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

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12
votes
1answer
110 views

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 ...
11
votes
5answers
868 views

Dependence of Axioms of Equivalence Relation?

This question is problem 11(a) in chapter 1 in 'Topics in Algebra' by I.N. Herstein. These are the properties of equivalence relation given in this book. Prop 1 $a \sim a$ Prop 2 $a \sim b$ ...
9
votes
1answer
414 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 ...
9
votes
1answer
113 views

Does the Trace product in a semigroup have any relation with Trace of a matrix / matrix product

I recently read an article on generalized inverses and Green's relations (by X.Mary). The framework is semigroups, but obviously it has a lot of application within matrix theory. In the article ...
8
votes
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 ...
8
votes
3answers
196 views

Can we take images of equivalence relations?

Given a function $f : X \rightarrow Y$, it is well-known that we can take the image under $f$ of any subset $A \subseteq X$, and we can take the preimage under $f$ of any subset $A \subseteq Y$. This ...
8
votes
1answer
117 views

When is the topological closure of an equivalence relation automatically an equivalence relation?

Let $X$ be a topological space, not necessarily nice, let $R$ be a subspace of $X \times X$, and let $\overline{R}$ be the closure of $R$ in $X \times X$. Then: If $R$ is a reflexive binary relation ...
7
votes
4answers
841 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 ...
7
votes
2answers
167 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 ...
7
votes
4answers
263 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 ...
6
votes
1answer
163 views

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 ...
6
votes
1answer
494 views

Is “have the same cardinality” a equivalence relation?

A relation is a subset of the Cartesian product of two sets, if "have the same cardinality", denoted as $R$, is a relation, then there must exist set $A, B$, such that $R \subset A \times B$. What are ...
6
votes
1answer
203 views

How to simply show that there are “78 'strict ordinal' 2x2 game matrices”

In "Theory of Moves", Steven J. Brams analyses two-player games with two strategies per player, where each player can totally rank his payoffs, although payoffs need not be comparable among players. ...
6
votes
1answer
161 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: ...
5
votes
3answers
2k views

Why is the empty set finite?

On page 25 of Principles of Mathematical Analysis (ed. 3) by Rudin, there is the definition (excluding the irrelevant parts for this question): Definition 2.4: For any positive integer $n$, let ...
5
votes
4answers
413 views

What exactly are equivalence classes

What exactly are equivalence classes? Suppose I have an equivalence relation $\sim$ on some set $X$ we denote this as $x \sim y$. The equivalence classes are then $[x] = \{y \in X : y \sim x\}$. ...
5
votes
1answer
122 views

Prove $x^2 + 2y^2 \neq 805, x,y\in\mathbb{Z}$, using equivalence classes modulo 5

Prove $x^2 + 2y^2 \neq 805, x,y\in\mathbb{Z}$. We did this in class and, for the life of me, I cannot remember how to finish the problem. It starts out by taking all of the values to be $\mod5$. ...
5
votes
2answers
127 views

Categorical description of equivalence relation generated by a relation?

The notion of an equivalence relation generated by a relation is widespread and useful. How can one categorically describe it?
5
votes
2answers
155 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 ...
5
votes
1answer
95 views

Are these two equivalence relations the same ? (Equivalence relation defined using subgroups of a group)

I was solving the book Abstract Algebra by Charles Pinter. In one of the questions I was stuck. The question is as follows :- $H$ is a subgroup of group $G$.Two equivalence relations on $G$ are ...
5
votes
1answer
290 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 ...
5
votes
3answers
486 views

Equivalence class of polynomials

$X$ is the set of all polynomials over $\mathbb{R}$. We define an equivalence relation on $X$ such that $p$~$q$ iff $p(0)=q(0)$. ($1$) What is the equivalence class of $p(x)=x$? ($2$) Give a ...
5
votes
1answer
94 views

Question about right and left cosets.

I want to do a question about how my algebraic structures professor defined left and right cosets. I'll write here his way to present them. We first talked about quotient group. Let $G$ be a group, ...
5
votes
2answers
665 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
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 ...
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
2answers
384 views

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 ...
4
votes
3answers
186 views

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!
4
votes
2answers
967 views

What is the standard notation for a set of equivalence classes?

What is the standard notation for a set of equivalence relations? Specifically, I have a pair of objects, call them $x$ and $y$ and I denote the ordered pair as $\left(x,y\right)$. I have a set of ...
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
4
votes
5answers
162 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 ...
4
votes
4answers
712 views

$5 \mid n^2 - m^2$ is an equivalence relation

How can I show this is an equivalence relation: $$ n \operatorname{R} m \Longleftrightarrow n^2 - m^2 \textrm{ is divisible by } 5 $$ I know equivalence relations are symmetric, reflexive and ...
4
votes
2answers
135 views

“Tricky” wording on Congruence Modulo Question?

I'm asked for all possible values, but I can only see one. The question on my practice exam reads: Consider the equivalence class [3] for the equivalence relation "congruence modulo $7$" on $\Bbb Z$. ...
4
votes
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, ...
4
votes
3answers
103 views

Is $\mathbb{Z}_p$ a Finite Field?

Denote the integers modulo $p$, $\mathbb{Z}$ mod $P$, as $\mathbb{Z}_P$. Denote the set of integers equivalent to $n$ mod $P$ - the equivalence class of $n$ as $\overline{n}$. We know that for any ...
4
votes
2answers
99 views

In topology $X$ is also $Y$ means homeomorphic?

E.g. $\mathbb{R}P^n$ is also the quotient space $S^n / (v \sim -v)$. And when is it safe to refer to a space as one of it's homeomorphic spaces and perform further deductions from that homeomorphic ...
4
votes
2answers
116 views

Uncountability of the equivalence classes of $\mathbb{R}/\mathbb{Q}$

Let $a,b\in[0,1]$ and define the equivalence relation $\sim$ by $a\sim b\iff a-b\in\mathbb{Q}$. This relation partitions $[0,1]$ into equivalence classes where every class consists of a set of numbers ...
4
votes
1answer
1k views

Myhill Nerode - is language regular or not?

I'm trying to understand how to find equivalence classes of a language to prove its regularity. I think that if I'm able to FULLY understand one example then I will get this topic right. Let's say I ...
4
votes
2answers
83 views

Using Logical Equivalences to prove $(((\neg r) \lor q) \lor ((q \lor (\neg p)) \land ((\neg p) \lor q)))$ is equivalent to $(\neg(r \land p) \lor q)$

I have been trying to solve the following proof: $$(((\neg r) \lor q) \lor ((q \lor (\neg p)) \land ((\neg p) \lor q)))\text{ is equivalent to } (\neg(r \land p) \lor q)$$ I am new to proofs and ...
4
votes
1answer
281 views

Proving that there are n Equivalence Classes Modulo n

For $a,b,n \in \mathbb{Z}$ and $n \geq 2$, I want to prove that there are $n$ equivalence classes mod $n$. I'm not sure how to do it - would I do it inductively? Any help would be appreciated.
4
votes
2answers
131 views

Some equivalence relation from flipping binary trees

I know almost nothing in combinatorics, so this question might be very easy, or well-known. Fix a number $n$. We will consider rooted planar binary trees with $n$ leaves. We will distinguish between ...
4
votes
2answers
220 views

Counting the number of functions

Let $A$ and $B$ be subsets of the set $\Bbb Z$ for all integers, and let $\mathscr F$ denote the set of all functions $f:A\rightarrow B.$ Assume $A = \{1,2,3\}$ and $B=\{1,2,...,n\}$ where $n\ge 2$ is ...
4
votes
1answer
37 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 ...
4
votes
2answers
142 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
2answers
700 views

Is $x = y \mod 7$ for a set of integers an equivalence relation?

Equivalence relation is the relation which is reflexive, symmetric and transitive. I have read somewhere that modulo operator defines an equivalence relation. But for this relationship I cant find ...
4
votes
1answer
144 views

Mean values theorem and countable sets

The mean values theorem says that there exists a $c∈(u,v)$ such that $$f(v)-f(u)=f′(c)(v-u)$$ My question is: Assume that $u$ is a root of $f$, hence we obtain $$f(v)=f′(c)(v-u)$$ Assume that $f$ is a ...
4
votes
1answer
155 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
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 ...
4
votes
0answers
46 views

Equiv Relation of Orbits - Group Action [duplicate]

Let $G$ be a group that acts on $X$. I want to show that the orbits of $G$ partition $X$. I am given the relation $x\sim y \iff x\in Orb(y)$. Now: $x\sim y\iff x\in Orb(y) \iff x=gy$ for some $g\in G ...