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

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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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4answers
77 views

Show that $R = \{ (x,y) \in \mathbb{Z} \times \mathbb{Z} : 4 \mid(5x+3y)\}$ is an equivalence relation.

Let $R$ be a relation on $\mathbb{Z}$ defined by $$ R = \{ (x,y) \in \mathbb{Z} \times \mathbb{Z} : 4 \mid (5x+3y)\}.$$ Show that $R$ is an equivalence relation. I'm having a bit of trouble ...
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2answers
22 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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3answers
85 views

Compute equivalence classes of equivalence relation

I have already proven that the relation $R=\{(x,y) \in \mathbb Z \times \mathbb Z \mid x+y\text{ is even}\}$ is an equivalence relation by showing reflexive, symmetric, and transitive properties of ...
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1answer
29 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 ...
2
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1answer
27 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
28 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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1answer
27 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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0answers
39 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
29 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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2answers
56 views

why are equivalence relations called so?

"an equivalence relation is the relation that holds between two elements if and only if they are members of the same cell within a set that has been partitioned into cells such that every element of ...
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1answer
16 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 = ...
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1answer
67 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 ...
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2answers
32 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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0answers
20 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
37 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
38 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
25 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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1answer
50 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
32 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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1answer
43 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
9 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
27 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 ...
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1answer
287 views

Equivalence relation and partitions [closed]

Define an equivalence relation on the set R that partitions the real line into subsets of length 1.
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1answer
22 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 ...
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2answers
19 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, ...
0
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0answers
101 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 ...
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1answer
42 views

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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0answers
17 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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0answers
33 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
64 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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3answers
5k views

why is frobenius norm of a matrix greater than or equal to the 2 norm?

How can you prove that: $$\|A\|_2 \le \|A\|_F$$ I cannot use: $$\|A\|_2^2 = \lambda_{max}(A^TA)$$ It makes sense that the 2-norm would be less than or equal to the frobenius norm but I dont know how ...
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1answer
19 views

Addition in $R_S$ is well defined

Let $R$ be a commutative ring with $1 \neq 0$ and suppose S is a multiplicatively closed subset of $R \backslash {\{\, 0 \,\} }$ containing no zero divisors. We have the relation ∼ defined on $R × S$ ...
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1answer
407 views

A relation that is Reflexive & Transitive but neither an equivalence nor partial order relation

Set $A = \{0,7,1\}$ 1. So for a relation that is reflexive and transitive but neither an equivalence relation nor partial order...Can a relation be both partial order and equivalence? Attempt: ...
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1answer
10 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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0answers
23 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. ...
0
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1answer
33 views

Solve the relation with congruence

On $\Bbb Z$ consider the relation $xRy \Leftrightarrow x-y \not\equiv 0 \mod 3$. Prove (with explanation), whether the relation reflexive, symmetric, antisymmetric transitive is and prove if they are ...
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1answer
23 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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3answers
48 views

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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3answers
46 views

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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3answers
55 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 ...
0
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3answers
56 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 ...
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1answer
77 views

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 ...
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80 views

Unitary Farey Sequence Matrices

Take the Farey sequence $\mathcal{F}_n$ with values $a_m\in \mathcal{F}_n$ and put them into a vector $$ \vec v_k=\frac1{\sqrt{|\mathcal{F}_n|}}\biggr(\exp(2\pi i k a_m)\biggr)_m $$ The dimension of ...
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0answers
25 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
44 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
30 views

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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1answer
35 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=\{ ...
1
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1answer
35 views

The relation $a_1 \sim a_2 \iff f(a_1 ) = f(a_2 )$ is an equivalence relation

Suppose a function $f : A → B$ is given. Define a relation $\sim$ on $A$ as follows: $a_1 \sim a_2 \iff f(a_1 ) = f(a_2 )$. Prove that $\sim$ is an equivalence relation on $A$. I know that in ...
0
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1answer
25 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 ...