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

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Equivalence Relation using Binary Operations.

Question: Let ∗ be a binary operation on a set A. Assume that ∗ is associative with identity. Let R be the relation on A defined on elements a,b ∈ R as follows: aRb if there exists an invertible ...
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51 views

Equivalence relations and equivalence classes

I dont know how to start this proof? Also, our professor did not explain equivalence classes fully so I am not understanding them very well.
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1answer
36 views

Show that this relation is an equivalence relation.

Given functions $$f_1 : A\to B$$and$$f_2 : A\to B,$$ let us write $f_1 \equiv f_2$ when there exist bijections $\alpha : A\to A$ and $\beta : B \to B$ such that $f_2(\alpha(a)) = \beta(f_1(a))$ for ...
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1answer
100 views

Does the Trace product in a semigroup have any relation with Trace of a matrix / matrix product

I recently read an article on generalized inverses and Green's relations (by X.Mary). The framework is semigroups, but obviously it has a lot of application within matrix theory. In the article ...
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2answers
65 views

For all $x,y∈\Bbb{R}$ define that $ x\equiv y$ if$ x^2=y^2$

For all $x,y\in\Bbb{R}$ define that $x\equiv y$ if $x^2=y^2$ . Then $\equiv$ is an equivalence relation on $\Bbb{R}$ , there are infinitely many equivalence classes, one of them consists of one ...
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276 views

How do I show this

Given invertible matrices $A,B$ and $P$ such that $A = PB$, then we say that $A$ is left equivalent to $B$. Show that left equivalence is indeed an equivalence relation.
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61 views

For all $x,y\in\mathbb{R}$ define that $x\equiv y$ if $x^{2}=y^{2} $; prove $\equiv$ is an equivalence relation. [duplicate]

For all $x,y\in\mathbb{R}$ define that $x\equiv y$ if $x^{2}=y^{2}$ . Then $\equiv$ is an equivalence relation on $\mathbb{R}$ , there are infinitely many equivalence classes, one of them ...
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44 views

equivalence classes of ∼ are left cosets of H in G - my attempt

Let $H$ be a subgroup of G, and define a relation $∼$ on G by the rules that $x∼y$ mean $x^{-1}y\in H $. Show that $∼$ is an equivalence relation and its equivalence classes are the left cosets ...
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1answer
21 views

Topologically equivalent metric spaces is an equivalence relation

I'm trying to prove that topological equivalence is an equivalence relation. Reflexivity was easy, and I'm sure transitivity is too, but I'm stuck on symmetry. My book's definition is that a metric ...
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0answers
77 views

Improper integral calculation and nature

I would like to know how to calculate this integral n check its nature : $$ A= \int_1^\infty \frac{e^{-t}}{1+t^{2}} dt . $$ I think I figured it out , we can do it with Equivalence like this (since ...
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1answer
129 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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1answer
25 views

Need help understanding transitive relations

My discrete math professor gave an example stating that the following relation is transitive, reflexive, symmetric, and antisymmetric. A = {a,b,c,d} R = {(a,a), (b,b), (c,c), (d,d)} I do not ...
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1answer
19 views

equivalence relation for $2a - b = 2c - d$ where $a,b,c,d$ are elements of $\mathbb{R}$

For $(a,b), (c,d)\in \mathbb{R}^2$ define $(a,b)\sim (c,d)$ to mean that $2a−b = 2c−d$. Prove that $\sim$ is an equivalence relation on $\mathbb{R}^2$. Reflexive: let $a$ be an element in ...
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1answer
25 views

partitioning a set using a non-equivalence relation

I understand that a set can be partitioned into equivalence classes by an equivalence relation (wikipedia article). However can we partition a set into a forest of trees by a relation that's simply ...
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1answer
44 views

How many equivalence relations there are on a set with 7 elements with some conditions

Calculate how many equivalence relations there are on $\{1,2,3,4,5,6,7 \}$ that include the set $\{(2,2),(1,3),(3,6),(7,5)\}$ and are foreign to the set $\{(1,7),(4,7),(4,3)\}$. Well I first drew ...
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1answer
110 views

Equivalence relations and their class

I am trying to figure out how to prove that an equivalence relation of the relation x~y defined on Z <=> x-y is a multiple of 3. My attempt was: 1) Reflexive: x = x => x ~ x 2) Symmetric: x ~ y ...
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1answer
27 views

Find a subspace homeomorphic to the quotient.

I'm giving the equivalence relation $V\sim W$ when $\exists\lambda>0:\lambda V=W$. I have to prove that $\sim$ is an equivalence relation and find a subspace of $\mathbb{R}^2$ homeomorphic to the ...
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1answer
20 views

Equivalence relations proof

I need to prove that if $R_1$ and $R_2$ are equivalence relations on the set $A$, then $R_1\cap R_2$ is an equivalence relation. Problem is I dont know how. Please help!
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1answer
35 views

Discrete Math Proofs, Partial Orders and Equivalence Relations

I am horribly stuck on $3$ proofs for my discrete math class. Any help would be greatly appreciated. Prove that if $R$ is a partial order, then $R^{-1}$ is a partial order Prove that if $R_1$ and ...
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1answer
36 views

Prove a relation is a equivalence

Let $\sim$ be defined so that $a\sim b$ when $a+b$ is even. Is this an equivalence relation? Equivalence relations confuse me a lot, so any help is appreciated!
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1answer
37 views

Why does x ~ y <=> [x] = [y]

So I read in wikipedia that "It follows from the properties of an equivalence relation that $x \sim y$ $⟺$ $[x] = [y]"$, but there seems to be no further elaboration on why $x \sim y$ $⟺$ $[x] = [y]$ ...
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2answers
54 views

aRb if and only if a=b or a-b=2^n for a natural number, n

Is R reflexive? Is R symmetric? Is R transitive? I know a=b is an equivalence relation so it is reflexive, symmetric, and transitive. I know a-b=2^n is reflexive but not symmetric or transitive. Not ...
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1answer
20 views

Openess of sets given by equivalence relations in the quotient topology.

I'm trying to prove this: Let $R$ be an equivalence relation in $X$. Show that $A$ is open in $X/R .\iff \bigcup_{[x]\in A}[x]$ is open in $X$ One of the first things that come to my mind is ...
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28 views

Equivalence Relation Proof for modular arithmatic

Given this modular relation: $x^3 \equiv y \pmod{3}$ how would you go about proving the transitivity of the system? I have proven the reflexivity, and symmetry pretty easily but the transitivity is ...
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1answer
21 views

Existence of neutral element at a certain position in subgroups

Given a group $G$ with neutral element $e$ and a subgroup $H \leq G$ as well as the equivalence relation $g_1 \sim g_2 \Leftrightarrow g_2^{-1} g_1 \in H$ (equivalence classes $[g]$). G be finite. ...
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1answer
29 views

Are these two summations equivalent?

Is $\sum_{y=2}^{\infty} (\frac{1}{y})(1-p)^{y-1}$ equivalent to $\sum_{y=1}^{\infty} (\frac{1}{y})(1-p)^{y}$ ?
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24 views

equivalence properties of $\equiv \pmod n$ proofs

Prove the identity: $$a \equiv b \pmod n \wedge a\equiv c \pmod n\implies b\equiv c \pmod n$$ I need to prove this property of $\equiv \pmod{n}$ along with a few others can someone link me to a ...
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3answers
32 views

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 ...
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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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33 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? ...
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4answers
45 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, ...
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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 ...
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4answers
361 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 $[x] = \{y \in X : y \sim x\}$. ...
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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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29 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
43 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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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 ...
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1answer
66 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 ...
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1answer
38 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
62 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 ...
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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 ...
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2answers
2k views

How many equivalence relations on a set with 4 elements.

Let S be a set containing 4 elements (I choose {$a,b,c,d$}). How many possible equivalence relations are there? So I started by making a list of the possible relations: ...
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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
58 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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70 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
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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1answer
42 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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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 ...
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2answers
1k 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 ...