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

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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
30 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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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
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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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2answers
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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
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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1answer
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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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331 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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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
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 ...
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2answers
160 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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0answers
22 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
126 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
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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
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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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609 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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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
286 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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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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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 ...
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3answers
40 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 ...
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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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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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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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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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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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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
18 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 ...
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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 ...
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1answer
21 views

A question on eqivalence relation

Please explain what the examiner means by asking: Also find [3,6] in Q 1 (a)(i)??
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40 views

Does this notion of “weak” isomorphism exist in literature?

Let $(M,\circ)$ and $(N,\ast)$ be two magmas. I'd like to relax the notion of isomorphism by defining a notion of "weak" isomorphism in the following way: $M$ and $N$ are "weakly" isomorphic if there ...
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About the group of units

I'm stuck at this section of the following problem: Let $R$ be a commutative ring and $S$ be a subset of $R$, with $S$ multiplicatively closed. Find the group of units of $S^{-1}R$. My try: ...
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Well-defined and Equivalence relations

I am wondering why the following is well-defined... The definition of well-defined is given as; $g:(X/\sim) \to Z$ is well-defined if a mapping $f:X \to Z$ can be found where $f$ has the property $x ...
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Proof of an equivalence relation

Let S be the relation on R defined by $xSy \Leftrightarrow x=|y|$, $\forall x,y\in\Re$ Is the relation reflexive, symmetric and/or transitive? By my proof that 1) $x=|y| \Rightarrow |y|=x$ ...
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Equivalence classes in $\mathbb{Z}_n$

I've the following exercise: Solve each of the following equations in the given set $\mathbb{Z}_n$: 1) $[5]+x=[1]$ in $\mathbb{Z}_9$ 2) $[2]\cdot x=[7]$ in $\mathbb{Z}_{11}$ For 1), is $x=5$ ...
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1answer
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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 ...
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1answer
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More clarification on an equivalence relation problem already answered

So this problem already has a solution: Problem with Equivalence Relations I'm good for the majority of it except for part c), I wasn't able to figure that out on my own or by looking at the ...
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simple clarification of equivalence relation and order relation notation meaning

So by definition an equivalence relation is a binary relation on a subset of the cartesian product of our set A, i.e $A\times A$ satisfying the 3 conditions. But I'm a little confused about ...
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1answer
23 views

Make the set $R$ transitive

Let $R$ be a relation on a set $A=\{w,x,y,z\}$ defined by $R=\{(w,x),(y,x),(x,y),(z,z)\}$. Using the original relation, $R$, make the necessary minimal additions to make $R$ transitive. I thought ...
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1answer
44 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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1answer
26 views

Intersection of two sets which are equivalence on set A is always equivalence?

If $R_{1}$ and $R_{2}$ are equivalence relations on set A ,then$ R_{1}\bigcap R_{2}$ must be equivalence relation. firstly, I am not understanding the function of R,I think that, this is only a ...