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

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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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60 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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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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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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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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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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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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39 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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66 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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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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51 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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12 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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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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309 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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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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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, ...
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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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43 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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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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35 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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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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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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22 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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452 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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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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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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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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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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$x\sim y$ 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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61 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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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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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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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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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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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=\{ ...
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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 ...
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28 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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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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2answers
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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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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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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 ...
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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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Is the relation $a $~$ b$ iff $ ab$ is square on $\mathbb{Z}$ transitive?

I'm trying to determine whether the relation given above is a equivalence relation. I've already proved it is reflexive and symmetric, but I'm stuck trying to prove (or disprove) its transitivity. I ...
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41 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, ...
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Does $R=\{(x,y) \in \mathbb{Z}\times\mathbb{Z} : 3|(x+y)\}$ define an equivalence relation?

Given $R=\{(x,y) \in \mathbb{Z}\times\mathbb{Z} : 3|(x+y)\}$, Is $R$ reflexive? Is $R$ symmetric? Is $R$ transitive? Reflexivity: Could $(1,1)$ be a counter-example because $3\nmid(1+1)$? Symmetry: ...
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2answers
112 views

Define a relation on the integers such that $a R b$ iff $\;3\mid (a + 2b)$?

I've seen relations defined as functions between sets and as sets of ordered sets; however, I've never seen a relation defined as $3\mid(a+2b)$. What does this mean? --Edit-- I'll try and express my ...