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

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Explicit Description for an Equivalence Relation

Given a set function $f : X \to X$ let $\sim$ be the equivalence relation $x \sim f(x)$. Contextually, I am working with the coequalizer of $f$ and $1_X$. I want to have as much information about the ...
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1answer
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Basic question on equivalence relations.

Show that the following relation is an equivalence relation on the given set. $m \sim n$ in $\mathbb{Z}$ if $m \equiv n\,(\text{mod}\,6)$.
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Finding the Equivalence of this Relation (Maximum)

I have this inequality relation where $\frac{\log(1+X_k^2)}{A_k} \geq \max_{m \in \mathcal{K} \setminus k} \frac{\log(1+X_m^2)}{A_m}$ Since the maximum is irrespective of the $k$'s, I reduce it to ...
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32 views

does the equivalence class of an element in a set is the set itself?

what does equivalence class mean? I am trying to understand I think that I am a little confused so let's take this example: Suppose that we have the relation $2x+3y$ is a number less than or equal ...
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1answer
25 views

Proving equivalence relations help [closed]

I need to show the three properties of an equivalence relation basically. (a.) Why is A~A for every set A? For this part I know that there exists a function f:A->A that is one to one and onto, ...
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1answer
26 views

Distinct elements of relations

R is the relation on Z (integers) given by xRy ( X is related to Y ) if 3 divides (x-y), what are the distinct elements of R?
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110 views

When is $(a+b)^n \equiv a^n+b^n$?

I remember a relation like $(a+b)^n \equiv a^n+b^n$, but I don't remember mod which numbers this is true. Where can I learn more about this?
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1answer
30 views

Finding a unique relation $T$

This is one question I have been solving from Velleman's How to prove book: Suppose $R$ and $S$ are relations on a set $A$, and $S$ is an equivalence relation. We will say that $R$ is compatible ...
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3answers
31 views

Why is this binary relation symmetric but not reflexive or transitive

Let $\def\Rthree{\,{\mathrm{R}_3}\,} \Rthree$ be the relation on sets $C$, $D$ of natural numbers such that $C \Rthree D$ iff $C \cap D$ is finite. Then $\Rthree$ is symmetric, but not reflexive or ...
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1answer
48 views

Proof of identity A = A or 1 = 1

Is 1 = 1 an assumption? I feel it's a very good assumption, but is there a proof for it? Imagine a world where people were contesting it, where equivalence wasn't a common sense concept. In reality no ...
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1answer
51 views

Why does this equivalence stand?

I am reading the proof of the following theorem: THEOREM A. Let $R$ be an integral domain of characteristic zero; then the diophantine problem for $R[T]$ with coefficients in ...
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44 views

Relations, Equivalence class

Define the relation $R$ on the set $\Bbb Z^+$ of all positive integers by: for all $a, b \in \Bbb Z^+$, $aRb$ if and only if the largest digit of a is equal to the largest digit of $b$. For example, ...
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When are algebraic expressions equivalent?

This question arose when I was going to determine the domain for $f \circ f(x)$. Let $f(x) = \dfrac{1-x}{1+x}$. $f \circ f(x) = x, \quad$ But the domain is not $\mathbb{R}$ because $f(x)$ is undefined ...
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2answers
27 views

Reflexive, Symmetric, and Transitive on a relationship defined as “m-n is odd” proof

Main question: Is my solution for this proof correct? Also, I have some questions about my solution and the definitions of Reflexive, Symmetric, and Transitive. Here is the question and here is my ...
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2answers
59 views

Proving $Y$ such that $Y \cap B = \emptyset$

I have been solving this problem from Velleman's How to prove book: Suppose $B \subseteq A$ and define a relation $R$ on $\mathcal{P}(A)$ as follows: ...
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0answers
16 views

The equivalence principle and experiments concerning it? [migrated]

Imagine that we are in a rocket accelerating with some magnitude $a_1 = dx^2/d^2y$, also imagine that we have a stationary rocket ship in close proximity to ours, stationary relative to our reference ...
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1answer
37 views

Can someone present an example of linear ordering less trivial than $A = [1,2,3,4,5,6…]$

A linear ordering (loset) is a poset that also satisfies the trichotomy law. For any $x,y \in A$, we have $x \leq y$ or $y \leq x$ A common example is presented as $A = [1,2,3,4,5,6...]$ Can ...
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27 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 ...
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1answer
48 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 ...
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2answers
28 views

Shouldn't this be transitive relation also?

$ R = \{ (a,b) ∈\Bbb R^2 ; 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)...\}$ ...
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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 ...
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3answers
31 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 ...
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1answer
42 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 ...
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25 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 ...
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3answers
37 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. ...
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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 ...
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1answer
27 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 ...
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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 $\sim$, ...
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1answer
34 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 ...
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4answers
52 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 $a-b\in\mathbb Z$. ...
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402 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 ...
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1answer
28 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 ...
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1answer
45 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 ...
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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
55 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 ...
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28 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 ...
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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 ...
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1answer
13 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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25 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 ...
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1answer
62 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 ...
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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 ...
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65 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. ...
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4answers
68 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 ...
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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.
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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
21 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 ...
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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 ...
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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 ...
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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 ...
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1answer
31 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 ...