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

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Proving projection map is onto

Let $X$ be a nonempty set and let $R$ be an equivalence relation on $X.$ Given $X/R={[a]:a \in X}$. Prove that there is a map called the projection where $p_x:X\to X/R$ given by $p_x(t)=[t].$ Then ...
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Orbits in G = $Z_6$ by listing 2 element subsets in G.

1) Let $G = \mathbb{Z}_6$. List all 2-element subsets of $G$, and show that under the regular action of G (by left addition) there are 3 orbits, 2 of length 6, one of length 3. Deduce that the ...
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29 views

How to show that the following relation is not an equivalence relation?

We have the relation $\sim$ in $\mathbb{R}^n$: $x\sim y \leftrightarrow d(x,y)\in \mathbb{Q}$, where $d(x,y)=\sqrt{\sum^n_{i=1}(x_i-y_i)^2}$. How do you prove that this isn't an equivalence relation ...
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24 views

Having trouble proving transitivity

We have a universal set of lowercase alphabet letters, $U = \{a, b, ... , z\}$ . For sets $A,B \subseteq U$ we can define a relation, $A \sim B$ as long as the number of elements that are in either ...
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How to prove $S=\{(x,y) \in \mathbb{R}\times \mathbb{R}|x - y \in \mathbb{Q} \}$ is an equivalence relation?

I am really stuck with this problem, and I cannot come out with a solution. I know that to prove a relation is an equivalence relation we have to prove that it's reflexive, symmetric and transitive, ...
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44 views

How many elements are there in the set R?

+Let A be a finite set with $n \geq 4$ elements and let R be an equivalence relation on A . Suppose that there are exactly $n-2$ equivalence classes and that no equivalence class can contain exactly ...
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49 views

How to prove the Cone is contractible?

Let $\sim$ be an equivalence relation on a topological space $X$ with a subset $A ⊂ X$ such that the equivalence classes are: $A$ itself and Singletons $\{x\}$ such that $x ∉ A$. Then define $X/A$ ...
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Describe an equivalence class

On the set N x N, define the following relation: (a, b) ~ (c, d) if and only if a + d = b + c (a). Show that this is an equivalence relation I have shown that this is an equivalence relation by ...
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1answer
10 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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28 views

Equivalence relation example. How is this even reflexive?

Is the below question a mistake? How is this an equivalence relation? For example, how would it even be reflexive? E.g if you pick any A $\subseteq$ $U$, say A = {a, b}, then A ~ A is not true, ...
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36 views

Question about notation of sequences and equivalence classes.

In these notes (see pg3 second-to-last paragraph), what does $d(x_k,x^\ast_{N_k})$ mean? The term $x_k$ lies in $X$, but $x^\ast_{N_k}$ is a class of Cauchy sequences in $X$. Should I take ...
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2answers
18 views

Describes Equivalence Classes

Let $R$ be the relation on the natural numbers defined by $(a,b)\in R$ if and only if $2\mid(a+b)$ I already proved this was an equivalence relation, but how do I determine the number of equivalence ...
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1answer
25 views

English wording around equivalence relation

What is the English word to mean an element of an equivalence class of an equivalence relation? In French we say "représentant".
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17 views

Showing relation is transitive (a,b) $\in$ R iff 2|(a+b)

Let R be the relation on natural numbers defined by (a,b) $\in$ R iff 2|(a+b) show it is transitive.
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1answer
25 views

Proving $S$ is the unique smallest relation on $A$ containing $R$

Suppose $R$ is a reflexive and symmetric relation on a finite set $A$. Define a relation $S$ on $A$ by declaring $xSy$ if and only if for some $n \in \mathbb{N}$ there are elements $x_1,x_2,\ldots,x_n ...
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1answer
15 views

Construct equivalence classes for a relation R

Define relation R as follows: xRy if x and y are bit strings with |x| >= 2 and |y| >= 2 such that x and y agree in their first two bits. Show that R is an equivalence relation. Construct the ...
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Consider P a partition of set A. Given relation R on A and xRy if and only if x, y $\in$ X for some X $\in$ P. Show R is equivalence relation on A

Consider $P$, a partition of a set $A$. Define a relation $R$ on $A$ such that $x\mathrel{R}y$ if and only if $x, y \in X$ for some $X \in P$. Show that $R$ is an equivalence relation on $A$. Next ...
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Equivalence modulo [duplicate]

I'm having trouble with this question... Suppose m is a positive integer. Show that equivalence modulo m is an equivalence relation on Z. That is, show the following for integers a,b, and c: - a=a(mod ...
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1answer
42 views

Prove that if R is a symmetric relation, so is R^2.

Prove that if R is a symmetric relation, so is R^2. My attempt : The Relation R has (a,b) provided (b,a) is a member of R. So if I go on to find R^2 it will always have element (a,a) that makes R^2 ...
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How to find $R^2$ given $S$ and $R$.

If $S = \{1,2,3\}$ has a relation $R = \{(1,2), (1,3), (2,3)\}$, find the relation $R^2$? I am not able to find $R^2$, can anyone please help me with this?
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Let E1, E2 Equivalence relations on A, Prove or disprove :

Let E1, E2 Equivalence relations on A, Prove or disprove : 1) E1 ∩ E2 an equivalence relation on A 2) E1 ∪ E2 an equivalence relation on A
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31 views

Proofs with Relations and functions

I need help with setting up a homework problem. I am having trouble finding where to start. Problem: Suppose A is a set. Show that $i_A$ is the only relation on A that is both an equivalence relation ...
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40 views

how to find relation R^2

Suppose S is a set of airports, and R is the following relation on S: aRb if and only if there is a direct flight from a to b. Explain your answers to the following questions and use common sense. a. ...
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25 views

Define f:Z/3Z→Z/3Z by f([a])=[2a+1]

Just finished proving this to be injective, and well-defined. How would you prove it to be surjective? I understand surjective means that every element in the codomain is being used, and thus is the ...
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1answer
25 views

Define $f : Z/3Z → Z/3Z$ by $f ([a]) = [2a + 1]$

Just finished proving this is well-defined, how do I prove it's surjective and injective? I know that injective means that if $x1 \neq x2$, then $f(x_1) \neq f(x2)$, i.e. each value in the domain is ...
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1answer
13 views

Determine whether the relation is an equivalence relation:

xRy in Z iff x,y > 0 Apparently this is the answer: This is not an equivalence relation since 0 ∈ Z and 0 is not related to 0. So I know that x relates to y iff x and y are in the same cell of the ...
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1answer
22 views

Define f: Z/4Z → Z/4Z by f([a]) = [3a+1]

I need to show this function is well defined For well defined, I was thinking something along the lines of: Assume [a1] = [a2] in Z/4Z. Then, a1 is congruent to a2(mod4). So, 4 | a1 - a2. Thus, 4 | ...
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1answer
24 views

Z / 6Z being a set of well dedfined equivalence classes, and a congruent to b(mod 6)

why is this = [0],[1],[2],[3],[4],[5],[6] and how would I define f Z/6Z - Z/6Z by f([a]) = ([2a]). I have the proof but I don't understand it. Proof: Assume [a1] = [a2] in Z/6Z. then a1 congruent to ...
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Define f: Z /6Z by g(5[a]) = [5a]

So, in our notes, we had an example where we defined f: Z / 6 Z by g([a]) = [5a] (where z is set of all integers) Already, I don't follow what the g([a]) = [5a] means, I'm assuming they are ...
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5 views

Finding the relation for the partitions { 2x } and { 2x + 1 }

I have to find an equivalence relation in the set of natural numbers which has the two partitions { 2x } and { 2x + 1 } My first thought was ...
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2answers
93 views

If $a+1/a$ is an integer, then so is $a^t+1/a^t$ for $t\in\mathbb N$

I need to show if $a$ is in $\mathbb{R}$ but not equal to $0$, and $a+\dfrac{1}{a}$ is integer, $a^t+\dfrac{1}{a^t}$ is also an integer for all $t\in\mathbb N$. Can you provide me some hints please?
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Help with logical equivalence proof regarding a lemma for equivalence relations.

So the question is this: Suppose A is a set. Let ~ be an equivalence relation on A and let a,b be elements of A. Then Ta = Tb if and only if a ~ b. I need to prove this statement to be true. I know ...
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1answer
33 views

What means $A \subsetneq X$ with A ~ X? [closed]

How it is possible to have a subset A, which is $\neq$ to X and at the same time they have an equivalence relation ~? When $A \subset X$ therefore a $\in$ A is also a $\in$ X. With A ~ X ...
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Finding Equivalence Classes for Infinite Sets

Let R be the relation on the set of rational numbers Q defined as follows: for all q, r ∈ Q, qRr iff q − r ∈ Z, where Z is the set of integers. R is an equivalence relation on Q. What is the ...
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1answer
44 views

What is wrong with the following argument?

Say we have a set $X$ and an equivalence relation $C$ on $X$. Why do we need reflexivity? Let $x,y \in X$ with $xCy $. By symmetry we obtain $yCx$. Applying now transitivity, we have $xCx$. So, we ...
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31 views

Is this undergrad equivalence class question solvable?

Let x,y be real numbers. Define the relation S as x S y if |x - y| $\epsilon$ Q where Q is the set of rational numbers. Find all equivalence classes of S. I work in the undergrad tutor center ...
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1answer
8 views

Prove that a partial equivalence relation in set Dom(A) is an equivalence relation

We know that $r$ is a partial equivalence relation in set $$Dom(r) = \{x|\exists y.(x,y)\in r\}$$ The problem is to prove that this is an equivalence relation. Here is my proof. Did I do it right? ...
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Define another Equivalence Relation from Geometric Figure?

Define relation $W$ on $R^2$ by $(x_1,y_1)W(x_2,y_2)$ whenever $x_1−y_1=x_2−y_2$. I have the equivalence classes to be given by $f^{-1}(r)=\{(x,y)\mid x−y=r\}$, where $r∈R$. So how would you define ...
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1answer
22 views

How many relations on a set with 6 elements?

I know there is a lot of information on this internet for this, I've been going through it the past 30 minutes. I'm getting confused to if the answer is actually 203 relations, because when I try to ...
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1answer
28 views

Relations on set based off Cardinality [closed]

Let A be a set with cardinality 6. How many relations on A are there? How many are reflexive? symmetric? Not sure where to go with only this information. Thanks!
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Show that $W$ equivalence relation on $\mathbb{R}^2$

Define relation $W$ on $\mathbb{R}^2$ by $(x_1,y_1)W(x_2,y_2)$ whenever $x_1-y_1=x_2-y_2$. Show that $W$ is an equivalence relation on $\mathbb{R}^2$. I believe it is reflexive, not sure about ...
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27 views

Determining equivalence classes

I have done (a), pretty straight forward. I understand an equivalence class as all the elements in the domain that map to the same result in the co-domain. For example in (mod 3), [|0|] would be the ...
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48 views

In $\mathbb{R}^{n}$ all norms are equivalent

While trying to prove the Theorem mentioned in the Title, I got stuck in the inequality shown below. I think that the proof uses the $\epsilon$ and $\delta$ definition of continuity but I am not ...
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36 views

Poker Hand Equivalent Relation

Let $P$ be the set of all possible poker hands. Define a relation $J$ of $P$ by $a$ is $J$-related to $b$ iff $a$ and $b$ have no cards in common. Is $J$ reflexive? Symmetric? Transitive? Having a ...
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1answer
37 views

Equivalence relation proof example. Starting off help

Now I need to prove its reflexive, symetric, and transitive! Now my biggest confusion is what do I let "a" equal? Obviously it will be an arbitrary element in N(sub 0). Any help would be great. ...
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Symmetric Relations and Cycles

Let's say I have the set: R = {(a,b),(b,c),(c,d),(d,a)} If you visualize this in graph form, it forms a cycle. My question is, is this already a symmetric relation, or do I have to add ...
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Finding the smallest binary relation in a given set including a relation

The following binary relation of the set {a, b, c, d, e} is given: R = { (a,b), (a,c), (b,c) } What I have to do is to find the smallest reflexive / symmetric / transitive / antisymmetric relation ...
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1answer
23 views

Determining the transversal for an equivalance relation

If you have an equivalence relation $c$ on $\Bbb{Z}$ defined by $$\{x,y\in\Bbb{Z}:p\in\Bbb{Z},x=5p+y\}$$ How would you proceed to determine if the following subset of $\Bbb{Z}$ $$\{-8,1,10,13,19\}$$ ...
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1answer
34 views

How many equivalence classes does this relation have?

I have this relation: $$A = \mathbb {R} \\ \quad\;\; x\sim y \iff x-y \in \mathbb {Z} $$ I have already proved if it is an equivalence relation. Now I am just searching for the equivalence classes ...
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1answer
39 views

Find the equivalence class of this relation!

I am having the following relation with the set A and B: $$ (x_1, y_1) \sim_{A\times B} (x_2, y_2) \iff\; x_1 \sim_A x_2\ \;\land\; \; y_1 \sim_B \; y_2 $$ I haved already proved, that it is a ...