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

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8
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4answers
4k views

Understanding equivalence class, equivalence relation, partition

Im having difficulty grasping a couple of set theory concepts, specifically concepts dealing with relations. Here are the ones I'm having trouble with and their definitions. 1) The collection of ...
1
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5answers
2k views

prove that $a \equiv b \mod m$ is an equivalence relation on the integers

prove that $a \equiv b \mod m$ is an equivalence relation on the integers I believe there are 3 properties that it must meet to prove and equivalence relationship. Any reference material would be ...
2
votes
1answer
167 views

Equivalence relation question with cardinality and countability $A=\mathbb R,\ aSb \iff a-b\in \mathbb Q $

Let $A=\mathbb R,\ aSb \iff a-b\in \mathbb Q $ What is the cardinality of $[\pi]_S$ ? Prove that the quotient group $\mathbb R/S$ is uncountable. Well I think that cardinality is ...
2
votes
3answers
568 views

Equivalence Relation problem [duplicate]

Let $S$ be the relation on $\mathbb{N} \times \mathbb{N}$ defined by $(a,b)S(c,d)$ if and only if $ad=bc$. Prove this is an equivalence relation on $\mathbb{N} \times \mathbb{N}$. I think I've found ...
1
vote
1answer
110 views

Let A be a non-empty set, and p an equivalence relation on A . Let a , b be an element of A . Prove that [ a ] = [ b ] is equivalent to apb

the question: a) Let A be a non-empty set, and p an equivalence relation on A . Let a , b be an element of A . Prove that [ a ] = [ b ] is equivalent to $apb$ b) If p is both an equivalence relation ...
2
votes
2answers
609 views

Distinguishing equality and isomorphism as relations

Is this relational characterization of equality in Wikipedia accepted? The identity relation is the archetype of the more general concept of an equivalence relation on a set: those binary ...
0
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2answers
132 views

How to calculate equivalence relations

How can I calculate how many equivalence relations can be defined on a given set? For example: How many possible equivalence relations can be defined on S = {a,b,c,d}?
0
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2answers
363 views

How do you show that $R$ is an equivalence relation and enumerate one bit string from each of the different equivalence classes of $R$?

If $A$ is the set of all bit strings of length $12$. Let $R$ be the relation define on $A$ where two bit strings are related if the first $2$ bits, the $4^{\text{th}}$ bit and the $7^{\text{th}}$ bit ...
9
votes
1answer
387 views

Is the quotient map a homotopy equivalence?

It is well known that, if $A \subset X$ is a reasonable contractible subspace, then the quotient map $X \to X/A$ is a homotopy equivalence ("reasonable" means that the pair $(X,A)$ has the homotopy ...
4
votes
2answers
127 views

Equivalence relations on classes instead of sets

Can someone please explain to me how to deal with equivalence relations on classes instead of sets? Is there some sort of generalisation of relations? Thank you
5
votes
1answer
288 views

Is this alternative definition of 'equivalence relation' well-known? useful? used?

I discovered that $$R \textrm{ is an equivalence relation on } A \;\equiv\; \langle \forall a,b \in A :: aRb \:\equiv\: \langle \forall x \in A :: aRx \equiv bRx\rangle \rangle$$ The nice thing about ...
4
votes
1answer
152 views

The class of equivalence.

Let $Q$ be the following set of $\mathbb{Z}\times \mathbb{Z}$ \begin{align*} Q=\{(a,b)\in \mathbb{Z}\times \mathbb{Z} | b\neq 0\} \end{align*} Define the relation $\sim$ on $Q$ as ...
4
votes
2answers
140 views

Prove that $\sim$ defines an equivalence relation on $\mathbb{Z}$.

For $a,b \in \mathbb{R}$ define $a \sim b$ if $a - b \in \mathbb{Z}$ I don't understand how I'm suppose to prove this: Prove that $\sim$ defines an equivalence relation on $\mathbb{Z}$ Also can you ...
4
votes
3answers
5k views

Prove that the intersection of two equivalence relations is an equivalence relation.

I am reading this chapter of the Book of Proof, and I'm stuck at the Exercise 10 of section 11.2. It is as follows. Suppose $R$ and $S$ are two equivalence relations on a set $A$. Prove that $R ...
4
votes
5answers
3k views

Equivalence relation $(a,b) R (c,d) \Leftrightarrow a + d = b + c$

Suppose $A$ is the set composed of all ordered pairs of positive integers. Let $R$ be the relation defined on $A$ where $(a,b) R (c,d)$ means that $a + d = b + c$. (a) Prove that $R$ is an ...
3
votes
1answer
107 views

Surjections and equivalence relations

(a) Let $f: A \to B$ be a surjective function. We define $a_1 \sim a_2$ if $f(a_1)=f(a_2)$. Prove that $\sim$ is an equivalence relation. Reflexivity: This comes for free. If $a_1 \sim a_1$, ...
2
votes
1answer
77 views

If $X,Y$ are equivalence relations, so is $X \times Y$

If $X,Y$ are reflexive, symmetric, and transitive, then $X \times Y$ is an equivalence relation where ${(a,b):a\in X, b\in Y}$. I am trying to self learn these topics. I do know what an ...
2
votes
3answers
106 views

Equivalence class for the relation of having the same set of prime divisors

For an integer $n\in \mathbb{N}$ define $P(n) = \{p : p \mid n \text{, where $p$ is a prime} \}$. For example $P(12)=\{2,3\} $ and $P(1)=\emptyset$. Question: Consider the relation $R$ on ...
2
votes
4answers
307 views

Showing that $R$ is an equivalence relation on $X \times X$

Let $X = \{1,2,3,..,10\}$ define a relation $R$ on $X \times X$ by $(a,b)R(c,d)$ if $ad=bc$. Show that R is an equivalence relation on $X \times X$. I know that the $R$ have to be reflexive (because ...
1
vote
2answers
70 views

Transitive, symmetric $R$ such that for all $x$, there is $y$ such that $xRy$, is an equivalence relation

I'm stuck at one particular task I'm working on. Here is the task: Let R be a transitive and symmetrical relation on $S$. Assume that for all $x \in S$ there is a $y \in S$ so that $xRy$. ...
1
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2answers
232 views

Is this alternative definition of 'equivalence relation' correct?

I was puzzling over another question: Let $R$ be an equivalence relation on a set $A$, $a,b \in A$. Prove $[a] = [b]$ iff $aRb$. And this made me discover that $$(0) \; \langle \forall a,b :: aRb ...
1
vote
2answers
919 views

Congruence Class $[n]_5$ (Equivalence class of n wrt congruence mod 5) when n = $-3$, 2, 3, 6

The question is, "What is the congruence class$[n]_5$ (that is, the equivalence class of $n$ with respect to congruence modulo 5) when n is a) 2? b)3? c) 6? d)−3?" I know this is more work ...
0
votes
2answers
68 views

Proofs on equivalence relations rational numbers

The relation R = {(x, y)|x − y is an integer} is an equivalence relation on the set of rational numbers. I'm kind of confused with this question and what it is asking me to do. In order to solve ...
0
votes
1answer
46 views

Equivalence relation for set given as matrix

I need a hint. My task is to proof that ((a11, a12), (b11, b12)) ((a21, a22), (b21, b22)) ∈ R ⇔ a11 + a12 + a21 + a22 = b11 + b12 + b21 + b22 R is equivalnce relation. My problem is that I ...
0
votes
1answer
59 views

Double check $G\sim H$ iff $G≈H$

Let $S$ be the collection of all groups. Define a relation on $S$ by $G \sim H$ iff $G ≈ H$. Prove that this is an equivalence relation. So $S$ is partitioned into isomorphism classes. Proof: Let ...
0
votes
2answers
67 views

For all $x,y∈\Bbb{R}$ define that $ x\equiv y$ if$ x^2=y^2$

For all $x,y\in\Bbb{R}$ define that $x\equiv y$ if $x^2=y^2$ . Then $\equiv$ is an equivalence relation on $\Bbb{R}$ , there are infinitely many equivalence classes, one of them consists of one ...
0
votes
0answers
63 views

For all $x,y\in\mathbb{R}$ define that $x\equiv y$ if $x^{2}=y^{2} $; prove $\equiv$ is an equivalence relation. [duplicate]

For all $x,y\in\mathbb{R}$ define that $x\equiv y$ if $x^{2}=y^{2}$ . Then $\equiv$ is an equivalence relation on $\mathbb{R}$ , there are infinitely many equivalence classes, one of them ...
0
votes
1answer
61 views

Equivalence relations and power sets.

Let $\mathcal{A}$ be the class of all sets and define the relation $R$ on $\mathcal{A}$ as: $A\space R\space B$ iff there is a bijective function $f:A \to B$. Prove that $R$ is an equivalence relation ...
0
votes
1answer
106 views

Equivalence relations on S given no relation?

On my assignment, one of the questions ask to list all equivalence relations on S and count how many are of partial orders. Let S = {u, v, w}. List all equivalence relations on S. How many of these ...
0
votes
2answers
113 views

Revisted$_2$: Are doubling and squaring well-defined on $\mathbb{R}/\mathbb{Z}$?

Define a relation $\sim$ on $\mathbb{R}$ as follows: for any $a,b \in \mathbb{R}$, $$a\sim b \iff a-b\in \mathbb{Z}.$$ Let $S=\mathbb{R}/{\sim}$. That is, $S$ is the set of equivalence classes of ...
0
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1answer
204 views

Rational Numbers and Equivalence Classes

Describe the rational numbers as the equivalence classes for an equivalence relation on certain pairs of integers.
0
votes
1answer
127 views

Proving $\;x-y \in2\pi \mathbb Z\;$ defines an Equivalence Relation

Prove that $\;x-y \in2\pi \mathbb Z\;$ defines an equivalence relation on $\;\mathbb R.$
0
votes
1answer
205 views

Equivalence Relation? Column-equivalence on the set of all $m\times n$ matrices.

How do I show that column equivalence on the set of all $m\times n$ matrices is an equivalence relation? I know that to show it is an equivalence relation, I need to show that column equivalence is ...
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votes
2answers
56 views

Is this relation reflexive?

Suppose $R$ is a relation on $N_4=\{1,2,3,4\}$ such that $R\circ R = R$. How would I prove that $R$ is reflexive? Could you give me some tips about how to start off?
-1
votes
2answers
141 views

Cardinality of “$x-y\in\Bbb Q$”-equivalence class of $1/\sqrt2$

For $x,y\in I:=[0,1]$ define the relation on $I$ as $x-y\in \Bbb Q$. How big (using cardinal number) is the cardinality of the equivalence class $[1/\sqrt2]$? I have tried to solve it by ...
-2
votes
1answer
77 views

Cardinality of “x−y∈Q”-equivalence class of 1/2 √ [duplicate]

For x,y∈I:=[0,1] define the relation on I as x−y∈Q. How big (using cardinal number) is the cardinality of the equivalence class [1/√2]? I have tried to solve it by finding the equivalence class but ...
7
votes
2answers
166 views

Is there a name for relations with this property?

Is there a name for relations $\rho : X \rightarrow Y$ such that for all $x,x' \in X$ and all $y,y' \in Y$ we have that the following conditions $$xy \in \rho$$ $$x'y \in \rho$$ $$xy' \in \rho$$ imply ...
6
votes
1answer
154 views

Solve for ? - undetermined inequality symbol

So I was solving a problem in Rudin (chapter 3 #16, to be specific) and I realized how convenient it would be to have a symbol that represented an undetermined equivalence relationship. As an example ...
5
votes
2answers
6k views

How many equivalence relations on a set with 4 elements.

Let S be a set containing 4 elements (I choose {$a,b,c,d$}). How many possible equivalence relations are there? So I started by making a list of the possible relations: ...
4
votes
4answers
2k views

Proving an equivalence relation on $\mathbb{Z}\times\mathbb{Z}$

I'm working on some discrete mathematics problems, and have run into an issue involving proving an equivalence relation. The relation I'm tasked with proving is the relation $R$ defined on ...
2
votes
0answers
206 views

Attempting to find a specific similarity (equivalence) matrix

Apologies if this has already been asked - I searched but couldn't find. My question is regarding matrix similarity (equivalence): I have a square matrix $A$ that is of permutation form and (square) ...
1
vote
1answer
171 views

I don't understand equivalence relations

I'm trying to show that the left coset with a subgroup is an equivalence relation. So taking some element $g \in G$ and a subgroup $H$, the left coset is defined as $gH = \{gh : h \in H\}$. That means ...
0
votes
1answer
146 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 $X/A$ ...
0
votes
1answer
108 views

Equivalence relation classifying the slopes of the tangent lines to the curve?

We apply the Mean value theorem to a real analytic function $f$ (defined on $\mathbb R$) in the interval $(u,a)$ such that $f(u)=0$ and $f(a)\neq 0$ to find a $c\in(u,a)$ such that: the expression ...
-1
votes
3answers
209 views

When can we modify $a \propto \ b$ adding $\delta $?

This is a general, theoretical question about formalization of concepts, it is difficult for me to explain it adequately, please, if I fail, tell me in a comment what is not clear or feel free to edit ...
5
votes
2answers
627 views

Sets, functions and relations problem

Yes this is a homework problem but I have attempted to solve it and my work is below, also this is my first question here so I'm sorry for any mistakes: Question: Context: Let $A$ and $B$ be subsets ...
4
votes
5answers
153 views

How do I work with a relation that is a set of 4-tuples?

Define the relation $\sim$ on $\mathbb{Z}\times\mathbb{Z}$ by $(a,b)\sim(c,d)$ if $a-c=b-d$. Show that $\sim$ is an equivalence relation. What is the equivalence class of $(1,2)$? I'm not sure ...
3
votes
1answer
185 views

$R$ is transitive if and only if $ R \circ R \subseteq R$

Question: Let $R$ be a relation on a set $S$. Prove the following. $R$ is transitive if and only if $ R \circ R \subseteq R$. Definition 6.3.9 states that we let $R_1$ and $R_2$ be relations on a ...
3
votes
3answers
388 views

Equivalence relation question with functions

We'll define on the set: $A=\Bbb R^{[0,1]}$ the relation $R$ by $fRg$ if $f(0)=g(0)$. Make sure it's an equivilence relation. What is $[\cos x]$ ? Describe all the equivalence classes ...
3
votes
2answers
221 views

Equivalence classes of “$x \sim y \Longleftrightarrow x -y $ is rational”.

Given the equivalence relation $x \sim y \Longleftrightarrow x -y $ is rational on the interval $[0,1)$. How do we reason* that there are uncountably infinite number of equivalence classes? *A ...