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

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Name of and references for the equivalence relation $x \sim y :\Longleftrightarrow x^2 = y^2$

Playing around with the concepts of negativity and positivity, I came across the following equivalence relation defined for all elements $x,y$ of a field $\mathbb{F}$: $$ x \sim y :\Longleftrightarrow ...
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1answer
39 views

Transitivity of a binary relation on the power set

I'm studying for a test and there's a question that I've tried and I don't understand: Let $E$ be a binary relation on a set $A$; let a binary relation $F$ on $\mathcal P (A) \setminus \{\emptyset\...
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37 views

Prove that multiplication is well defined

Let $M = \mathbb{N} \ \mathbb{x} \ \mathbb{N}$. We define the following relation on $M$. Let $(a,b)R(a',b')$ iff $a + b'=a'+b$ We define the set of intergers $\mathbb{Z}$, to be the set of ...
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1answer
31 views

equivalence relation and quotient set, Given $A = \{0,1,2,3,4,5\}$

Given $A = \{0,1,2,3,4,5\}$, Write the appropriate equivalence relation of this quotient set: $$A/_R = \{\{1,2\},\{3\},\{4,0,5\}\}$$ Well, if it was to compute $$A/_R = \{\{0\},\{1\},\{2\},\{3\},\{4\}...
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Prove equivalence ratio and find class of $B\subseteq A$ , $X \sim Y \Leftrightarrow X \cap B = Y \cap B$

Prove equivalence ratio and find class of $B\subseteq A$ , $X \sim Y \Leftrightarrow X \cap B = Y \cap B$. Well, I've proved really easily that it is reflexsive, symmetrical and transitive. But I'm ...
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48 views

Give a complete set of equivalence class representatives for an equivalence relation on the natural numbers (including zero)

The full question: Having an equivalence relation $\sim$ on $\Bbb N$ defined by: $a \sim b$ meaning $a,b\in\Bbb N$ such that $a=b*10^k,$ for some $k\in\Bbb Z$, give a complete set of equivalence ...
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1answer
30 views

Relation $ (x,y) \in \rho \Leftrightarrow (\exists k \in \mathbb{Z})\mid x- y=3k$

I know that there is a similar question here, but it's about classes of equivalence of this relation. I would like to know how to prove that this is an equivalence relation. It seems simple, but the ...
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1answer
29 views

congruent modulo S

This is most likely a silly English question, but in Roman's "Advanced Linear Algebra," on page 21, he writes that: Let $S$ be a subset of a commutative ring $R$ with identity. Let $\equiv$ be the ...
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1answer
54 views

Show that the equivalence classes give a partition of $X$. [duplicate]

"... the following subset of $X$ denoted by $[x]$ $(x\in X)$ is called the equivalence class generated by $x$: $$[x]:= \{y \in X; x\sim y \}$$ Show that the equivalence classes give a partition of $...
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41 views

Show that $X$ can be represented as a union of disjoint equivalence classes

Let $X$ be a set. Let $\sim$ be an equivalence relation on $X$. Show that $X$ is the union of disjoint equivalence classes $\{x\}$ for $x \in X$. What I have tried: claim: $X = \bigcup_{x\in X} \{ x ...
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41 views

relations and equivalence classes

$S$ is the set of all equivalence relations on set $A = \{1,2,3,4,5,6,7,8,9\}$ in which one of the equivalence classes is $\{1,3,5,7,9\}$. What is the size of $S$? I don't even know how to start... ...
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1answer
38 views

Propositional Equivalence

Are the following two propositions equivalent? p IMPLIES (q IMPLIES r) p IMPLIES (q AND r) From what I can tell, using the logical equivalences, this should be false, correct? p ...
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1answer
18 views

Is this relation symmetric?

While solving questions related to reflexivity, symmetricity and transivity of relations, I came across this question: Show that the relation $R$ in the set $A = \{1,2,3\}$ given by $R = \{(1,2),(...
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70 views

Isn't x/E = y/ E ⇔ x E y deduced, not just x/E = y/ E ⇒ x E y from (a) x/ E≠ Ø, (b) x/E ∩ y/ E ≠ Ø ⇔xEy?

"Theorem 3. Let $\mathscr E$ be an equivalence relation on a nonempty set $X$. Then (a) Each $x/\mathscr E$ is a nonempty subset of $X$. (b) $x/\mathscr E \cap y/\mathscr E \neq \emptyset$ if and ...
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104 views

Total Number of Equivalence classes of R

I was given the following question for homework: Let P denote the set of all compound propositions involving the simple/atomic propositions p, q, and r and the logical connectives ∨, ∧, and ¬ (...
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35 views

Determine which of the following are equivalence relations:

Let $\mathbf{X}$ be the set of all residents in New Jersey. Determine which are equivalance relations: a) $x\sim y$ provided $y$ has the same natural parents as $x$ b) $x\sim y$ provided $y$ lives ...
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1answer
44 views

Addition in the space of orbits (under group action)

This question may be very trivial in this area, but I am a beginner to this and I stuck here. Could anyone please help me here! Let $\Gamma$ be a group whose identity is $e$. Let $X$ be a set and $∗:...
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1answer
28 views

Prove that $\sigma^*$ is the least group congruence on $S$

Let $S$ be an inverse semigroup and consider the relation $\sigma$ on $S$ given by $$a \sigma b \iff ab^{-1} \in E(S)$$ Consider the congruence generated by $\sigma$, say $\sigma^*$. Prove that $\...
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75 views

How to prove equivalence relations

I'm going through Pinter's "A Book of Abstract Algebra" and I'm currently on the topic of Partitions and Equivalence Relations. I'm having a little trouble understanding the way he (and apparently ...
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1answer
40 views

Equivalence class of functions with commutative diagram.

Let $S$, $T$ be sets, and $f,g: S \to T $ be function satisfying a condition that, there exist $\phi : S \to S, \rho : T \to T$, bijections, such that $f = \rho^{-1} \circ g \circ \phi$. Then we call ...
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1answer
61 views

Geometric description of equivalence classes.

For $X = R^2$ define the relation $R$ on $X$ by $(x_1, y_1)R(x_2, y_2)$ if $x_1 = x_2$. a). Verify that $R$ is an equivalence relation on $X$. I've already shown that this is reflexive, symmetric, ...
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63 views

Is the relation on a singleton set an equivalence relation?

So I understand that for a relation on a set to be an equivalence relation, it must satisfy three axioms: For all $x, y, z \in X$ and the relation $R$ on $X$, $(x,x) \in R$ if $(x,y) \in R$ then $(...
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1answer
13 views

What is the meaning of cyclically equivalence classes of multiple indices?

Can you please give me an example for the cyclic equivalence classes of multiple indices on the following set? $$ I(w,d)=\{\ (k_1,...k_d)\mid k_1+\ldots+k_d=w, k_1,...k_d \ge 1\,\}$$ where $ w$: ...
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1answer
28 views

given definition of a relation $R$, prove that $R$ is an Equivalence Relation

The relation is on set $\mathbb{R}^\mathbb{R}$ and the definition of the relation $R$ is: $f \mathop{R} g \iff \exists _{y\in \mathbb{R}} \forall_{x\in \mathbb{R}}\ ((x>y)\to(f(x)-g(x)\in \mathbb{...
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2answers
26 views

Relation of equivalence with sgn

Test if the relation $$(x, y)ρ(a, b)\Leftarrow\Rightarrow sgn(y+\pi x) = sgn(b + \pi a)$$ is a relation of equivalence on $R^2$ and if so, determine the quotient set and $C_{(2, \pi)}$. Also, $sgn(x)...
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1answer
24 views

For each $x \in X$, $x/ \mathscr E=\{y \in X \mid y\mathscr Ex\}$? Shouldn't it be for each $y \in X$?

"Definition 6. Let $\mathscr E$ be an equivalence relation on a nonempty set $X$. For each $x \in X$, we define $x/ \mathscr E=\{y \in X \mid y\mathscr Ex\}$ which is called the equivalence ...
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Where in (a) and (b) above shows that $x/\mathscr E = y/\mathscr E \Rightarrow x \mathscr E y$?

"Definition 6. Let $\mathscr E$ be an equivalence relation on a nonempty set $X$. For each $x \in X$, we define $x/ \mathscr E=\{y \in X \mid y\mathscr Ex\}$ which is called the equivalence class ...
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23 views

Equivalence relation on $S^2$ and unit square make same space

I think I understand a bit of this task, but I hope someone can look critical to my answers: Let $X$ be the space obtained from the sphere $S^2$ by gluing the north and the south pole (with the ...
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104 views

Is there any partial order that extends $\delta$?

Let $M = \{(x_n)_{n\ge1} | x_n \in \mathbb Z, \forall n \in \mathbb{N}^{*}\}$ We define relations $\delta$ and $\sim$ on $M$ as: $(x_n)_{n\ge1}\ \delta\ (y_n)_{n\ge1} \iff \forall n \in \mathbb{N}^{...
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36 views

Is there a relation name that only applies to equal vectors if they also have the same origin?

Vectors $\vec{A}$ and $\vec{B}$ are equal if they have the same magnitude and direction. Is there a mathematical relation defined to apply to them only if they also have the same origin? Actually, I ...
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2answers
70 views

Show that a relation is a equivalence relation

I have a infinite set $A$, and $F$ is the set of all functions $g \colon A \to A$. Let the equivalence relation $\sim$ on $F$ be defined such that $f \sim g$ if only if the set $D_{fg} = \{ a \in A | ...
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1answer
15 views

Canonical injection is compatible with induced equivalence relation?

Let $R$ be an equivalence relation on a set $E$, let $A$ be a subset of $E$, let $j : A \to E$ be the canonical injection, and let $R_A$ be the equivalence relation induced by $R$ on $A$ (that is, the ...
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28 views

Relations consisted of triples

A relation $m$ is defined on the set of nonnegative real triples as follows: $(a_1,a_2,a_3)\,m\, (b_1,b_2,b_3)$ if two of the inequalities $a_1>b_1,a_2>b_2, a_3>b_3$ are satisfied. ...
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1answer
24 views

Does symmetric property of Equivalence hold when the left hand side of the equation is in indeterminate form?

The Symmetric Property of Equivalence is a=b implies b=a. This property does not have any conditions on a or b. But what if through manipulation I get $a=\frac00$ where $a$ is a real number does this ...
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Find the equivalence classes for $a T b \iff \frac a b \in \Bbb Q$

Given the set $S = \{ x − \sqrt 5 y : x,y \in \Bbb Q, \ x − \sqrt 5 y \ne 0 \}$, assume the relation $T$ is defined on $S$ by $a T b \iff \frac a b \in \Bbb Q$. How can I find the distinct ...
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56 views

$x\sim y$ if $|x-y|\le 3$, then is $\sim $ or R an equivalence relation?

Let R or $\sim$ be the relation defined on Z by $$x\sim y\text{ if } |x-y| \le 3$$ Is $\sim$ an equivalence relation? It is reflexive and symmetric if I did it correctly. However, I am having doubt ...
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3answers
68 views

Equivalence relation and Distinct equivalence classes

Given the set $S = \{x-y \sqrt5: x, y$ are rational numbers and $x-y \sqrt5 \neq 0\}$. Assume the relation $T$ is de fined on the set $S$ by $a T b$ if $a/b$ is a rational number. Question has two ...
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Finding distinct equivalence classes

Q: Given the set $S = \{x - y\sqrt 5 : \text{x, y are rational numbers and }x - y \sqrt5 \neq 0 \}$. Assume the relation defined on the set $S$ by $a\ T\ b$ if $a/b$ is a rational number. Find the ...
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Equivalence relation: $aRb$ iff $2a+3b$ is divisible by $5$

Prove that $R$ is an equivalence relation: $aRb$ iff $2a+3b$ is divisible by $5$. Here $a,b\in \mathbb{Z}$ (set of integers). I can prove that $R$ is reflexive and transitive. How to prove it's ...
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38 views

E1 and E2 are equivalent then they are “almost equivalent”

Given : 2 statements E1, E2 in relational algebra are "almost equivalent" if every phase in the database D ,except finite number of D's E1(D)=E2(D). E(D) means the result of activating the statement E ...
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2answers
81 views

What is the symbol for “coincident” in geometry?

I am looking for a symbol to say that one geometrical figure coincides with another without writing the phrase "is coincident with." For example, the altitude $a$ of an equilateral triangle coincides ...
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1answer
36 views

Can we always define a congruence category?

In Awodey's Category Theory the congruence category is defined as follows... We have a congruence ~ on a category $C$. Then $C^\tilde{}$ is defined as: $(C^\tilde{})_0=C_0$ $(C^\tilde{})_1=\{ \left&...
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Suppose $(a+i)^{a+i}\equiv a^a \mod p$ for all $a=1,2,..$ Then $i$ is divisible by $p(p-1)$

Suppose $(a+i)^{a+i}\equiv a^a \mod p$ for all $a=1,2,\dots$ Then $i$ is divisible by $p(p-1)$. Solution: Take $a=p$ then we see that $(i+p)^{p+i}\equiv p^p \equiv 0 \mod p$ Since $i+p\equiv 0 \...
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29 views

Finding distinct equivalence classes.

I am going through some practice questions and am having trouble to finding distinct equivalence classes and this is my understanding so far. Let S be a nonempty subset of Z, and let R be a relation ...
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1answer
33 views

First-order equivalence formulas in Logic

Can someone help me understand as to why the following are equivalent when x is a bound variable that does not occur free in A? $\forall x (A \lor B) \iff A \lor \forall x B$ $\exists x (A \...
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1answer
20 views

On an exercise that asks for a homeomorphism between a quotient space and a metrizable space.

I have the solution to the exercise but have a doubt on one thing, I state the exercise: Given $$ X = \{ (x,y) \in R^2 | x = \frac{1}{n}, n \in N \}$$ and $Y = X/_{\sim}$ where the equivalence ...
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2answers
40 views

Partition into “fibers” $f^{-1}(y) \in Y$

Consider any surjective map f from a set X onto another set Y. We can define an equivelance relation on X by $x_1Rx_2$ if $ f(x_1)=f(x_2)$. Check that this is an equivelance relation. Show that the ...
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68 views

Given $S=\{0,1,2,3,4,5\}$, find the partition induced by the equivalence relation $R$

I am currently taking discrete math and have been given the following question to answer. Given $S=\{0,1,2,3,4,5\}$, find the partition induced by the equivalence relation $R$ where $R=\{(0,0),(0,4),(...
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31 views

Cardinality of this equivalence class

I'm looking at the following equivalence relation on $\mathbb{Z}$: $a \sim b$ if and only if there exist $n,m \in \mathbb{N}_{>0}$ so that $a^n = b^m$ I'm trying to determine what the cardinality ...
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3answers
43 views

Construct an equivalence relation on a given set

can anyone help me on this problem? I have the set $\{0,1,3,8,9\}$ and I want to define an example of an equivalence relation. I know that to be an equivalence relation it needs to be reflexive, ...