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

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Logical Equivalence with →

I am given the problem of proving: $p → (q\land r) \equiv (p→q) \land(p→r)$ Using known logical equivalences. I'm not well practiced in transforming logical statements that contain →'s in them into ...
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1answer
45 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 $G$.
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2answers
107 views

Conformal Equivalence of two Riemann metrics

I'm reading a paper and encountered a concept of conformal equivalence between two Riemannian metrics on a differentiable $2$-manifold $M$ : Two Riemannian metric $g$ and $f$ are conformally ...
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2answers
64 views

equivalence classes and cardinality

I need to prove that every equivalence class created by the equivalnce relation $\sim$ on $\mathbb{R}$, that is defined by: $a\sim b \Leftrightarrow (a-b) \in \mathbb{Q}$, is $\aleph_0$. Furthermore, ...
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0answers
34 views

The null set as the underlying set of a relation structure

Can the null-set underly a relation structure? Can it underly a well-order? If so, would such a well-order have order-type $0$? (Can one even have an order-type of $0$?) My motivation for this ...
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1answer
71 views

Transitive relations on a set of n elements

How to find out the total number of transitive relations in a set of n elements? I am facing a problem in finding all the possible cases, is it not possible to find all cases? If not possible, why?
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1answer
35 views

Proof that a given relation is an equivalence relation

Can someone can tell me if my proof of the next propostion is correct? Define the following relation: $$a\sim b \iff a-b=km, m\in \mathbb{Z}$$ Show $\sim$ is an equivalence relation And so here's my ...
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1answer
25 views

Show that f is a symmetric relation on A

I am learning about relations and I come across this exercise. And I don't understand the problem. Let me first state the problem here: Let $f: A \rightarrow A$ be a function for which $f(f(x))=x$ ...
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2answers
46 views

Equivalence and Order Relations

I have the following problem: Provide an example of a set $S$ and a relation $\sim$ on $S$ such that $\sim$ is both an equivalence relation and an order relation. Conjecture for which sets and this ...
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3answers
43 views

Equivalence Relation Statements Proof

Let $\sim$ be an equivalence relation on a class $X$. The following are equivalent for $x,y \in X$. 1) $[x]=[y]$ 2) $x \sim y$ 3) $x \in [y]$ 4) $y \in [x]$ 5) $[x] \bigcap [y] \neq \emptyset$ ...
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1answer
67 views

Show that homeomorphism is an equivalence relation in metric spaces

It needs to be shown that homeomorphism is reflexive, symmetric and transitive in all metric spaces. Reflexivity seems to be easy to show, but I'm not sure how to do the rest. Any help?
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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 ...
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3answers
118 views

Can a premise imply contradictory statements?

Can a premise imply contradictory statements? Can two contradictory premises imply the same conclusion? Determine the answers to these questions by doing the following. Prove or disprove: the ...
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1answer
63 views

Sets Modulo Equivalence Relations

I am stuck on this question and would greatly appreciate any help: Recall, for an arbitrary set $S$ and equivalence relation $\equiv$ on $S$, $S/\equiv$ denotes the set of equivalence classes in $S$. ...
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0answers
154 views

Colored Picture for Equivalence Classes, Relations, Partitions, .. [closed]

Origin — A Book of Abstract Algebra — Charles Pinter — p120. I'm trying to sketch a colored picture for the ideas from equivalence classes, equivalence relations, partitions, etc... underneath. ...
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2answers
107 views

What does an equivalence class look like?

Let $\mathbb{R}^2 = \{Q = (a,b) | a,b\in \mathbb{R}\}$. Prove that if $q_1 = (a_1,b_1)$ and $q_2=(a_2,b_2)$ are equivalent, meaning $a_1^2+b_1^2 = a_2^2 +b_2^2$, then this gives an equivalence ...
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1answer
65 views

Shortcut method for proving equivalence relations

Define the relation R on N*N by: (x,y)R(z,w) if and only if x-z = w-y. Check whether R is an equivalence relation. Explain your answer My teacher answer is: Using the shortcut method: ...
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1answer
46 views

Equivalence class for a relation

Consider the equivalence relation on Z ! Z given by (m, n)R(p, q) if and only if mq = np: (a) Find the equivalence class represented by (2, 5). (b) Describe the set S of the equivalence classes ...
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1answer
68 views

Equivalence Relations

I would appreciate any help available for the following problem: Let $S$ be a set. Let $T$ be the set of all relations on $S$. Construct a relation $\equiv$ on $T$ in the following way: for $\sim, ...
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0answers
31 views

Prove that $[x]_{R}=[y]_{R} \Rightarrow g(f(x))=g(f(y))$

Let $f:\mathbb{Z} \to \mathbb{N}$ and $g:\mathbb{N} \to \mathbb{N}$ be functions. And let $R$ be a equivalence relation on $\mathbb{Z}$, defined by: $$xRy \Leftrightarrow f(x)=f(y)$$ For any ...
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1answer
44 views

Very Abstract Relation with points

So I have this question on relations, that I really cant understand. I mean, I cant understand the question to be honest. Suppose a set $X$ of points on the plane and we "stabilize" a point $O ∈ X$. ...
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3answers
163 views

Can we take images of equivalence relations?

Given a function $f : X \rightarrow Y$, it is well-known that we can take the image under $f$ of any subset $A \subseteq X$, and we can take the preimage under $f$ of any subset $A \subseteq Y$. This ...
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1answer
31 views

How can be a class of paths be open for a connected open set?

I have the following excercise: Let $A$ be an open set. If $x,y\in A$ we write $x\sim y $ when there is a path from $x$ to $y$, this is, $\exists P=\bigcup_{i=0}^{n} [r_{i-1},r_i]$ with $a=r_0$ and ...
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1answer
28 views

Equivalence involving diagonalization of block matrix

Given $T^\top L T = diag(A, B, C)$ being a block diagonalization with $L$ symmetric $A,B$ positive definite (p.d.) $T = \left( \begin{array}{ccc}I & 0 & T_2 \\ T_1 & I & T_3 \\ 0 ...
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1answer
70 views

How to show or prove equivalence relation?

I have this relation : for all integers m and n so : m R n ⇔ m ≡ n mod(3) How can I show that R is an equivalence relation
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2answers
90 views

3-dimensional cube shortest path question

Let Q be the graph consisting of vertices and edges of a 3-dimensional cube. Two relations are defined on the vertices of Q. • R1={(v,w):the shortest path from v to w has an odd number of edges}. ...
5
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1answer
104 views

How to simply show that there are “78 'strict ordinal' 2x2 game matrices”

In "Theory of Moves", Steven J. Brams analyses two-player games with two strategies per player, where each player can totally rank his payoffs, although payoffs need not be comparable among players. ...
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1answer
38 views

If $E_1$ and $E_2$ are equivalence relations, is $E_1\circ E_2$ an equivalence relation?

I'm given two equivalence relations $E_1$ and $E_2$ over a set A and need to show whether the composition $E_1 \circ E_2$ is reflexive, symmetric and transitive. I only managed to show that $E_1 ...
2
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1answer
97 views

Equivalence relation question with cardinality and countability $A=\mathbb R,\ aSb \iff a-b\in \mathbb Q $

Let $A=\mathbb R,\ aSb \iff a-b\in \mathbb Q $ What is the cardinality of $[\pi]_S$ ? Prove that the quotient group $\mathbb R/S$ is uncountable. Well I think that cardinality is ...
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1answer
116 views

Prove or disprove question with equivalence relation, classes and quotient group

Let $A$ be a set and $R$ an equivalence relation on $A$. Prove or disprove: If $A$ is countable then all the equivalence classes of $R$ are countable. If $A$ isn't countable then the ...
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1answer
55 views

Equivalence classes of points of R^2

Let $A$ and $B$ be two sets of points in $\Bbb R^2$. We define an equivalence relation on the powerset of $\Bbb R^2$, by saying that $R(A,B)$ iff there is a translation $f$ on $\Bbb R^2$ such that the ...
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3answers
254 views

Equivalence relation question with functions

We'll define on the set: $A=\Bbb R^{[0,1]}$ the relation $R$ by $fRg$ if $f(0)=g(0)$. Make sure it's an equivilence relation. What is $[\cos x]$ ? Describe all the equivalence classes ...
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3answers
59 views

Proof of equivalence relation on a set

Let $X = \{(a,b) \mid a,b \in \Bbb Z; b \ne 0\}$. We define $(a,b)\mathrel R (c,d)$ iff $ad = bc$. Prove that $R$ is an equivalence relation on the set $X$. Which known set do the equivalence classes ...
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2answers
16 views

Having trouble with symmetry (equivalence relation)

Define a relation of $x,y \in R$ when $x = |y|$. I know this is reflexive as $x = |x|$ holds true because the relation has to have x as positive since $x = |y|$ which makes $x$ have to be positive ...
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4answers
51 views

Bijection from set of equivalence classes to $\mathbb R$

In $\mathbb R[X]$, define an equivalence relation ~ by $P_1$~$P_2$ if $P_1-P_2$ is divisible by X. I have shown that $X$ is an equivalence relation. Let $\mathscr Q$ denote the set of equivalence ...
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2answers
75 views

Properties of the relation $R=A\times B \cup B\times A$

A is a set. Let $B\subsetneq A$. $R=A\times B \cup B\times A$ Determine if the relation is (a)reflexive, (b)symmetric, (c)transitive, (d)anti-reflexive, (e)anti-symmetric, (f)asymmetric, ...
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1answer
61 views

Properties of the relation $R=\{(x,y)\in\Bbb R^2|x-y\in \Bbb Z\}$

$A= \Bbb R \\ R=\{(x,y)\in\Bbb R^2|x-y\in \Bbb Z\}$ Determine if the relation is (a)reflexive, (b)symmetric, (c)transitive, (d)anti-reflexive, (e)anti-symmetric, (f)asymmetric, (g)equivalence ...
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3answers
149 views

Define a relation on the set of all real numbers $x,y \in \mathbb{R} $ as follows:

Define a relation on the set of all real numbers $x,y \in \Bbb{R} $ as follows: $x \sim y$ if and only if $x - y \in \Bbb{Z}$ Prove this is an equivalence relation and find the equivalence class of ...
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1answer
65 views

Equivalence relation $g\sim h :\Longleftrightarrow h \in \{g,g^{-1}\}$

Let $(G, \cdot)$ be a group with an Identity element $e$. (i) A relation on $G$ is defined through $g\sim h :\Longleftrightarrow h \in \{g,g^{-1}\}$. Show that $\sim$ is a equivalence relation and ...
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3answers
224 views

Is this relation reflexive, symmetric and transitive?

Define a relation $R$ on the set of functions from $\mathbb{R}$ to $\mathbb{R}$ as follows: $(f, g) \in R $ if and only if $f(x) - g(x) \geq 0$ for all $x \in \mathbb{R}$ Is this relation ...
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3answers
28 views

Relation symmetric confusion

So Symmetric = (a,b), (b,a) Set = {<1, 1>, <1, 2>, <1, 4>, <2, 1>, <2, 2>, <3, 3>, <4,1 >, <4, 4>} I understand ...
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1answer
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Formulate a relation $R$ between $2$ sets $A$ and $B$

Let $A$ and $B$ be $2$ sets of real numbers. How can I formulate the following entence, in mathematical terms, not plain english. IF At least one Element $x$ of $A$ is equal to one element $y$ of ...
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0answers
58 views

What is this equivalence relation explicitly?

Let $S \colon = \{ \ (x,y) \in \mathbf{R}^2 \ | \ \ y = x +1, \ \ 0 < x < 1 \ \}$, and let $T$ be the intersection of all the equivalence relation on the plane that contain $S$. Then how ...
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2answers
131 views

Difference between Reflexive and Symmetric in Discrete Maths

Difference between Reflexive and Symmetric in Discrete Maths? This is what I understand: Reflexive -> <a,a=a>, <b,b=b> uses ...
2
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1answer
101 views

equivalence relation composition problem

Let $R_1$, $R_2$ be two equivalence relations on $X$, prove that $R_1\circ R_2$ is an equivalence relation if and only if $R_1\circ R_2= R_2\circ R_1$ First I´m trying to prove that $R_1\circ R_2= ...
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1answer
37 views

equivalence relation difficult problem

Let $R_1$, $R_2$ be 2 equivalence relations on $X$; prove that $R_1\cup R_2$ is an equivalence relation on $X$ if and only if $R_1\cup R_2=R_1\circ R_2$ I really don´t have any idea how to do it, I ...
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1answer
43 views

equivalence relation and quotient set problem

Let $R$ and $S$ be equivalence relations on X so that $X/R$=$X/S$, prove that $R=S$ how can I solve this problem?
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2answers
47 views

Equivalence Classes of a Relation Given as a Set of Ordered Pairs

Question: The relation R is an equivalence relation on the set A. Find the distinct equivalence classes of R. A = {a, b, c, d} R = {(a, a), (b, b), (b, d), (c, c), (d, b), (d, d)} My work: So when ...
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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 ...
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1answer
17 views

How to prove this proposition that has to do with elements and equivalence relations

Every element $z$ in $X$ is in exactly one equivalence class. Not sure how to prove this. I proved that every element $z$ in $X$ is in some equivalence class by using the definition of $[x]$. How ...