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

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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
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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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Equivalence relation in the collection of all normed linear space over some field K.

The concept of equivalent norm produce an equivalence relation in the collection of all normed linear space over some field K. I've no idea how to make a start... Please help
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1answer
22 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
57 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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2answers
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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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1answer
61 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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450 views

Equivalence relation and equivalence class

I have looked around the internet for an easy to approach - down to earth explanation of equivalence relation & equivalence class but having no success. If any of you can explain in very basic ...
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4answers
33 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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Proving equivalence relations

I just started my abstract algebra class and I am struggling with the concept of equivalence relations. I know that in order to prove equivalence relations, I have to prove the reflexive, symmetric, ...
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1answer
18 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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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 ...
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1answer
76 views

Optimization: KKT conditions statement

I'm currently following this material Optimization Theory: Chapter 2 Theory of Constrained Optimization And I can't understand why the following statement is true, between the equations (2.9) and ...
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Construction from a semiring

Hello guys! Reading about abstract algebra,I came across something I didn't really understand and don't even know where to look for it,as I don't know the name. can you please help me,? Thanks!
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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 ...
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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 ...
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3answers
35 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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1answer
26 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 ...
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1answer
299 views

equivalence relation composition problem

Let $R_1$, $R_2$ be two equivalence relations on $X$, prove that $R_1\circ R_2$ is an equivalence relation if and only if $R_1\circ R_2= R_2\circ R_1$ First I´m trying to prove that $R_1\circ R_2= ...
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1answer
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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
36 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 ...
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2answers
152 views

Why are equivalence classes called “classes” and not “sets”?

Why are equivalence classes called so and not equivalence sets? I am kind of not able to find the difference between a class and a set. What properties that a set have that a class cannot have? It ...
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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?
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4answers
123 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 ...
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1answer
26 views

Verify Equivalence Relations

Find an example of three relations $R_{1}$, $R_{2}$, and $R_{3}$ on the set $S=\{1,2,3,4,5\}$ such that $R_{1}$ is reflexive but not transitive, $R_{2}$ is transitive ...
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1answer
12 views

Equivalence Relation Properties

I have to define whether the following relation is symmetric, reflexive, and transitive. Define a relation R on Z as follows: (x, y) ∈ R if and only if x = |y|: This is my answer so far: Is ...
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1answer
65 views

The relation $a + d = b + c$ between pairs $(a, b)$, $(c, d)$ is an equivalence relation

Let R be the relation on $Z × Z$, that is elements of this relation are pairs of pairs of integers, such that $((a, b),(c, d))\in R$ if and only if $a + d = b + c$. Show that R is an equivalence ...
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4answers
445 views

If R is $(a,b)R(c,d) \iff a+d =b+c$ show that R is an equivalence relation.

The relation R is defined n all positive integers such that, $(a,b)R(c,d) \iff a+d =b+c$ . Show that R is an equivalence relation. In order to be an equivalence relation, R has to be reflexive, ...
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61 views

Equivalence relations for $\mathbb{N} \times \mathbb{N}$ defined as $(m, n) \sim (k, l)$ if $m+l=n+k$

On the set $\mathbb{N} \times \mathbb{N}$ define $(m, n) \sim (k, l)$ if $m + l = n + k$. Show that $\sim$ is an equivalence relation on $\mathbb{N} \times \mathbb{N}$. Draw a sketch of $\mathbb{N} ...
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373 views

The fibres of a map form a partition of the domain.

This is a question from the free Harvard online abstract algebra lectures. I'm posting my solutions here to get some feedback on them. For a fuller explanation, see this post. This problem is from ...
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4answers
285 views

Equivalence Relations: $(a,b)\sim(c,d)$ if and only if $a^2+b^2=c^2+d^2$

I need someone to check my proof. Question: On the set $\mathbb{R}^2$ of ordered pairs define the 2-plane relation $\sim$ as follows $(a,b)\sim(c,d)$ if and only if $a^2+b^2=c^2+d^2$. Prove that ...
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What's the name for the equivalence induced by a function on its domain?

Any function $f$ with domain $X$ induces an equivalence relation on $X$, with classes $$\{f^{-1}(\{y\})\,:\, y \in \operatorname{im}f\;\} .$$ Is there a name for this equivalence? Thanks!
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2answers
30 views

How to prove that a1 ∼ a2 ⇔ f(a1) = f(a2) is an equivalence relation?

Suppose a function f : A → B is given. Define a relation ∼ on A as follows: a1 ∼ a2 ⇔ f(a1) = f(a2). a) Prove that ∼ is an equivalence relation on A. I know that I have to prove for the reflexive, ...
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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 ...
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3answers
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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 ...
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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 ...
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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 ...
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60 views

Finding Equivalence Classes for Infinite Sets

Let $R$ be the relation on the set of rational numbers $\Bbb Q$ defined as follows: for all $q, r \in \Bbb Q$, $qRr$ iff $q − r \in \Bbb Z$. Then $R$ is an equivalence relation on $\Bbb Q$. What is ...
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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 ...
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1answer
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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 ...
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1answer
159 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 ...
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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 ...
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Is $R=\left \{ (a,a),(a,b),(b,a),(b,b),(c,c),(c,d),(d,c),(d,d) \right \}$ an equivalence relation on $X$?

Let $X= \left \{ a,b,c,d \right \}$ and $R=\left \{ (a,a),(a,b),(b,a),((b,b),(c,c),(c,d),(d,c),(d,d) \right \}$. I want to show that $R$ is an equivalence relation on $X$. My work: $R$ is reflexive: ...
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89 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 ...
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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 \\ ...
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2answers
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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 ...
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1answer
71 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?
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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 ...
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1answer
32 views

Equivalence relations regarding binary relations

Let $R \subseteq X \times X$ be a binary relation for $X = \{a, b, c, d\}$. $R = \{(a, a), (b, c), (c, d), (b, d)\}$. Is the relation an equivalence relation? I don't know if I am proving it correctly ...