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

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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
62 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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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
64 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
43 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
85 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}. ...
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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
37 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 ...
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1answer
96 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
248 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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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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139 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
195 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
31 views

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
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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 ...
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1answer
98 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
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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
70 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
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Proving rational numbers [closed]

Definition 33: Define a relation $\sim$ on $\mathbb Z \times (\mathbb Z \setminus \{0\})$ by setting $(a,b) \sim (c,d)$ if $ad - bc = 0$. Proposition: The relation $\sim$ defined above is an ...
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1answer
16 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 ...
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1answer
65 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 ...
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4answers
66 views

Equivalence Relations

Review for Group Theory Final Exam: Define a relation on $\Bbb{R}^2 \setminus (0, 0)$ by letting $(x_1, y_1) \sim (x_2, y_2)$ if there exists a nonzero real number $\lambda$ such that $(x_1, y_1) = ...
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1answer
435 views

Quotient set definition

so, i search and search about quotient set and cant figure out what is this. At the beginning i think it was the same of partitions, but now i'm confuse. Can someone show some examples and explain? ...
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2answers
148 views

Reflexive, Symmetric, Anti-Symmetric relations

Let $A = \mathbb Z \times ( \mathbb Z\setminus {0} )$. A binary relation $R$ on $A$ is defined as follows: For all $(a,b),(c,d) \in A$ $$(a,b) \,R\,(c,d) \iff ad = bc$$ now how do I find if $R$ is ...
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1answer
65 views

Equinumerosity between equivalence classes set and power set

I´m currently working on the following problem: "Let $\xi$ = $\{ $ $\bot$ $\mid $$\bot$ is a equivalence relation over $\mathbb{N} $$\} $ Show that $\xi$ and $2^\mathbb{N} $ (power set) are ...
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2answers
124 views

Discrete Math - Equivalence Classes

I'm trying to understand a problem that my textbook gives me. Here is the problem: The relation $R$ is an equivalence relation on the set $A$. Find the distinct equivalence classes of $R$. $A = \{0, ...
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1answer
45 views

how many elements does Ia have?

Let $A=\{1,2,3,4\}$. Let $F$ be the set of all functions from $A$ to $A$. Let $R$ be the relation on $F$ defined by $f,g \in F$ $f R g \Leftrightarrow |f(A)|=|g(A)|$ $f(A)=\{f(x): x\in A\}$ ...
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1answer
40 views

Need help to understand equivalence class

This is in my note Let S={1,2,3,4} Let R be the relation on P(s) defined by xRy <=>|x|=|y| how many equivalence classes are there ? 5 [∅]={∅} [{2}]={{1},{2},{3},{4}} [{2,3}]={{1,2},.......... ...
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0answers
26 views

Finding equivalence relations containing specific equivalences

"Find the number of equivalence relations on the set $\{1,2,3,\ldots,7\}$ such that: a) $1\sim2$ and $3\sim4$. b) $1\not\sim2$, $1\not\sim3$ and $3\not\sim2$." Solving this problem is equivalent to ...
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2answers
60 views

Determine all equivalence classes of $xy>0$

Define the equivalence relation $R$ as follows: For $x,y\in\mathbb R$, $x$ is equivalent to $y$ if and only if $xy\geq 0$. Determine all of the equivalence classes of this equivalence relation.
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1answer
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Basic Equivalent Relations Question [duplicate]

For $x,y\in \mathbb R$ $x\sim y$ if and only if $x-y \in \mathbb Q$. I need help with the following questions: If $a \in \mathbb Q$, what is the equivalence class of $a$? If $a \in \mathbb Q$, prove ...