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

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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
18 views

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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2answers
29 views

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
45 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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27 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 ...
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1answer
39 views

How to understand this definition of equivalence relations

I often see this type of definitions of equivalence: Suppose that $f$ and $g$ are differentiable on $\mathbb R$. We can define an equivalence relation on such functions by letting $f(x) \sim g(x)$ ...
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1answer
38 views

Defining Equivalence relations

So I am not really comfortable with equivalence relations, so this example from Wikipedia gives me trouble. Here is what it says: Let the set $\{a,b,c\}$ have the equivalence relation ...
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36 views

why are equivalence relations called so?

"an equivalence relation is the relation that holds between two elements if and only if they are members of the same cell within a set that has been partitioned into cells such that every element of ...
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1answer
16 views

Transitive Relations Problem,

Let S be the set of all three-digit numbers, and define x~y to mean that x and y have the same first and last digit. (i) Show that the relations ~ is transitive. (ii) List two numbers in the ...
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2answers
28 views

Equivalence Classes Output

I understand that an equivalence relation is a set that is reflexsive, symmetric, and transitive. I don't quite understand equivalence classes though. For example: What would the equivalence class be ...
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1answer
34 views

How to determine an equivalence class?

Let $A$ be a nonempty set and let $B$ be a fixed subset of $A$. A relations $R$ is defined on the power set $\mathcal{P}(A)$ by $X\mathrel{R}Y$ if $X \cap B = Y \cap B$. Let $A=\{1,2,3,4,5\}$ and ...
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38 views

Why is this an equivalence relation, and what does the equivalence classes contain?

I'm doing some discrete mathematics exercises, but I can't seem to wrap my head around this relation: $$R(x, y) \text{ if } \exists z(\text{LiesInPart}\circ\text{LiesInCountry}(x,z) \wedge ...
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2answers
24 views

Relations and Equivalence - numbers are related if they have the same floor

$S$ is defined on $\mathbb{Q}$ by $xSy$ if and only if $⌊x⌋=⌊y⌋$ (Note that$⌊q⌋$is defined to be the largest integer less than or equal to q. You can think of it as “$q$ rounded down”.) We've been ...
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1answer
20 views

Equivalence relations and classes

$T$ is defined on $\mathbb{N} \times \mathbb{N}$ by $(a,b)\mathrel{T}(c,d)$ if and only if $a \leq c$ and $b \leq d$. I know this is a partial order relation as it is Transitive, Anti Symmetric and ...
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1answer
22 views

Describing equivalence classes over the set of natural numbers

So for this problem we are supposed to describe the equivalence class for the following relation. $x \thicksim y$ iff $x$ mod 2 = $y$ mod 2 and $x$ mod 4 = $y$ mod 4. I am confused on what is meant ...
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1answer
36 views

verifying properties of relations to test equivalence

We are doing some more with relations and this time we are given a relation and told that it is not equivalent. We need to find out which property it does not fulfill. So we at least know there is ...
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2answers
39 views

Can someone explain how to find the number of equivalence classes and elements?

I am struggling so much with this topic. Trying to do some practice questions but I don't seem to get it. What I'm working on is Let $A = \{ 1, 2, 3, \dots, 2014 \} = \{ x \mid 1 \le x \le ...
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1answer
30 views

Proving that two equivalence classes are disjoint?

I am having trouble with the following proof: Define the relation $R$ on $\mathbb{Z}$ by $nRm$ if $n-m$ is divisible by $2$. Prove that the equivalence class for $0^{(\bar{0})}$ and the equivalence ...
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1answer
40 views

Which equivalence class represents the zero element $0_{\mathbb Q}$ in $\mathbb Q$?

The Statement of the Problem: We identify $\mathbb Q$ with the set of equivalence classes $[a,b]$, where $(a,b) \in \mathbb Z \times \mathbb N^+$ and $(a,b) \sim (a'b')$ iff $ab'=ba'$. We define ...
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1answer
24 views

Union of equivalence relations

As part of a HW assignment in the course elementary set theory, I was given the following question: Let $A$ be a set and let $T$ and $S$ be two equivalence relations on $A$. Prove: $S\circ ...
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1answer
49 views

Composition of equivalence relations

As part of a HW assignment in the course elementary set theory, I was given the following question: Let $A$ be a set and let $T$ and $S$ be two equivalence relations on $A$. Prove: $S\circ ...
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1answer
24 views

Equivalence Relation on the set of ordered pairs of positive integers

Have a homework question, but how can I show that the given relation R is reflexive, symmetric and transitive, so that it is an equivalence relation. Appreciate assistance from anyone. "Let R be the ...
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858 views

“$ x $ is a brother of $ y $.” Why is this not transitive?

I am working on a problem set at the moment, and while checking my answers I realized that I have listed "x is a brother of y" as a transitive relation, while the answers say that it is not. EDIT: I ...
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1answer
32 views

Help with equivalence classes for $x\sim$y iff $x-y\in\mathbb{Q}$.

Help with equivalence classes for $x\sim$y iff $x-y\in\mathbb{Q}$. I need to show the equivalence classes for $[0]_{\sim}$ and $[\sqrt{2}]_{\sim}$. Here is what I did: $[0]_{\sim}$ = ...
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0answers
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Is bisimilarity an equivalence relation

I want to know if bisimilarity is an equivalence relation. I need to make a proof showing that this is true but I have searched and I can only find for branching bisimilarity.
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1answer
31 views

Determining a relation if reflexive, symmetric, and transitive

I just get stuck in this relation and need to find if this relation is Reflexive/ Irreflexive or Neither, Symmetric/ Antisymmetric or Neither, Transitive or Not. $$W_1 = \{(a , b) \in \mathbb ...
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1answer
40 views

Partition and equivalence relation

Consider the equivalence relation between non-empty subsets $A , B$ of $\{ 1,2,3, 4,\dots,100\}$ defined by the condition: the greatest element of $A$ is the same as the greatest element of $B .$ ...
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1answer
24 views

The equivalence relation generated by a relation

Let $X$ be a non-empty set and let $r\subseteq X\times X$ be a relation on $X$. Let $R$ be the intersection of all equivalence relations on $X$ that contain $r$. Prove that if $xRy$, then one of the ...
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1answer
96 views

Klein bottle and real projective two-space

I've been looking at equivalence relations on the unit square: $[0,1] \times [0,1]$ that give rise to various surfaces such as the m$\ddot{\mathrm{o}}$bius strip, but I'm not too sure about the Klein ...
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quick question about a definition of homeomorphism set classes

Take $S$ a surface of general type. I want to define $Q$ the set of homeomorphism classes determined by the surface $S$. How can i define $Q$? I think that $Q$ is determined by all surfaces $S^{'}$ ...
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83 views

Finding the equivalence classes of a relation R

Let A = {0,1,2,3,4} and define a relation R on A as follows: R = {{0,0},{0,4},{1,1},{1,3},{2,2},{3,1},{3,3},{4,0},{4,4}}. Find the distinct equivalence classes of R. How do I solve this problem? ...
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1answer
46 views

Proving Equivalence Relations On A Set

Let X be the set of all nonempty subsets of {1,2,3}. Then X = {{1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}} Define a relation R on X as follows: for all S and T in X, SRT if, and only if, the least ...
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1answer
35 views

Describe the equivalence classes for each equivalence relation

Let ~ be an equivalence relation on $\mathbb R^2$ such that $\left( x_1, y_1 \right)$ ~ $\left(x_2, y_2 \right)$ iff $y_1=y_2$. Let ~ be an equivalence relation on $\mathbb R^2$ such that $\left(x_1, ...
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0answers
37 views

Show that same cardinal number defines and equivalence relation

Same cardinal number must satisfy all three properties : symmetry, reflexivity, transitivity Symmetry Suppose bijection $f:A→A$, Then by definition, |A| = |A| Reflexivity Suppose bijections ...
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1answer
33 views

How to show the existence of a binary chain

Suppose A, B are finite binary chains that hold AB = BA (* is the concatenation operator). How can I show there exists a binary chain C such that A, B are of the form CCC...C?
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1answer
99 views

Notions of consistency / heterogeneity in sets of vector values

The problem Let us consider a row vector u of size $n\in\mathbb{N}$, containing only binary values (0,1): $$u=(u_1 \cdots u_n), n\in\mathbb{N}$$ $$\forall i \in \{1\ldots n\}, u_i \in\{0,1\}$$ I ...
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1answer
23 views

Relations and Equivalence Sequences

A relation is defined on the set $A=\{a + b\sqrt{2} \; : \; a, b \in \mathbb{Q} \text{ and } a + b\sqrt{2} \neq 0\}$ by $xRy$ if $x/y$ is in $\mathbb{Q}$. Show that $R$ is an equivalence relation and ...
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0answers
41 views

The relationship between an equivalence relation, equivalence classes, and partitions?

So I'm struggling right now to understand the concepts behind equivalence relations, equivalence classes, and partitions. This is what I understand about all these topics right now: Equivalence ...
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1answer
25 views

Equivalence relation, product and quotient spaces

I have a problem with the following: "Define a relation $\sim$ on $R^2$ by $(u,v) \sim (x,y)$ if and only if both $u-x$ and $v-y$ are integers. Show that for each point $(x,y) \in R^2$ there exists ...
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0answers
27 views

Help Representing Equivalence Classes

In the set $\mathbb{Z}$, we define two integers $x$ and $y$ to be equivalent ($x ≈ y$) if and only if $x \operatorname{div} 10 = y \operatorname{div}10$. How would one select a representative from ...
0
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1answer
22 views

Complementary Relation Proof

If $R$ is reflexive, prove that $R^c$ is irreflexive. If $R$ is asymmetric, prove that $R^c$ is reflexive. Where $R^c$ = complement of $R$. I just can't figure out anything to say for the first one ...
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1answer
38 views

Given the following, is there equivalence relation?

Let $n$ be an integer. On the set $F$ of all integer-valued functions of a set $A$, suppose we define $f$ and $g$ to be related if $f(a)\equiv g(a)\pmod{n}$ for every $a\in A$. Is this an equivalence ...
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139 views

Number of elements in an equivalence class

Let a set X = {1, 2, 3, 4, ... , 2015} and a set Y = {1, 2, 3, 4, ... , 271}. Let S be the relation on P(X) defined by: For all sets A, B, that are elements of P(X), (A,B) are elements of S if and ...
0
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1answer
27 views

Decimals and equivalence relations

I am told that decimals set up an equivalence relation on the Reals and that decimal numbers and the Reals are not the same thing. I believe this also clarifys the famous $.\bar{9}=1$. That $1$ and ...
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1answer
46 views

Prove that S is an equivalence relation on P(X)

Let $X = \{ 1, 2, 3, \ldots , 2015\}$ and $Y = \{ 1, 2, 3,\ldots, 271 \}$. Let S be the relation on power set (X) defined by For all A, B in $\mathcal{P} (X)$, $(A, B) \in S$ if and only if $|A \cap ...
2
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1answer
72 views

Measure spaces s.t. $\mathcal{L}^1 = L^1$

I have two questions: 1, Give an example of a measure space such that $L^{1}(X,\mathcal{A},\mu) = \mathcal{L}^{1}(X,\mathcal{A},\mu)$. 2, State, and prove, a condition on $\mu$ which is equivalent ...
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286 views

How to find the appropriate equivalence class?

I would like find the system of representatives & equivalence class for $[(1,-1)]_{\equiv 1}$ and $[(1,-1)]_{\equiv 2}$ given: $R_1=_{def.} (x_1,y_1)\equiv(x_2,y_2) \Leftrightarrow ...
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4answers
55 views

Prove $R$ is an equivalence relation.

I think I'm on the right track. Set $S = N \times N$, and for any two members $(a,b),(c,d)$ of $S$, define $(a,b) \simeq (c,d)$ provided that $ad = bc$. Prove that $\simeq$ is an equivalence ...
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1answer
50 views

cardinality of polynomial

What is the cardinality of the following sets? (Choose from finite, countably infinite, or uncountably infinite.) The set of polynomials of the form $ax+b$ with $a \in\Bbb N$ and $b \in\{0,1\}$ ...