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

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50 views

Can a relation from A to some other set B also be considered symmetric?

Note: This definition is from Discrete Mathematics and Its Applications [7th ed, page 577]. This is my book's definition of a relation R on a set A My ...
0
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4answers
97 views

Is antisymmetric the same as reflexive?

Note: The following definitions from my book, Discrete Mathematics and Its Applications [7th ed, 598]. This is my book's definition for a reflexive relation This is my book's definition for a anti ...
0
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1answer
41 views

What is the equivalence class of this equivalence relation?

Note: This problem is from Discrete Mathematics and Its Applications [7th ed, prob 26 pg 645]. Problem: What is the equivalence class of this equivalence relation? Relation {(0, 0), (0, 1), (0, 2), ...
0
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1answer
22 views

Last step in proof of equivalence results in partition

Prove that: Given an equivalence relation $∼$ on a set $X$, the equivalence classes of $X$ form a partition of $X$. Well, I first define $O_x ={\{y:x∼y}\}$. If $O_x ∩ O_y \neq ∅$, so there is some ...
3
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0answers
139 views

Arrow kernel in category theory and generalized equivalence relation

let $F : \mathcal{C} \rightarrow \mathcal{D}$ be a functor from two categories. It looks like that there are various notions of kernels one could define for a functor. One could define the arrow ...
0
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0answers
21 views

About finding a binary relation

Let $δ_{n},β_{n}$ two sequences of rational numbers. Assume that the points $$P_{p}=(δ_{p-1},β_{p-1})$$ $$Q_{p}=(δ_{p},β_{p})$$ $$R_{p}=(δ_{p+1},β_{p+1})$$ are colinear and assume also that the ...
0
votes
2answers
33 views

Equivalence relation on a set of integers

I was wondering if the relation $X$ would be an equivalence relation only if the result is an even number. For example the relation $X$ is given by $a\ X\ b$ only if $a+b$ is even. Would this be ...
0
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1answer
27 views

Is transitive Relation closed under composition?

it's true that equivalence relations is closed under composition, i.e., if R is a equivalence relation RoR is so.(Because RoR =R) But this not imply that any transitive Relation is so. Briefly; i ...
0
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3answers
39 views

Equivalence Relation with dividing x and y integers

Define $x\sim y$ means 5 divides $(x - y)$ for $x$ and $y$ integers. Show that is an equivalence relation. Equivalence relation means it satisfies reflexity, symmetry, and transitivity. reflexive: ...
5
votes
1answer
88 views

Question about right and left cosets.

I want to do a question about how my algebraic structures professor defined left and right cosets. I'll write here his way to present them. We first talked about quotient group. Let $G$ be a group, ...
0
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1answer
31 views

Equivalence relations between ordered pairs of natural numbers

I have been looking into equivalence relations and trying to figure out if certain relationships would be considered an equivalence relation. Lets using the following relation X between ordered pairs ...
0
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2answers
47 views

If $R=K[X]/(X^n)$, can represent any element as polynomial with degree $<n$

Let $K$ be a field and $R=K[X]/(X^n)$ where $n \in \mathbb{Z}_{n\geq1}$ and $(X^n)$ is the ideal generated by $X^n$. We denote $x:=X+(X^n) \in R$, any equivalence class $r$ in $R$ has a representing ...
1
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1answer
34 views

Proving symmetry and transitivity

I want to prove $\mathbb{N} \sim \mathbb{Z}$ by indication of a bijection, thus the equipotency of the two sets. I know that I have to prove reflexivity, symmetry and transitivity. The reflexivity ...
0
votes
1answer
54 views

How to show a relation is/isn't reflexive, transitive, or symmetric

I was tasked with this: Define a relation on Z by setting x R y if xy is even. (a) Give a counterexample to show that R is not reflexive. How do I go about proving this? Do I express this ...
0
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1answer
21 views

Question regarding symmetric difference - Please check my work

Let $S$ be a relation on $P(\Bbb{R})$, such that $\displaystyle S=\{<A,B>\in\left(P(\Bbb{R})\right)^2.\left|A\triangle B\right|\le \aleph_0$ Is $S$ an equivalence relation? My try: ...
1
vote
1answer
165 views

If S is finite and $x/\rho$ is an idempotent of $S/\rho$ then $x/ \rho$ contains an idempotent

Currently working on the following problem, need a little help with the solution to the last part, any hints? Q: Let S be a semigroup, and let ρ be a congruence on S. Prove that if e ∈ S is an ...
0
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1answer
33 views

How to define a relation

I was tasked with this question: Let X = {0,1,2,3,4}. Define a relation R on X such that x R y if x + y = 4. I don't understand/know what syntax I should use to define this as a relation. What ...
0
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1answer
22 views

Relations: Transitivity, Symmetry, and Reflexivity

Each of the following subsets $R$ of the $(x, y)$-plane defines a relation on the set $\mathbb{R}$ of real numbers. Determine which of the axioms (transitivity, symmetry, reflexivity) are satisfied: ...
1
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0answers
39 views

Showing properties of equivalence relation

I just started my abstract algebra class and I am struggling with the concept of equivalence relations. I know that in order to prove equivalence relations, I have to prove the reflexive, symmetric, ...
0
votes
1answer
62 views

Vector Spaces: difference between Equivalence Classes and Quotient Spaces

I'm reading Herstein's Topics in Algebra and Halmos's Finite Dimensional Vector Spaces. I think I understand that if $V_F$ is a vector space over $F$ and $W$ a subspace of $V$, then there is an ...
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0answers
52 views

Equivalence Relations with equivalence classes

Consider the following relation on $\mathbb{Z} \times \mathbb{Z}$: $(x, y)\sim (x', y')$ iff $xy' = x'y$. (a) Prove that it is an equivalence relation. (b) Consider the set of equivalence classes ...
1
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1answer
37 views

Proof of a equivalence relation

A set $A$ is equipotent to a set $B$ $(A\sim B)$, if a bijection $f: A \rightarrow B$ exists. How to prove, that $\sim$ is a equivalence relation? EDIT: I understand the concept of reflexivity, ...
2
votes
3answers
55 views

Trouble understanding equivalence relations and equivalence classes

I've read up a bit on equivalence relations and equivalence classes, but I'm a bit unsure on the whole concept. From what I've read and equivalence relation, ~, between two mathematical objects $a$ ...
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1answer
10 views

For a preorder $R$ on a set $A$, show that $S=R \cap R^{-1}$ is an equivalence relation on $A$

Let $R$ be a preorder on a set $A$, and let $S$ be the intersection of $R$ and $R^{-1}$ (the relational inverse of $R$). Show that $S$ is an equivalence relation on $A$.
0
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1answer
21 views

Equivalence relation - Equilavence classes explanation

I have the following equivalence relation problem. $Let \ R\subseteq 2^S*2^S = \{(A,B):|A\cap T|=|B \cap T|\}\ where\ S=\{0,1,2\} \ and \ T=\{1,2\} \ Show \ that \ R \ is \ a \ equivalence \ ...
3
votes
4answers
383 views

When a determinant is zero

Is it true that if $C$ is a square matrix of size $n$ and $\det(C) = 0,$ then $C^n = O_n$ or the $0$ matrix? If yes, then why is that? I know that the reverse is obviously true, so I wondered if ...
0
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1answer
58 views

How to show the simple equivalence?

I come across the following equivalence of integrations: $$\int\left[-I \left(h\right){\partial h \over \partial \tau}+{\partial h \over \partial x}{\partial \over \partial \tau} \left({\partial h ...
0
votes
2answers
33 views

How many equivalence relations over $\mathcal P(\mathbb N)$ satisfy: $[\{8\}]_S=\{A\in \mathcal P(\mathbb N)|A\neq \{1\}\wedge A\neq \{2\}\}$

How many equivalence relations $S$ over $\mathcal P(\mathbb N)$ satisfy: $$[\{8\}]_S=\{A\in \mathcal P(\mathbb N)\mid A\neq \{1\}\wedge A\neq \{2\}\}$$ Just to make sure I understand, the ...
0
votes
1answer
22 views

Determine if “$n \sim m$ iff $nm>0$” is an equivalence relation on $\Bbb Z$

Determine whether the given relation is an equivalence relation on the set. $n$ is related to $m$ in the set of integers if $nm>0$. So my teacher said this set is not an ...
2
votes
1answer
75 views

Using first order sentences, axiomize the $\mathcal{L}$-theory of an equivalence relation with infinitely many infinite classes.

Problem Let $\mathcal{L} = \{E\}$ where $E$ is a binary relation symbol. Let $T$ be the $\mathcal{L}$-theory of an equivalence relation with infinitely many infinite classes. Current Solution ...
1
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1answer
19 views

Is it possible for a relation to be transitive and symmetric but not reflexive with only one element?

E.g. On the set $A = \{1,2,3,4,5,6\}$, is the relation set $R = \{(1,1)\}$ a transitive and symmetric relation but not reflexive?
0
votes
1answer
27 views

Proving $|S/R^2|=\aleph$ , $(x_1,x_2)S(y_1,y_2)\iff x_1^2-x_2=y^2_1-y_2$

Let $S$ be an equivalence relation over $\mathbb R^2$ such that: $(x_1,x_2)S(y_1,y_2)\iff x_1^2-x_2=y^2_1-y_2$ Prove that $|S/R^2|=\aleph$ One side is pretty simple: $|S/R^2|\le |\mathbb ...
0
votes
1answer
34 views

equivalence relation - checking

At $\mathcal P(\mathbb{Z})$ we define equivalence relation $\equiv$ : $A \equiv B \iff (A=B \vee (A \cup B)\cap\mathbb{N}=\emptyset)$ a)show that $\equiv$ is a equivalence relation at $\mathcal ...
0
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1answer
32 views

Equivalence Relation of Dice

Suppose you are rolling two dice, one red and one white. Two rolls of the dice are considered equivalent if the dice sum to the same number. The dice are six sided. a) Give the partition induced by ...
1
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1answer
40 views

How to find a representative of an equivalence relation on $\omega\times\omega$

In one of my exercises we are shown that the relation "$\sim$" is defined as the following: $$\langle n,m \rangle \sim \langle k,l \rangle \iff |(n\setminus m)| = |(k\setminus l)|$$ in which $|X|$ ...
1
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0answers
27 views

equivalence relation problem - checking

We define eqiuvalence relation $\equiv$ at set $P(\mathbb{Z})$ as follow: $A \equiv B \iff |A \setminus \mathbb{N}| = |B \setminus\mathbb{N}|$ a)find equivalence class for $\emptyset$ b)find ...
2
votes
2answers
44 views

For what $z\in\mathbb{N}$ is “$x\equiv y\iff xyz$ is a square” an equivalence relation?

Consider the set $\mathbb{N}\cup\{0\}$ and fix $z\in \mathbb{N}\setminus\{0\}$. Define the relation $x \equiv y \iff xyz$ is a square number. I am trying to verify that this is an equivalence ...
2
votes
1answer
59 views

Equivalence relation: cardinality of quotient set

At $A=\mathbb{Z}^{\mathbb{N}}$ we define equivalence relation $\equiv$ by: $$f\equiv g \iff \forall n\in \mathbb{N} ((f(2n)=g(2n)) \wedge(f(n)\cdot g(n)> 0 \ \vee f(n)=g(n)=0)) $$ a) ...
0
votes
1answer
21 views

Equivalence Classes and Relations of Hexagons

Suppose there is a hexagon in the plane. Consider two colorings of the edges of the hexagon equivalent if you can rotate the hexagon so that edges of the same color map to each other. Suppose you ...
1
vote
3answers
28 views

Prove a relation using an equivalence relation

Prove the relation define on $\mathbb{R} \!\,^2$ by $$(x_1,y_1) \sim(x_2,y_2) \Leftrightarrow x_1^2+y_1^2=x_2^2+y_2^2$$ is an equivalence relation Ok, so I know what an equivalence relation is. It ...
0
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1answer
41 views

Proving something is an equivalence relation

Problem: For two subsets $A$ and $B$ of some set $X$, we define \begin{align*} A \triangle B = (A \cup B) \setminus (A \cap B). \end{align*} We now define a relation $R$ on $P(X)$ (power set of $X$) ...
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1answer
67 views

Functions that preserve equivalence relations

Quick question: Let $X$ and $Y$ bet two sets and $\sim$ an equivalence relation on $X$. I was wondering what it means to say that a function $f$: $X\to Y$ 'preserves' $\sim$ in this case. Does it ...
0
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1answer
19 views

how many reflexive relations but not equivalence, are in a set with 4 elements?

I know that for reflexive relations on a set with n elements the formula is: $2^{(n^2-n)}$ So for a set with $4$ elements: $2^{(4^2-4)}$ = $2^{12}$ But I don't know how to find the relations that ...
0
votes
0answers
44 views

Why relation “parallel” on the set of lines in a plane not transitive?

My book says relation "parallel" on the set of lines in the plane not transitive. And the definition in the book given is : A relation $R$ on a set $A$ is transitive if whenever $aRb$ and $bRc$ ...
0
votes
2answers
25 views

Equivalence Relations and 1-1 Correspondences

I have an engineering instructor who has claimed that "equivalence relations and one-to-one correspondences are pretty much the same thing". However, I believe the answers to both of the following ...
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1answer
35 views

Finding the relation between on set of natural numbers?

Each case below gives a relation on the set of all nonempty subsets of the natural numbers. Determine whether the relation is transitive,symmetric, or reflexive. Case 1: $R$ is defined by $ARB$ if ...
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1answer
32 views

Equivalence Classes with 1 or 2 elements?

Let ~ be the relation on R defined by a ~ b if and only if |a| = |b|: (a) Prove that is an equivalence relation. (b) Give an example of an equivalence class with two elements. (c) Give an example ...
0
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1answer
72 views

Discrete Math dealing with Partition of Ordered Pairs. [closed]

Given the partition $\{a,b,c\}$ and $\{d,e\},\,$ of the set $S=\{a,b,c,d,e\},\,$ list the ordered pairs in the corresponding equivalence relation. How can I determine which elements are related to ...
2
votes
3answers
31 views

Not a precise question on equivalence class.

Consider $f:X\longrightarrow Y$. Define a relation $\sim$ on $X$ by $a\sim b$ iff $f(a)=f(b)$. I proved that $\sim$ is an equivalence relation and that if $f$ is onto and $X/\sim$ is the set of ...
2
votes
1answer
23 views

Equivalence class of $(x_1,y_1)\sim(x_2,y_2)$ iff $ x_1=x_2$

I proved that $x\sim y$ iff $x-y\in \mathbb Z$ is an equivalence relation on $\mathbb R$. I'd like to know if $[x]=\{x+n:n\in\mathbb Z\}$ is an equivalence class for every $x\in \mathbb R$ (if it is ...