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Questions tagged [quotient-set]

This tag should be used for questions about quotient sets that might not have any other quotient structure or quotient structures that don't have their own tag.

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1answer
117 views

Thinking about the cosets of $\mathbb{Z}[X]/(f(x))$

Let's think about $$\mathbb{Z}[X]/(f(x))$$ First of all, (f(x)) is the ideal generated by $f(x)$ on $\mathbb{Z}[x]$. Let's consider $f(x) = x^d+1$. Then, if we ...
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1answer
63 views

How to prove that a function on a quotient set $\mathbb{Z}/n\mathbb{Z}$ is well defined?

I am asked to prove that a function under a quotient set is well-defined. I know that for a function to be well-defined, two mappings or images found in the co-domain/range can't be mapped from the ...
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0answers
45 views

Special case where $I\subseteq J$ implies $R/J\subseteq R/I$ seems to hold

Consider a ring $R$ and two ideals $I,J$ such that $I\subseteq J$. There is a natural projection $R/I\to R/J$ by the isomorphism theorem. There is, in general, no homomorphism going the other way ...
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1answer
32 views

In $k[x,y]/(xy^2-x)$, $(x)=(xy)$

Let $k$ be a field In $R=k[x,y]/(xy^2-x)$. I want to check $(x)=(xy)$ in $R$. I understand $(x)=(xy^2)$ but I cannot proceed from here. Thank you in advance.
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1answer
32 views

Constructing a quotient set for negative numbers

There's a relation $\rho$ defined as $\forall a,b \in \mathcal{A}$ such that $\mathcal{A} = R - \{0\} $ and $(x,y)\in \rho = x \cdot y > 0$. I proved that this is in fact an equivalent relation and ...
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2answers
59 views

How to show $\frac{R}{J} \subset \frac{R}{I}$ when $I \subset J$?

I know the elements of $\frac{R}{J}$ are like $r+J$ when $r \in R$. So I have to show that there is some $s \in R$ that $r+J = s+I$. But I don't know how to do that. I actually want to show that $\...
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0answers
21 views

How do I construct setoids in terms of coequalizers?

Many constructive theorem provers don't have native support for quotient types. In order for presheafs over Set to work properly I need some sort of workaround. Currently I'm using a simple approach ...
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1answer
46 views

Quotient set of the relationship $x,y\ \in \mathbb{Q}, \ x \sim y \iff x-y \in \mathbb{Z}$

$x,y\ \in \mathbb{Q} \ \\ x\sim y \iff x-y \in \mathbb{Z}$ I already know this is an equivalence relationship, in fact: reflexive: $x-x =0 \in \mathbb{Z}$ symmetric: $x-y\in \mathbb{Z} \implies y-x \...
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0answers
19 views

Non-equivariant map that may descend to the quotient

Let $X$ and $Y$ be sets equipped with an action of a group $G$ (the same group for both sets). Let now $f:X\to Y$ be a function with the following property: For every $g\in G$, there exists $h\in G$ ...
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1answer
34 views

Show $\mathbb R^{\ge 0}$ can be taken as a model for $\mathbb R^2/\sim$

Paolo Aluffi says something to the effect of what's in the title in his intro to math notes. Link below. $\mathbb R^2/\sim$ is a quotient of $\mathbb R^2$ modulo the equivalence relation given by $(...
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1answer
31 views

A consequence of quotient map from unit circle to itself

In Part II of General Topology, Munkres, Theorem 57.1 states that if $h:S^1\to S^1$ is continuous and antipode-preserving, then $h$ is not nulhomotopic. There are some definitions I should clarify ...
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0answers
85 views

How general is the fundamental theorem of equivalence relations?

We know that the fundamental theorem of equivalence relations can be stated without reference to sets by using congruence as a conjunction of two propositions as following: If $R$ together with ...
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3answers
68 views

Composition of equivalence relations and quotient set

Let $R$ be an equivalence relation on non empty set $X$. Prove that $X/(X/R\circ R \circ \dots R) = R$ Since $R$ is an equivalence relation we have $$R\circ R \circ \dots R = R$$So we should prove ...
2
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1answer
63 views

Visual understanding of $\frac{\mathbb R} {\mathbb Q} $

Consider the quotient set $\frac{\mathbb R} {\mathbb Q} $ obtained by the equivalence relation $$ x \equiv y \mod \mathbb Q \quad \text{iff} \quad x-y \in \mathbb Q $$ I was wondering if there exists ...
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0answers
16 views

If $R \subseteq R\,'$ are equivalence relations on $X$ then $\left|\frac{X}{R\,'}\right|\le\left|\frac{X}{R}\right|$

Let $X$ be a set and let $R, R'$ be two equivalence relations on $X$. Statement. If $R \subseteq R\,'$ then $\left|\frac{X}{R\,'}\right| \le \left|\frac{X}{R}\right|$. Proof. We have to exibit an ...
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1answer
42 views

Let $X = [0,1]$ and $\sim$ have the following equivalence relation over $X$: is the quotient first countable?

Let $X = [0,1]$ and $\sim$ have the following equivalence relation over $X$: $$x\sim y\iff x = y\text{ or }x, y \in {]}0, 1{[}$$ Show that $X/{\sim}$ is not first countable. =========================...
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0answers
24 views

Let K = B [0, 1] be the ring of the continuous functions of [0, 1] in $ \mathbb{R}$ with the usual sum and... [duplicate]

Let K= B [0, 1] be the ring of the continuous functions of [0, 1] in $ \mathbb{R}$ with the usual sum and multiplication operations. For each a $\in$ [0, 1], give the set $I_a$ = {f $\in$ k; f (a) = ...
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0answers
25 views

What does it mean for $g_{1}G^{'} = g_{2}G^{'}$ in cosets?

I learned from here Are all normal subgroups Abelian? what does it mean for the left coset to be equal to the right coset. But I still do not understand what does it mean for a coset of two different ...
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1answer
53 views

Comparing 2 universal properties.

Here is the first universal property and the map required to prove it: And here is the second universal property: My questions are: 1- Are they the same or not? I am guessing that both of them are ...
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1answer
49 views

A proof that if $M’$ is a matroid quotient of $M$, each base in $M$ contains a base of $M’$ that doesn’t use rank function?

I'm a math student and I'm studying matroids. I tried to prove it myself, but I just couldn’t do it. Just note that the book I'm following, called Coxeter Matroids by A. V. Borovik, I. M. Gelfand and ...
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0answers
25 views

What is $G/Z(G)$ in case of $G$ is the group of quaternions?

What is $G/Z(G)$ in case of $G$ is the group of quaternions?where $Z(G)$ is the center of $G,$ which is $\{\pm 1\}$ my guess They are the cosets $i + Z(G),j + Z(G), k + Z(G), (-i) + Z(G),(-j) + Z(G), (...
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0answers
35 views

When is the center of the quotient equals 1? [duplicate]

I want to prove or disprove this statement: If $H = Z(G),$ then $Z(G/H) = 1.$ I have seen an example here Between the center of a quotient group and the total center that shows that in some cases that ...
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1answer
83 views

Proof of the existence of a well-defined function $\bar{f}$.

Given $f: X \rightarrow Y$ there exists a well-defined function $$\bar{f}: X /\sim \rightarrow Y,$$ $$[x] \mapsto f(x).$$ Such that $\bar{f}$ is injective and $f = \bar{f} \circ \pi,$ where $$\pi : ...
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0answers
32 views

Find equivalence classes and quotient set of $(x,y)R(w,z)\iff2(x-w)=z-y$

Let $R$ be an equivalence relation defined on $\Bbb{N}^2$, where $$(x,y)R(w,z)\iff2(x-w)=z-y.$$ Find equivalence classes and quotient set. First, we can express $2(x-w)=z-y$ as $$2x+y=2w+z.$$ So, for ...
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0answers
52 views

Number of equivalence classes A = $\{100000,...,999999\}$

I am asked to find the number of elements in the quotient set of $A\times A \quad$ $A=\{100000,...,999999\}$ considering that the defined relationship n~m $\Leftrightarrow$ n and m they have exactly ...
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0answers
19 views

$f^{-1}f(L)=L+\text{Ker}f$

Let $f:M \longrightarrow N$ be a module homomorphism and let $L$ be a submodule of $M$. I would like to prove that $f^{-1}f(L)=L+\text{Ker}f.$ I tried using the first isomorphism theorem, but I cannot ...
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1answer
128 views

What is the quotient of the absolute value metric in $\Bbb Z[\frac16]^+/\langle2,3\rangle$?

What is the quotient of the absolute value metric in $\Bbb Z[\frac16]^+/\langle2,3\rangle$? I'm somewhat baffled by Wikipedia's definition of a quotient pseudometric. How does it apply to the example ...
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0answers
13 views

Generating the blocks of a partition from a family of filaments

Is the following theory/proof correct? Let $X$ be a nonempty set. A family of subsets $\mathcal K$ of $X$ is said to be a family of filaments for $X$ if the following conditions hold: $\tag 1 \...
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1answer
43 views

Cardinal of quotient set

Consider a finite set $X$ and let's denote by $F$ the set of all maps defined on $X$ with values in $\mathbb{N}$. How to prove that $F$ is countable. Let $\pi_1, \pi_2 \in F$. We define the binary ...
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1answer
114 views

Cardinality of a quotient set of [0,1]

Let $[0,1] \subset \mathbb{R}$. Let $x,y \in [0,1]$ and $q \in \mathbb{Z}$, $k \in \mathbb{N}$. Define the equivalence relation $$x \sim y \iff x-y = \frac{q}{2^k}$$ for some $q,k$. How do I find ...
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3answers
70 views

Find the cardinality of the quotient of $\mathbb R$ in respect to R

R is an equivalence relation defined as $xRy \Leftrightarrow a - b$ is an integer. What is the cardinality of the quotient of $\mathbb R$ in respect to R? How would you prove it? I thought about a ...
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1answer
57 views

Is this quotient mapping bijective?

Let $V$ be a vector space (with respect to some scalar filed $\mathcal{F}$), and let $M$ be a linear vector space of $V$. Consider the equivalence relation $\sim$ such that for any $x,y\in V$, $x\sim ...
3
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1answer
54 views

$p: \mathbb{R} \to \mathbb{R}/A$ Prove $p$ is an open map $\iff$ $A$ is open.

$A$ is a subset of $\mathbb{R}$ with more than 1 point and $p: \mathbb{R} \to \mathbb{R}/A$ is the quotient map. Prove that $p$ is an open map $\iff$ $A$ is open. I know if $A$ is open then for each ...
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1answer
34 views

What does it mean "Lifting" from $Z_3[x]/\langle x^3-x-1\rangle$ to $Z[x]/\langle x^3-x-1\rangle$ exactly?

I have been reading a paper of a cryptographic algorithm. At some point, algorithm takes a polynomial f(x) that belongs to $Z_3[x]/\langle x^p-x-1\rangle$ and takes the lifting of this polynomial to $...
-1
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1answer
35 views

"Kind" of Elements of the Quotient Ring $\Bbb Z[x] / (6,x)$ [closed]

I need some help with this: Let $I$ be the ideal generated by $(6,x)$, the ring $R=\Bbb Z[x]$ and the polynomial $p(x)=132−3x \in R$. Say whether: a. $\;p(x)+I$ is a zero divisor. b. $\;...
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0answers
43 views

$\Bbb Z / 3 \Bbb Z[x]/I$ and $(\Bbb Z / 3 \Bbb Z[x]/I)/J$ quotients rings. Prime elements.

I need some help with this exersise because it mixes quotients rings, concruences, ideals, polynomials and it mess me up. I don't even know how to start. Any help will be appreciated. Let $R=\Bbb Z / ...
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1answer
66 views

Equivalence relation and bijection

I am completely stuck with this exercise. We have an equivalence relation $\mathcal R$ of $\mathbb{R}$ in $\mathbb{R}$ such as : $\ x \mathcal{R} y \iff x - y \in \mathbb{Z} $ The question I am ...
2
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4answers
122 views

What do elements of this set look like?

I was under the impression that elements of quotient rings such as $\mathbb F[x]/(f)$ were of the form $h(x)+(f)$ where $h(x)\in\mathbb F[x]$. Is this correct? If so could somebody explain why ...
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3answers
19 views

How many equivalence classes have the quotient set of this relation $R$?

How many classes have the quotient set defined on $\Bbb{Z}^2$ by the following?: $$(x,y)R(u,v)\iff x\equiv u\pmod{4}\quad\wedge\quad y\equiv v\pmod{2}.$$ I think they are asking for $|\Bbb{Z}^2/R|$. ...
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1answer
186 views

Prove equivalence kernel of an injection is the identity relation

Given a function $f$ and an equivalence relation $\sim$ on the domain of $f$, if two equivalent elements with respect to $\sim$ have the same image under $f$, then $\sim$ is called the equivalence ...
0
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1answer
53 views

Tensor products and the elementary notion of the quotient set

I know that the Tensor Product is a well structured mesh of concepts, which relates the notion of Universal Property (UP) (which defines the required algebraic structure) and the existence of such ...
-1
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1answer
57 views

Recommended Exercise Lists for Quotient Objects [closed]

I understand the different types of quotient objects, but I still struggle with them - which I think is a result of me neglecting to work on them despite knowing it's a weak area of mine. How I ...
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1answer
64 views

Why $[0,3]/(1,2)$ is not homeomorphism to $[0,1]$ [closed]

Clearly $[0,3]/(1,2)=[0,1]+ {\rm point} A+[2,3]$. How can I prove those are not homeomorphic
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0answers
37 views

Quotient set of associate elements

Let $R$ be a ring with $1$. We say that $a,b\in R$ are associate if $a|b$ and $b|a$, i.e. there exist $r,s\in R$ such that $a=rb$ and $b=sa$. This is an equivalence relation on $R$, and I would like ...