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

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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
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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, ...
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1answer
29 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 ...
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2answers
46 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 ...
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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 ...
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1answer
46 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 ...
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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: ...
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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 ...
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1answer
31 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 ...
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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: ...
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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, ...
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1answer
59 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
50 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 ...
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1answer
35 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, ...
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3answers
53 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
8 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 $A$.
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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 \ ...
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4answers
382 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 ...
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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 ...
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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 ...
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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 ...
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1answer
73 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 ...
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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?
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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 ...
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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 ...
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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 ...
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1answer
39 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 $|X|$ ...
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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 ...
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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
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1answer
58 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) ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $bRc$ ...
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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
34 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
31 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 ...
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1answer
71 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 ...
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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
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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 ...
0
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1answer
12 views

Equality: transitive property

Is the following relation a valid example for the transitive property of equality? If not, what is/are the name(s) of the property/ies involved? Given A, B, C, D. Given A = B, A = C, B = D. Then C ...
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0answers
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Trying to prove Equivalency using Boolean Algebra

The question presented was to use boolean algebra to show that XY’Z + X’Y’Z’ + XY’Z’ + X’YZ’ ≡ XYZ’ + XY’Z + XY’Z’ + XYZ’ I've tried using various laws of Boolean algebra, but the answer that I ...
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3answers
84 views

What’s wrong with this proof that $5$ is prime?

I’m reading How To Prove It and I’m confused as to how the proof of “$x$ is prime” is correct. I've written proof given below and also my conclusion after substituting in values for $x$, $y$ and $z$: ...
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1answer
39 views

Number of equivalence classes of binary sequences which differ only by finitely many elements.

This question rose up when i was reading a problem the author used to argue against the axiom of choice. Consider the set of all (infinite) sequences of 0's and 1's. Q1) How many such sequences are ...
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1answer
50 views

Is $R = \{ (x, y) \in \mathbb{R} \times \mathbb{R} \mid x - y \in \mathbb{N}\}$ an equivalence relation?

Note: I am also trying to answer my own question, but I am not sure if it is correct, please correct it, if it's wrong. Thanks :) I have an exercise where I have to say if a relation $R$ is or not an ...
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2answers
39 views

Pass from partition to the equivalence relations

I have a set $A = \{ 1, 2, 3\}$, which possible partitions are: $P_0 = \{ \{1, 2, 3 \} \}$ $P_1 = \{ \{1, 2 \}, \{3 \} \}$ $P_2 = \{ \{1 \}, \{2, 3 \} \}$ $P_3 = \{ \{1, 3 \}, \{2 \} \}$ $P_4 = ...
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1answer
56 views

Basic group algebra exercise

I'm working on this abstract algebra problem that was given as homework to me, I'm in my first year of university and have a semester's background in basic abstract algebra. Here is the problem: Let ...
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2answers
14 views

Equivalence relations-Discrete Math

Here is an equivalence relation R={ (x,y) | x-y is an integer} My question is: what is the equivalence class of 1 for this equivalence relation? Can say indicate the equivalence class of 1 as ...