0
votes
2answers
122 views

Why do we use the term “equivalent” with Operators but “equal” with Functions?

Why do we speak in terms of "equality" when we deal with functions but "equivalence" when dealing with operators? To elaborate: Two functions, f and g are equal to each other (denoted: f=g) if: ...
0
votes
1answer
36 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 ...
3
votes
4answers
45 views

Equivalence Relation definitions of Coset - looks like 1-step Subgroup Test? [Fraleigh p. 97 theorem 10.1]

p. 4 We are especially interested in the case where the set is a group, and the equivalence relation has something to do with a given subgroup. That is, we want to partition a group G into subsets, ...
5
votes
4answers
361 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 $[x] = \{y \in X : y \sim x\}$. ...
1
vote
1answer
66 views

Equivalence relation and equivalence class question

Show that the relation $\sim$ defined on the set $X = \mathbb{N} \times \mathbb{N} = \{(a, b) : a \in \mathbb{N}; b \in \mathbb{N}\}$ as $(a,b) \sim (c,d)$ if and only if $a + d = c + b$ is an ...
1
vote
1answer
42 views

How to prove that equality is an equivalence relation?

Probably, it's a elementary question, but I would like some explanation. Everyone knows that equality relation is (i) reflexive, (ii) symmetric and (iii) transitive, that is, satisfies (i) $x=x$; ...
0
votes
1answer
36 views

Show that the equivalence classes of $\sim$ are left cosets of $H$ in $G$.

Let $H \leq G$ and define a relation on $G$ by $x \sim y$ if $y^{-1}x \in H$. Show that $\sim$ is an equivalence relation on $G$ and then show that the equivalence classes of $\sim$ are left cosets ...
0
votes
0answers
21 views

Question about Equivalence relation and partition

This is my first time of Abstract algebra, and I don't know how to solve this problem. Although I have an idea to solve this problem, I can't assure whather it is correct or not. Please show me how ...
2
votes
1answer
105 views

The closures of a binary relation

I have the following dilemma concerning the equivalence closure of a binary relation. Let $X$ be a nonempty set and $R\subseteq X\times X$ (i.e., a binary relation over $X$). Consider the following ...
0
votes
0answers
18 views

Bijection between partial order and $<$

How to show the correspondence between a less than relation and partial orders ? here A less than relation $<$ on a set $S$ is a relation that satisfies If $a < b$ , then $a \neq b$. ...
2
votes
4answers
89 views

Equivalence relation and subgroup

I am taking abstract algebra now, and there's a lemma: Let $H$ be a subgroup of group $G$, for $a,b \in G$,define $a\sim b$ if $ab^{-1}\in H$, then it is an equivalence. I know how to prove it and how ...
0
votes
1answer
592 views

Proving equivalence relations

I just started my abstract algebra class and I am struggling with the concept of equivalence relations. I know that in order to prove equivalence relations, I have to prove the reflexive, symmetric, ...
-2
votes
1answer
46 views

For a group $G$, show the relation $x\sim y$ defined by $\exists a(y=axa^{-1})$ is an equivalence relation on $G$.

Let G be a group. For $x,y\in G$, define $x\sim y$ if there exists some element $a\in G$ such that $y=axa^{-1}$. Show that ~ defines an equivalence relation on $G$.
0
votes
0answers
34 views

A question on Partitioning regarding equivalence relations

Let $S$ be the Cartesian coordinate place $\mathbb R \times\mathbb R$ and define the equivalence relation $R$ on $S$ by $(a,b) R (c,d)$ iff $b-3a = d-3c$ Find the partition $D$ determined by $R$ by ...
1
vote
1answer
72 views

Understanding Pushouts in Top.

The Pushout of $X \leftarrow Z\rightarrow Y$ with $f:Z\rightarrow X$ and $g:Z\rightarrow Y$ in $\mathbf{Top}$ exists and is given by $X\coprod Y/\sim,$ where "$\sim$ is the equivalence relation ...
0
votes
1answer
66 views

Prove this is an equivalence relation

$A$ is related to $B$ if $M_n(A)\simeq M_m(B)$ for some integers $m$ and $n$. Clearly reflexivity and symmetry are trivial. It's transitivity that I am struggling with. Is it the case that if ...
0
votes
1answer
58 views

Equivalence relation of a group acting on a set

Let A be a set and G be any subgroup of S(A). G is a group of permutations of A. Assume that G is a finite group. If u∈A, the orbit of u is the set O(u)={g(u): g∈G}. Define a relation ~ on A by u~v ...
1
vote
3answers
59 views

Partition of an equivalence relation

I am having a hard time with the following problem: In F(R), let f~g iff f(x)=g(x) for all x>c where c is some fixed real number. I proved that it was a equivalence relation by the following: ...
0
votes
1answer
45 views

Properties of equivalence relations

Let $\sim_1$ and $\sim_2$ be distinct equivalence relations on $A$. Define $\sim_3$ by $a\sim_1 b$ and $a\sim_2 b$. Let $[x]_i$ denote the equivalence class of $x$ for $\sim_i$ ($i=1,2,3$). Prove ...
1
vote
0answers
33 views

Congruence induced by a subset.

Consider the additive monoid of natural numbers and calculate the congruence generated by $\{(2,3)\}$. I know the answer is that the congruence has 3 classes which are $\{0\}$, $\{1\}$, and the rest. ...
0
votes
1answer
42 views

Show that $s_1(\text{equivalence class}) = s_2(\text{equivalence class})$ iff $s_1\mathrel{R}s_2$.

Let $R$ be an equivalence relation on $S$. Show that for all $s_1, s_2$ elements of $S$ we have $s_1(\text{equivalence class}) = s_2(\text{equivalence class})$ iff $s_1\mathrel{R}s_2$. I understand ...
1
vote
1answer
38 views

Equivalence relations, Cosets

Let G be a group and for elements a,b (elements of)G let a R b mean that there exists an element x(element of)G such that a=xbx^(-1). Show that R is an equivalence relation on G. Not really sure how ...
1
vote
2answers
59 views

Understanding concept of an operation being well defined for an equivalence relation

Let $I$ be an ideal in a ring $R$. Define the relation (congruence modulo $I$) by $a \equiv b$ if $b - a \in I$ Denotes the equivalence class containing $a$ by $\bar{a}$. Define $$\bar{a} + ...
2
votes
1answer
44 views

Let $S=\{a,b\}$. Of all the relations on $S$ which are symmetric? Reflexive? Transitive?

The relations are as follows: 1.) $\{(a,a)\}$ 2.) $\{(a,b)\}$ 3.) $\{(b,a)\}$ 4.) $\{(b,b)\}$ 5.) $\{(a,a),(a,b)\}$ 6.) $\{(a,a),(b,a)\}$ 7.) $\{(a,a),(b,b)\}$ 8.) $\{(a,b),(b,a)\}$ 9.) ...
-1
votes
1answer
62 views

Show that ≡ is an equivalence relation, Show that ⊕ is well-defined, and Show that ⊕ is a commutative and associative operation.

Let $(a,b),(x,y) \in\Bbb R\times\Bbb R$ and define $(a,b) \equiv (x,y)$ iff $a+b = x+y$. a. Show that $\equiv$ is an equivalence relation. Define the operation $\oplus$ on the equivalence classes as ...
0
votes
1answer
117 views

Is a congruence (equivalence) class modulo n a group?

I know that the set of all equivalence classes Z/nZ is a group (with identity element the equivalence class [0], inverse element -[a]=[n-a]=[-a], etc.). However, is a single equivalence class modulo ...
9
votes
1answer
100 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 ...
2
votes
1answer
118 views

Clarification needed on natural projection

My book defines natural projection as such: Let $S$ be a set and let ~ be a equivalence relation on $S$. The function $\pi(x)=[x]$ for all $x\in S$ is called the natural projection from $S$ onto the ...
3
votes
1answer
57 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 $a_1 \sim a_1$, ...
1
vote
2answers
546 views

Variations : Anti-Symmetric Relations on an $n$-Element Set : Graph Theoretic Elucidation

Question: How many antisymmetric relations are there on an $n$-element set? Guess: I suspect that there are $2^n$ such relations. Discussion: I'm told that anti-symmetric relations on a ...
-1
votes
1answer
82 views

$\mathbb{R}/{\sim}$: A Question about the Formal Definition of a Quotient

For an equivalence relation $\sim$ what is $\mathbb{R}/{\sim}$? I mean explicitly and formally...
0
votes
3answers
70 views

An Equivalence Relation: Introspection into a Particular Well-Defined Quotient

DATA: Let $f:\mathbb{Z}\setminus \{0\}\rightarrow \mathbb{N}$ be a function defined by $$f(n) = \{k~:~n=2^km,~m\in \cal{O}\},$$ where $\cal{O}$ is the set of odd integers. Let ...
0
votes
1answer
53 views

For a group $G$ and subgroup $H$, is $a \sim b \iff a^{-1}b\in H$ an equivalence relation even when $H$ is not normal?

Is it true or false that defining a relation on the group $G$ based on the definition $a \sim b$ if and only if $a^{-1}b\in H$ defines an equivalence relation regardless of whether $H$ is a normal ...
2
votes
1answer
113 views

Equivalence relation on a group

I have the following question: Let $G$ be a finite group. We define a relation $\sim$ on $G \backslash \left\{e\right\} = \left\{ g \in G : g \neq e \right\}$ by $g \sim h$ if and only if there ...
1
vote
4answers
207 views

Prove that if G is an abelian group then the conjugation equivalence relation is the identity relation ($x$ $\thicksim$ $y$ if and only if x=y).

Can anyone please hint me out on how to prove that if the group G is abelian,then the conjugation equivalence relation is the identity relation ($x$ $\thicksim$ $y$ iff x=y).
1
vote
1answer
247 views

What is the difference between a binary relation and an equivalence class?

Is an equivalence class essentially a binary relation whose elements have an equivalence relation?
0
votes
1answer
80 views

Does every equivalence relation on set $S$ containing binary relation $C$ contain equivalence relation $E$?

Problem The following is a problem from Jacobson's Basic Algebra I: Let $C$ be a binary relation on $S$. For $r=1,2,3,\dots$ define $C^r=\{(s,t)|\text{ for some } s_1,\dots, s_{r-1}\in S,\text{ ...