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

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Equivalence relation in a commutative ring

Let $R$ be a commutative ring and $I \neq R$ ideal of $R$. For $x,y \in R$, define $$ x \sim y \iff \exists\ a,b \in I\ \text{such that } x(1+a) = y(1+b). $$ One can easily see this is an equivalence ...
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51 views

For a set of symmetric matrices $A_i$ of order p, show that if the sum of their ranks is p, $A_iA_j=0$

Here's what I know. Matrices $A_i$ for $i=1,...,k$ are all symmetric p by p matrices. $\sum\limits_{i=1}^k A_i = I_p$ where $I_p$ is the p by p identity matrix $\sum\limits_{i=1}^k rank(A_i) = p$ ...
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1answer
49 views

Equivalence relation $a\sim b$ iff $a=10^kb$

Consider the relation on the set of all real numbers $\mathbb R$, defined by $a\sim b$ if, and only if, there exists an integer $k$ so that $a = 10^k b$. Prove or disprove: this is an equivalence ...
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116 views

Proving Logical Equivalence with Laws of Logic

I'm working on Logical Equivalence problems and I'm having trouble understand what to do with this first problem. The problem is to show that these two statements are equivalent to one another ...
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1answer
27 views

Prove that there exists a unique injective map $\gamma:X/_R\to Y$ with image equals $f(A)$ with $f=\gamma \circ \phi.$

So recently I have established an equivalence relation $R$ on $X$, with $xRx'$ iff $f(x)=f(x')$. Let $\phi:X\to X/_R$ be the map of sets sending $x\mapsto \bar{x}$. Prove that there exists a unique ...
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221 views

Equivalence relation

So I'm pretty new to abstract mathematics being a biologist an all. My biggest issue is that I can't really wrap my head around how to solve problems. So I have the problem: Let $X$ be the set of ...
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Let $A$ be an equivalence relation on the set $X$ and $B$ be the equivalence on $Y$. Find equivalence class of $A\times B$.

The definition of $A\times B$ is given to be a condition on pairs. Let $a, c \in X$ and $b, d \in Y$, then $(a, b)\sim_{A\times B}(c, d)$ if $a\sim_Ac$ and $b\sim_Bd$. I have shown that this is an ...
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76 views

Set question, relations, equivalence classes

Let S be the set of all bit strings (a sequence of 1s and 0s) of length 3 or more. Let R be a relation on S of all pairs (x, y) where x and y are in S if x and y have the same first two bits. Is R ...
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1answer
32 views

Equivalence Classes, Power Sets, and basic set theory question

Fixing $N \in P(X)$ (the power set of X), we say that $A,B \in P(X)$ agree away from $N$ if $A - N = B - N$. We denote $A \sim B$ if $A - N = B - N$. I have to show that every equivalence class has a ...
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21 views

Quotient Construction of a Power Set, equivalence relations, and symmetric difference.

Fixing $N \in P(X)$ (the power set of X), we say that $A,B \in P(X)$ $agree \ away \ from \ N$ if $A - N = B - N$. I have to show that $A \sim A', B \sim B'$ implies that $(A \Delta B) \sim (A' ...
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2answers
26 views

Equivalence relations

Having trouble proving this is an equivalence relation. Is it suffice to say that let $x y z$ be any string in $\Sigma^*$, $(xz \in L \iff yz \in L) \rightarrow (yz \in L \iff xz \in L)$ shows ...
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1answer
53 views

Prove that $R$ given by $L(a,b)\cap W =\varnothing$ is an equivalence relation.

Let $V$ be a real vector space and suppose $W$ is a subspace of $V$ with $\dim(W)=n-1$. Now define a relation $R$ on $V-W$ s.t. $$aRb\iff L(a,b)=\{ra+(1-r)b | 0\leq r \leq 1\}\text{ has the property ...
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1answer
19 views

Addition in $R_S$ is well defined

Let $R$ be a commutative ring with $1 \neq 0$ and suppose S is a multiplicatively closed subset of $R \backslash {\{\, 0 \,\} }$ containing no zero divisors. We have the relation ∼ defined on $R × S$ ...
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2answers
118 views

Equivalence relation on class of all sets

Let $\mathcal{A}$ be the class of all sets. Prove that "has the same cardinality as" defines an equivalence relation on $\mathcal{A}$. I know that equivalence relations must be reflexive, symmetric ...
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37 views

Prove that $M/{\sim}$ forms a partition of $M$.

My attempt: Suppose $\sim$ is an equivalence relation on $M$. If $a \in M$, let $\bar{a}=\{m \in M \,|\, m\sim a\}$. Since each element $a$ of $M$ is in its own equivalence class $\bar{a}$, the union ...
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5answers
52 views

Prove that $(\mathbb{R}\setminus \{0\},\sim):= ab>0$ is transitive.

I have the following problem: A relation $\sim$ on $\mathbb{R}\setminus\{0\}$ is defined by $a\sim b$ if $ab>0$. Show that $\sim$ is an equivalence relation and identify the equivalence ...
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1answer
30 views

Equivalency of two definitions of WARP (Weak Axiom of Revealed Preference)

I have two definitions for WARP as follows. How can I prove they are equivalent? First Definition: $C(A) \cap B \neq \emptyset \Rightarrow C(B) \cap A \subset C(A)$ Second definition from ...
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1answer
28 views

Prove relation $\rho:(\forall x,y \in G)(x\rho y \Leftrightarrow x * y^{-1}\in H),H\le G$ is equivalence relation

Prove relation $\rho:(\forall x,y \in G)(x\rho y \Leftrightarrow x * y^{-1}\in H),H\le G$ is equivalence relation $H\le G\Rightarrow(\forall a,b\in H)a*b^{-1}\in H$ $(\forall x \in G)x\rho x$ ...
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1answer
43 views

Equivalent vs equal. When to use what?

When do you use equal and when equivalent? Why do I see on this site: (this is a random formula taken from this site): $\frac{\partial}{\partial \mu}F_X(x; \mu, \sigma^2) =\frac{\partial}{\partial ...
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0answers
12 views

Equivalence Relation on the common restriction of two functions

I need to determine whether the following is an equivalence relation on $X$: $X = \{ f|f: A \to \mathbb{R}, A \subseteq \mathbb{R}\}$ (the set of functions from a subset of $\mathbb{R}$ to ...
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1answer
92 views

Proof that every open set in $\mathbb{R}$ is a countable union of disjoint open intervals using equivalence classes

We were told to prove that every open set in $\mathbb{R}$ is a countable union of disjoint open intervals using equivalence classes in the following way: Consider the following relation on a given ...
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1answer
33 views

Question on equivalence relations

In set theory a relation is said to be equivalence if the relation is,. Reflexive Symmetric Transitive I would like to know if the following relation is an equivalence one. $R = \{ (m,n) \in Z ...
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1answer
26 views

Proving symmetry in an equivalence relation

Let $H$ be a subgroup of a group $G$ and let $a,b \in G$. Define the relation $\equiv$ on $G$ by $a\equiv b$ if and only if $ab^{-1} \in H$. Show that $\equiv$ is an equivalence relation on $G$. Well ...
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1answer
34 views

Logical equivalence with truth tables

I'm trying to solve the highlighted part. I understand how they got the left side of the equation. They wrote out the truth table for a or b and then negated it. Easy. However for the right side I ...
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1answer
48 views

Defining an Operation

Let $R$ be the relation on $X=\mathbb Z\times \mathbb N$ such that $(a,b)R(c,d)$ if and only if $ad=bc$. Define an operation $\bullet$ on $X/R$ as follows: for $x=(a,b)$ and $y=(c,d)$ let: ...
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1answer
15 views

Proving equivalences and bijection

Define $\equiv$ and $\sim$ on $\mathbb{R}$ by $x\equiv y$ if $x-y \in \mathbb{Z}$ and by $x\sim y$ if $x-y\in \mathbb{Q}$. a) Show that $\equiv$ and $\sim$ are equivalences. b) Show that ...
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85 views

Prove the relation ~ on $A$ defined by $a$~$b$ if and only if $a = hb$ is an equivalence relation.

Let $H$ be a group acting of a set $A$. Prove that the relation ~ on $A$ defined by $a$~$b$ if and only if $a = hb$ for some $h \in H$ is an equivalence relation. (For each $x\in A$ the equivalence ...
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1answer
25 views

Labeling of axes in quotient space

It seems to be trivial, but I couldn't find anything. I have a metric space, let us say $\mathbb{R}_{>0}^3$, and a equivalence relation $\sim$, let's say $(x_1,y_1,z_1)\sim(x_2,y_2,z_2)$ if ...
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1answer
30 views

Equivalences and bijections

I have to show that the following are an equivalence relation on $A$ and find a bijection between $A/\sim$ and $B$. I know that to show something is an equivalence relation it needs to satisfy the ...
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63 views

Help with partitions, equivalence classes, equivalence relations.

The following definitions and results are from my textbook. A partition $\mathcal{P}$ of a set $X$ is a collection of nonempty sets $X_1, X_2, \dots$ such that $X_1 \cap X_j = \emptyset$ for $i ...
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4answers
51 views

Into how many equivalences classes does $R$ partition $\mathbb{Z}$?

Let $R= \{ (a,b) \in\mathbb{Z}\times\mathbb{Z} \mid a^2\equiv b^2 \bmod 7\}$. Into how many equivalences classes does $R$ partition $\mathbb{Z}$? My best guess is that there are $7$ equivalence ...
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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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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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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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32 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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122 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
31 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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57 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
91 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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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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65 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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546 views

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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205 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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63 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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1answer
39 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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1answer
34 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
59 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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29 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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1answer
41 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 ...