For reflexive, symmetric and transitive relations. Use it with the tag (relations).

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2
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1answer
22 views

Can someone present a visualization of the partitioning of a $L^p$ space into equivalent classes?

I am a bit confused by what it means for a $L^p$ space to be partitioned into equivalent classes instead of functions. I understand that give two or more functions $f$, $g$, $h,\ldots$ of which are ...
2
votes
1answer
43 views

Questions about the Identification Topology and Equivalence Class from “Introduction to Topology” by Mendelson

I am currently reading Introduction to Topology by Bert Mendelson, and I have some questions regarding the topic on Identification Topology in his book. Let $(X,\tau)$ and $(Y,\gamma)$ be topological ...
0
votes
2answers
21 views

Shouldn't this be transitive relation also?

$ R = \{ (a,b) ∈ R ; 1 + ab > 0 \} $ It is clearly reflexive and symmetrical but I feel that it is transitive also because the relation R can be stated as $ R = \{(0,0), (0,1), (1,2)...\}$ and as ...
1
vote
1answer
35 views

Is this relation symmetric and transitive?

Set A is given as $A = \{1,2,3,4,5,6,7,8,9,10,11,13,14\} $ And is defined as $R = \{(x,y) : 3x = y\}$ The relation that I'm getting is: $ R = \{(3,3), (6,6), (9,9), (12,12)\} $ Over here, it is ...
0
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3answers
29 views

Proving $a=b \bmod 19$ is an equivalence relation

My question is: $a \equiv b \bmod {19} \iff aRb$ (prove that $R$ is an equivalence relation) Before that, I already know that equivalence relation is when $R$ is reflexive, symmetric and ...
1
vote
1answer
38 views

Discrete mathematics, equivalence relations, functions.

I'd like some insight on how to 'solve' this problem (more towards understanding what the problem is asking) Suppose that $A$ is a nonempty set and $\mathcal{R}$ is an equivalence relation on ...
0
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0answers
19 views

How to show the equivalence of an Euclidean norm to Euclidean distance?

I have the following problem: I have a set of $N$ vectors. I have some a priori information where the information for each vector comes from. If the two vectors $x_1$ and $x_2$ have been sampled ...
0
votes
3answers
34 views

Compute equivalence classes of equivalence relation

I have already proven that relation R={($x,$y) $\in$ $\mathbb Z$ x $\mathbb Z$ | $x+$y is even} is a equivalence relation by showing reflexive, symmetric, and transitive properties of the relation. ...
0
votes
1answer
37 views

Equivalence Relation, Is [15]r = [-13]r

For the equivalence relation on the integers given by $(x, y) ∈ R$ if and only if $7$ divides $x - y$, $7 | (x - y)$. Is $[15]_R = [-13]_R$? Is $[15]_R = [13]_R$ (the $R$'s would be subscripts ...
1
vote
1answer
25 views

Ternary equivalence relations that are not equivalent to some binary equivalance

1.Is there such a thing as ternary equivalence that is not equivalent or cant not be expressed as binary equivalence? 2.If there is such a thing as expressed in 1, are there any practical uses for ...
1
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0answers
14 views

Is there a term for irreflexive relations whose reflexive closures are equivalence relations?

Consider a relation $\sim$ which is irreflexive, that is: $$ \forall A \:A\nsim A $$ Further suppose the reflexive closure of $\sim$ is an equivalence relation. The reflexive closure of $\sim$, ...
1
vote
1answer
32 views

Proving reflexivity from transivity and symmetry.

Property 2 of an equivalence relation states that if $a\sim b$ and $b\sim c$ then $a\sim c$. What is wrong with the following proof that properties 2(symmetry) & 3 (transitivity) imply ...
1
vote
4answers
48 views

What is the structure of $S$?

Suppose we define an equivalence relation on $\mathbb R$ by $aRb$ iff $\{a\}=\{b\}$ for $a,b\in\mathbb R$. Here $\{.\}$ defines the fractional part. In other words, $aRb$ iff $a-b\in\mathbb Z$. ...
3
votes
7answers
389 views

Grasping the concept of equivalence classes more concretely

I have read about equivalence classes a lot but unable to understand exactly what they are. I know about equivalence relations, but I am having a hard time understanding what equivalence classes are ...
1
vote
1answer
27 views

Let $R$ be an equivalence relation: How many elements are in $R$?

Let $R\subset X\times X, |X|=30$. Supposing there are only 3 distinct equivalence classes and all of these have the same amount of elements, find $|R|$. I didn't get very far on this, I thought that ...
1
vote
1answer
41 views

$f : A \to B$ s.t. for all $x, y \in A, x R y \iff f(x) S f(y)$

Theorem. A relation $R$ on a set $A$ is reflexive and transitive if and only if there is a set $B$ with a partial order $S$ and a function $f : A \to B$ such that for all $x, y \in A, x R y \iff ...
2
votes
5answers
73 views

Can't determine if given relation is equivalence relation

Definition of relation ~ $(a,b)$ ~ $(c,d)$ $\iff$ $bc^2=da^2$, where $(a,b),(c,d)$ are from $\mathbb{R}\times\mathbb{R}$ and $(a,b),(c,d)$ are different from $(0,0)$ First of all, I wonder if ...
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4answers
53 views

Conjugacy Classes of a group G - Intuitive Understanding

How can I intuitively understand conjugacy classes of a group G. I feel I have a strong understanding of Equivalence Relations, and just completed the proof showing that conjugacy is an equivalence ...
0
votes
1answer
26 views

Establishing an equivalence relation for a particular question

Condition:For $x,y \in R^{n+1} \setminus \{0\} $ define: $x\sim y$ iff $y = \lambda x $ for some $\lambda \in \Bbb R$, $ \lambda \ne 0$ $x = \lambda x \implies \lambda = 1$ which is a scalar so ...
0
votes
1answer
12 views

These maps from the components into a directed system are injective when the directed system maps are.

Let $I, p_{ij} : A_i \to A_j$ be a directed system of maps such that $p_{jk}\circ p_{ij} = p_{ik}$ whenever $i \leq j\leq k$ and $p_{ii} = \text{id}$. Directed means that for any $i,j \in I$ there ...
0
votes
1answer
12 views

Proving that the direct limit of a directed system is an equivalence relation.

From Dummit & Foote, pg. 268: Let $I$ be an index set with a partial order. Suppose for every pair of indices $i, j \in I$ with $i \leq j$ there is a map $\rho_{ij} : A_i \to A_j$ such that ...
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votes
1answer
23 views

How to prove, that $\sim$ is an equivalence relation? (affine equivalence)

Two quadrics $Q_1$ and $Q_2$ in $\mathbb{R}^n$ are affine equivalent, $Q_1\sim Q_2$, if there exists an affine map $f:\mathbb{R}^n\rightarrow\mathbb{R}^n$ with $Q_2=f(Q_1)$. How do I prove, that ...
3
votes
1answer
60 views

Isomorphic or equal?

Let $\sim_n$ be the usual equivalence relation of congruence modulo $n$ in $\mathbb{Z}$, i.e., for $a,b\in\mathbb{Z}$, $a\sim_nb\Leftrightarrow a-b=k\cdot n$ for some $k\in\mathbb{Z}$. For $n=0$ the ...
0
votes
2answers
25 views

Partition and Equivalence Relation: In $\mathbb{Z}$, let $m\sim n$ iff $m - n$ is a multiple of 10

Prof. Pinter's "A Book of Abstract Algebra" presents this exercise: Prove that each of the following is an equivalence relation on the indicated set. Then, describe the partition associated with ...
1
vote
1answer
64 views

Facing problem to understand the equivalence class concept.

Currently I am trying to understand the equivalence class from the Herstein's book. And I am facing problem to understand the concept and also can't understand the theorem given in the page number 8. ...
2
votes
4answers
40 views

Proof of theorem about equivalence classes

I am looking to understand the following theorem, and I am also wondering what is meant by "mutually disjoint", or at least how it's to be understood in the following context: The distinct ...
0
votes
1answer
40 views

Equivalence Relations OF sets [duplicate]

We define the relationship between the two sets $S_1≡S_2$ if and only if $|S_1|=|S_2|$. How to show that $≡$ is an equivalence relation ? sorry I'm from Iran and Basic my English is poor.
0
votes
1answer
23 views

Determine whether the given relation is an equivalence relation

Problems: 1.) R = {(0, 0), (0, 1), (0, 2), (1, 1), (1, 2), (2, 2)} 2.) R = {(x, y) ¦ y - x is an odd integer} 3.) R = {(x, y) ¦ y - x is a multiple of 3} Attempt: By definition, a relation is ...
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2answers
20 views

Simple verification: is this equivalence always true?

I have a constrained optimization problem and I am trying to reduce the number of contraints and am afraid to be losing information by doing so. If we have two constraints as the following $$A \geq B ...
1
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0answers
19 views

Defn of invariant SET under an equivalence relation?

This is, I am sure, an easy question but something I did not see in my undergrad years and wikipedia does not define it. For a function or a relation or a property to be invariant under an equivalence ...
1
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3answers
37 views

Prove: In $\mathbb{Z}$, m ~ n iff $|m|=|n|$

Prof. Pinter's "A Book of Abstract Algebra" presents this exercise: Prove that each of the following is an equivalence relation on the indicated set. Then describe the partition associated with ...
-1
votes
1answer
47 views

Prove $\lbrace A_n : n \in \mathbb{Z}\rbrace$ is a partition of $\mathbb{Q}$

Dr. Pinter's "A Book of Abstract Algebra" presents the following exercise: Prove that each of the following is a partition of the indicated set. Then describe the equivalence relation with that ...
1
vote
1answer
29 views

Prove: $\langle A_0, A_1, A_2, A_3, A_4 \rangle$ is a partition of $\mathbb{Z}$

The following exercise from Dr. Pinter's "A Book of Abstract Algebra" presents this exercise: Prove that each of the following is a partition of the indicated set. Then describe the equivalence ...
1
vote
1answer
37 views

How do I show that the equivalence relation defining the rational numbers is transitive?

I apologize if this is a super easy question, but there is something fishy about my proof. I was to show: $$(p,q) \sim (m,n) \wedge (m,n) \sim (a,b) \implies (p,q) \sim (a,b) $$ under the ...
0
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0answers
21 views

Why is equivalence 'class', not equivalence 'set'? [duplicate]

Why do we call it a class, not a set? Is it not a set? Can it be a proper class?
2
votes
4answers
125 views

In what mathematically rigorous sense does $ \mathbb{Q}$ extend $\mathbb{Z}$?

I was trying to understand rigorously what the word "extends" means in this context, pin it down formally with the correct mathematical language. First, let me explain some of my thoughts and the ...
12
votes
1answer
110 views

What do you get with this equivalence relationship for all $\mathbb{Q}$ sequences

Consider all $\mathbb{Q}$ Cauchy sequences with this equivalence relationship $\{x_n\} \sim \{y_n\} \iff \{x_n-y_n\} \rightarrow 0$ Then you get all real numbers as an equivalence class with this ...
0
votes
0answers
50 views

Quotients vs equivalence relations

In a recent preprint I defined the category of quasi-frames (qframes for short) as follows: a qframe is a modular and upper continuous complete lattice; a morphism of qframes $f:L_1\to L_2$ is a map ...
1
vote
1answer
26 views

Best approach to determine the equivalence classes of a formal language

I created a minimum automaton for a formal language using the Myhill-Nerode theorem. The language for which I created the automaton is defined by $L=\{w \in \{a,b\}^*:w=av \text{ for a word } v ...
0
votes
3answers
38 views

Linear/Discrete Math Equivalence Classes

I am confused on this question: For each of the following binary relations on $\Bbb R$, state whether or not the relation is an equivalence relation. If it is an equivalence relation, describe the ...
3
votes
0answers
33 views

Proving Equivalence Relation $xPy$ iff $ y = x + n\pi$ on The Reals

I am trying to prove that the relation P on $\mathbb{R}$ by the rule $\forall x, y \in \mathbb{R}, xPy \text{ if and only if } \exists n \in \mathbb{Z} \text{ such that }y = x+ n\pi$ From what I can ...
0
votes
2answers
35 views

If S is finite then the equivalence classes will not exceed the size of set S

If S is a finite set and $\sim$ is an equivalence relation on it, then the total number of equivalence classes can never exceed $\vert S\vert$ and it can be any integer number $1\leq k\leq\vert ...
0
votes
1answer
31 views

Equivalence relation for which there are infinitely many equivalence classes.

On set $\mathbb{R}$ and the relation on it where $x\sim y$ if $x^{4}=y^{4}$. Then $\sim $ is equivalence relation for which there are infinitely many equivalence classes, one of which consists ...
6
votes
1answer
163 views

Confusion between an element and its preimage

Let $X$ be a set and $\sim$ is an equivalence relation on $X$, so that the quotient set $X/_\sim=\bigcup_{x\in X}{[x]}$ with $[x]=[y]$ if and only if $x\sim y$. Consider the quotient map $f:X\to ...
0
votes
2answers
23 views

How to proof equivalence relation?

I need help with this problem: Let $S=\left\{\left[\begin{matrix} a & b \\ c & d \end{matrix}\right] : a,b,c,d \in \mathbb{C}\right \}$ and $M=\left\{\left[\begin{matrix} a & b \\ ...
0
votes
2answers
22 views

What is an algorithm for describing the partition of this equivalence relation?

Let $\mathbb{R}$ be the set of real numbers,$ f : \mathbb{R} \to \mathbb{R}$ a map, and $E$ the equivalence relation on $ ℝ $ defined by $E = \{(x,y) \in \mathbb{R} \times \mathbb{R} \mid ...
3
votes
2answers
31 views

Test for symmetric property of this ordered pair

Suppose the set $$S={1,2,3}.$$ I must show that the equivalence relation $$R=\{(1,1),(1,3),(2,2),(3,1),(3,3)\}$$ is on the set. The reflexive property states that: $$(a,a) \in R \;\forall a \in ...
2
votes
1answer
96 views

Can a relation be both anti-reflexive and anti-symmetric?

Is it possible for a relation to be both anti-reflexive and anti-symmetric?
2
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0answers
90 views

Relative Identity vs Set Theory

The last complete, but unpublished, paper by the late Tom Etter titled "Three-Place Identity" purports to prove that all of mathematics can be expressed in terms of relative identity. In his own ...
0
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1answer
16 views

Anti-symmetric or asymmetric for a relation between pairs in the set of Z x Z?

Is this anti-symmetric or asymmetric? I at first thought asymmetric because anti-symmetric would mean a = c and b = d which would not be true. But because the domain is the Cartesian product of ...