws of set theory

Let $A$ and $B$ be any sets. Prove the following set identity using the laws of set theory (set identities). So I am trying Justify each step with the law I used. $A\cap(B\cup A')\cap B'=\emptyset$ ...
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2answers
55 views

In set theory, what is the difference between $\emptyset \subseteq A$ and $\emptyset \in A$?

I know that sometimes In set theory, $\emptyset \subseteq A$ is true, whereas $\emptyset \in A$ is false. What is the difference between $\emptyset \subseteq A$ and $\emptyset \in A$ ?
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3answers
28 views

If $A = \{a,b,c,d\}$, is $\{b,c\} \subseteq \mathcal P(A)$?

If $A = \{a,b,c,d\}$, then is $\{b,c\}\subseteq \mathcal P(A)$ ? I thought the answer was true, but I am not totally sure on that, because I am not sure if it is correct to say that $\{a,b\}$ is a ...
0
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2answers
28 views

Proving De Morgan's law with the minus sign

So I know how to prove De Morgan's Law in this form: $A\cap (B\cup C)^{c}$, what I'm trying to do for practice is prove it in the slightly different notation: $A- (B\cup C)$ I get everything except I ...
0
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2answers
37 views

Why is the number of subsets equal to the number of relations?

In this question I don't understand why the number of subsets is equal to the number of relations. Any help is welcome.
1
vote
1answer
29 views

Proving $(A\times B) \cap (C\times D) = (A\cap C) \times (B\cap D)$

So there is a similiar question in the archives which I looked at after I attempted my proof: Proving that for any sets $A,B,C$, and $D$, if $(A\times B)\cap (C\times D)=\emptyset $, then $A \cap C = ...