1
vote
1answer
21 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
24 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
20 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
102 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
17 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
73 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
132 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
42 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
32 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 ...
2
votes
1answer
65 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
63 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
46 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
49 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
42 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
32 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
41 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
35 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 ...
0
votes
0answers
42 views

Equivalence relations, cosets.

Let R be an equivalence relation on S. Show that for all s1, s2 elements of S we have s1(equivalence class) = s2(equivalence class iff s1Rs2. Not really sure how to go about this problem.
1
vote
2answers
55 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
42 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
60 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
101 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 ...
2
votes
1answer
99 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
52 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
501 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
81 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
68 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
48 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
103 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
195 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
206 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
76 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{ ...