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

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Why is max($\frac{2}{||w||}$)= min($\frac{1}{2}$)($||w||^2$)?

I was watching a video on machine learning. The instructor says that maximizing ($\frac{2}{||w||}$)is difficult (why?) so instead we prefer to minimize $\frac{1}{2}||w||^2$. $w$ is a vector. How are ...
0
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1answer
36 views

Equivalence relations and power sets.

Let $\mathcal{A}$ be the class of all sets and define the relation $R$ on $\mathcal{A}$ as: $A\space R\space B$ iff there is a bijective function $f:A \to B$. Prove that $R$ is an equivalence relation ...
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0answers
32 views

Transitive, Reflexive, Symmetric

So i know what each of these properties are..but this question does not provide any information on a 3rd variable so i was wondering how i would do it? ...
3
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4answers
42 views

Equivalence Relation definitions of Coset - looks like 1-step Subgroup Test? [Fraleigh p. 97 theorem 10.1]

p. 4 We are especially interested in the case where the set is a group, and the equivalence relation has something to do with a given subgroup. That is, we want to partition a group G into subsets, ...
0
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2answers
32 views

Define an equivalence relation on $\Bbb{Z}$ by $x\sim y$ if $x+2y$ is divisible by $3$

How do I approach this problem? I know how to approach on the equivalence relation of triangles(finding $x\sim x$, finding if $x\sim y$ then $y \sim x$, and finding if $x \sim y$ and $y \sim z$, then ...
4
votes
1answer
123 views

Picture - Equivalence Relation & Classes, Partitions, Quotient Set, & other related ideas

To get intuition for them and to remember them, I'd be grateful for a picture that combines and embodies the key definitions regarding Equivalence Relations & Classes, Quotient Sets, and ...
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votes
4answers
358 views

What exactly are equivalence classes

What exactly are equivalence classes? Suppose I have an equivalence relation $\sim$ on some set $X$ we denote this as $x \sim y$. The equivalence classes are then $[x] = \{y \in X : y \sim x\}$. ...
2
votes
1answer
78 views

Is there a name for this property: If $a\sim b$ and $c\sim b$ then $a\sim c$?

Let $X$ be a set with a binary relation $\sim$, such that for all $a$, $b$, and $c$ in $X$: If $a\sim b$ and $c\sim b$ then $a\sim c$ Is anyone familiar with this property of a binary relation? ...
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2answers
28 views

Set of equivalence classes equivalent to preimage of image of collapsing map proof

It says in a book I'm reading on topology that if $\mathit{R}$ is an equivalence relation on a space $X$, $p$ is the collapsing map $x \mapsto [x]$ and $A \subseteq X$ then: $$x \in A, y \mathit{R}x ...
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0answers
42 views

Good topologies on $\mathcal{P}(X)$

Let $X$ be a topological space, and let $\mathcal{P}(X)$ (resp. $\mathcal{P}_0(X)$) be the set of all subsets of $X$ (resp. the set of all non empty subsets of $X$). Finally, let ...
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1answer
63 views

Equivalence relation and equivalence class question

Show that the relation $\sim$ defined on the set $X = \mathbb{N} \times \mathbb{N} = \{(a, b) : a \in \mathbb{N}; b \in \mathbb{N}\}$ as $(a,b) \sim (c,d)$ if and only if $a + d = c + b$ is an ...
0
votes
1answer
57 views

Give an example of a relation R on $A^2$ which is reflexive, symmetric, and not transitive

I am just looking for some clarification on this exercise: Let $A = \{a,b,c,d\}$. Give an example of a relation $R$ on $A^2$ which is reflexive, symmetric, and not transitive. I understand that if I ...
1
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2answers
62 views

How to find the Equivalence class for a given set?

I'm really having trouble understanding these equivalence classes. Could someone please guide me through the following problem step by step, and help explain this. I have a final exam next week, and ...
1
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1answer
37 views

Equivalence Class.

Let R be the relation of congruence mod 4 on Z: a R b if a - b = 4k, for some k in Z. What integers are in the equivalence class of 31? How many distinct equivalence classes are there? What are ...
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1answer
41 views

list all the equivalence relation [duplicate]

list all the equivanlance relations in the set A={1,2,3,4) so there should be 15 right? so what I got so far (1 1) (22) (33) (44) (12) (13) (14) (21) (23) (24) (31) (32) (34) (41) (42) (43) these ...
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1answer
53 views

Prove that if the relation (R) is symmetric and antisymmetric on the set X, there exists a Y subset of X such that R is the = relation on Y.

Prove that for any relation R on a set X that is both symmetric and antisymmetric, there is subset Y \subseteq X for which R is the relation = on Y. I will tell you what I know. I know that ...
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0answers
69 views

Power Set, Bijection Function, Equivalence Relation

Let $S$ be a set and $P(S)$ the power set of $S$. For sets $A,B⊆P(S)$, we say that $A \sim B$ if there exists a bijective function $f: A \rightarrow B$. Show that $\sim $ is an equivalence relation.
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1answer
24 views

Show that $R \cap R^*$ and $R \cup R^*$ are equivalence relations.

Let $R$ be a reflexive and transitive relation on a set $S$. Let $R^*$ be the dual relation, $(a,b) \in R^*$ if and only if $(b,a) \in R$. Show that $R \cap R^*$ and $R \cup R^*$ are equivalence ...
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1answer
37 views

How to prove that equality is an equivalence relation?

Probably, it's a elementary question, but I would like some explanation. Everyone knows that equality relation is (i) reflexive, (ii) symmetric and (iii) transitive, that is, satisfies (i) $x=x$; ...
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0answers
65 views

On $N \times N $ define the relation R, setting $(a,b),(c,d) \in R$ if and only if $a+d=b+c$. Show that $R$ is an equivalence relation.

On $N \times N $ define the relation R, setting $(a,b),(c,d) \in R$ if and only if $a+d=b+c$ a. Show that $R$ is an equivalence relation. My attempt: By definition 6.2.3 $R$ is an equivalence ...
0
votes
1answer
36 views

Show that the equivalence classes of $\sim$ are left cosets of $H$ in $G$.

Let $H \leq G$ and define a relation on $G$ by $x \sim y$ if $y^{-1}x \in H$. Show that $\sim$ is an equivalence relation on $G$ and then show that the equivalence classes of $\sim$ are left cosets ...
0
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1answer
41 views

Equivalence relation: prove that $(X \cap Y) $\ $E $ $\subset (X$ \ $E) \cap (Y$ \ $E)$

I need to prove that $(X \cap Y) $\ $E $ $\subset (X$ \ $E) \cap (Y$ \ $E)$, where $E$ is an equivalence relation over $A$ and $X,Y \subset A$. I don't know where to begin. I know that $X$ \ $ E$ ...
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0answers
24 views

What does a null relation on all real numbers look like?

I'm being asked to test for reflexivity, symmetry, transitivity, and antisymmetry on a null relation for all real numbers, i.e. $X=\mathbb{R}; R = \emptyset $. What would such a resulting set look ...
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1answer
18 views

Determining equivalence classes on $\Bbb{R}$.

Say we have the following equivalence relation on $\Bbb{R}$: $$a\sim b\iff a-b\in\Bbb{Q}$$ What do the equivalence classes look like? On a preliminary investigation I got the following equivalence ...
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2answers
26 views

Proving that if $E, F$ are equivalence relations on $A$ and $E \subseteq F$, then there is a surjection from $A\setminus E$ to $A\setminus F$

Proving that if $E, F$ are equivalence relations on $A$ and $E \subseteq F$, then there is a surjective function from $A\setminus E$ onto $A\setminus F$. What does $E \subseteq F$ even mean? Does it ...
0
votes
1answer
31 views

Proof of equivalence relation on set.

I'm new to the whole relations topic and stumbled upon a problem. I know that an equivalence relation is a relation that is symmetric, transitive, reflexive, (and not usually anti-symmetric). But ...
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2answers
54 views

theorems on equivalence classes

I have a few short proofs that I wish to be checked regarding equivalent classes. Suppose that R is an equivalence relation on set X. If $a, b \in X$, then $a\in [a]$ $[a] = [b] \iff (a, b) \in R$ ...
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1answer
44 views

Equivalence Relation on R (real numbers)

Let R be the relation on R(real numbers) defined by: For all x, y (that belong) to R(real numbers), x relates y <=> x-y (that belongs) to Z. (a) Is R an equivalence relation? Prove your answer. ...
0
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3answers
54 views

Describing Equivalence Classes using set builder notation

How would you describe all the equivalence classes for the relation: $congruence$ $modulo$ $5$ over $Z$, using set builder notation?
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1answer
24 views

Size of partitions

I am looking at a problem and really confused. Let $X$ be a set and $R$ a subset of $ X×X $. We write $x1 ∼ x2$ if and only if $(x1, x2) ∈ R$ Suppose now that $R$ defines an equivalence relation and ...
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2answers
38 views

How to show properties of a given relation?

I am given that R is a relation on the given set X, and I have to show if the relation is (i) reflexive, (ii) symmetric, (iii) transitive, (iv) asymmetric, and (v) give an example of an element of ...
0
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1answer
69 views

Show that if $R$ is a strict partial order on $X$, and $R$ is not linear, then there exists a strict partial order $R'$ and $R' \supsetneqq R$.

Question: Show that if $R$ is a strict partial order on $X$, and $R$ is not linear, then there exists a strict partial order $R'$ and $R' \supsetneqq R$. My attempt: By definition 6.23,6.3.1, and ...
1
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1answer
62 views

An empty relation on a non-empty set — can it be an equivalence relation?

Given a non-empty set, A, and an empty relation, R, on that set A, can it be the case that the relation R is an equivalence class? Transitivity. (a,b) in R, (b,c) in R ===> (c,a) in R. This is ...
0
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1answer
29 views

How to determine a equivanelce relation?

I have a problem to understand the following output: Determine "representative system" or a "system of representatives" :).....for the following equivalence relation ...
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2answers
31 views

Sets equivalence relations

I seem to be having a hard time understanding some basic sets concepts. In week 5 of my class, I learnt about the cross product of 2 sets to be the following $A \times B = \{(a,b) : a \in A, b \in B ...
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0answers
24 views

A proper term for equivalence relation with a finite quotient set

Is there a proper term for an equivalence relation $\sim$ on some set $M$ such that it partitions $M$ into finitely many equivalence classes? Finite equivalence relation? or co-finite? or equivalence ...
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0answers
33 views

Equivalence Set, Subsets

Just a quick question: If a = [A] and a belongs to N (set of all natural numbers) doesn't that mean that A is a subset of N? The reason I'm asking this is because I'm trying to prove the theorem ...
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1answer
138 views

I don't understand equivalence relations

I'm trying to show that the left coset with a subgroup is an equivalence relation. So taking some element $g \in G$ and a subgroup $H$, the left coset is defined as $gH = \{gh : h \in H\}$. That means ...
0
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1answer
45 views

Determining Equivalence Relation on $\Bbb{Z}$

Alright, I have a homework problem which I have researched, read up on and I (think) solved. I just need someone to either confirm my answer (and re-affirm my knowledge) or explain why I am wrong. ...
1
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1answer
80 views

Determine if R is an equivalence relation

I've got this question: Consider the relationship $R$ between ordered pairs of natural numbers such that $(a, b)$ is related to $(c, d)$ (denoted by $(a, b) R (c, d)$) if and only if $ad = bc$. ...
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0answers
78 views

About Kernel and the coimage of a function

Introduction I was serching for a concept of "equivalence relations" induced by an arbitrary function in a "natural" way and I found the concept of Kernel. But I'm not sure that I understand it and ...
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0answers
28 views

Prove congruence relation.

Let $\sim$ be a relation where $x \sim y \Leftrightarrow x = y \lor (x \in I \land y \in I)$. Show $\sim$ is a congruence relation on $S$ where $S= \{a,b, c, d, e\}$ and $I = \{a, d\}$. One of the ...
2
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1answer
44 views

Prove transitivity of relation.

Let $\sim$ be a relation where $x \sim y \Leftrightarrow x = y \lor (x \in I \land y \in I)$. Show $\sim$ is a congruence relation on $S$ where $S= \{a,b, c, d, e\}$ and $I = \{a, d\}$. One of the ...
0
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4answers
47 views

Is $x \sim y \Leftrightarrow x,y$ are even an Equivalence relation?

The following instruction defineds an Equivalence relation on the set of natural numbers. $x \sim y \Leftrightarrow x,y$ are even My idea: Reflexivity: $x \sim x \Leftrightarrow x,x$ is even ...
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1answer
63 views

Number of Equivalence relations of $\{1,2,3\}$

Let $M$ be the set $\{1,2,3\}$. How many Equivalence relations $R \subset M \times M$ exists? My idea is to count the disjoint partitions of M: $K_1= ...
2
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1answer
55 views

$R$ is transitive if and only if $ R \circ R \subseteq R$

Question: Let $R$ be a relation on a set $S$. Prove the following. $R$ is transitive if and only if $ R \circ R \subseteq R$. Definition 6.3.9 states that we let $R_1$ and $R_2$ be relations on a ...
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2answers
53 views

Finding an equivalence relation on $\{1, 2, 3\}$ with two equivalence classes [closed]

I need some help on a particular question, this one: Describe an equivalence relation on $\{1, 2, 3\}$ that has exactly two equivalence classes. Regards.
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2answers
36 views

Let $x\in \mathbb{R}$ and $z\in\mathbb{C}$. And define equality ($x=z$) iff $x=(0,x)$. Is this equality well defined?

Let $x\in \mathbb{R}$ and $z\in\mathbb{C}$. And define equality ($x=z$) iff $x=(0,x)$. Is this equality well defined ? Okay. It is easily shown that something goes wrong if you define equality in ...
0
votes
2answers
59 views

Is $a \sim b$ such that $\gcd(a,b) > 1$ an equivalence relation?

Is $a \sim b$ such that $\gcd(a,b) > 1$ an equivalence relation? I know that it's reflexive, since $\gcd(a,a) > 1$. It's also symmetric since $\gcd(a,b) > 1$ iff $\gcd(b,a) > 1$. ...
1
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3answers
46 views

Equivalence Relation on $\mathbb{R}$ and its partition $\mathbb{R}/\sim$

Define the equivalence relation $\sim$ on $\mathbb{R}$ as follows: $$\forall a,b\in\mathbb{R},\ a\sim b\ \Leftrightarrow\ b-a\in\mathbb{Z}$$ I can prove that this is an equivalence relation, but I ...