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

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explain transitive nature of the given set

A relation $R$ in the set of human beings in a town given by $$R = \{(x,y):x \text{ is wife of } y \}. $$ How is it transitive? Can you explain?
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2answers
86 views

Proofs on equivalence relations rational numbers

The relation R = {(x, y)|x − y is an integer} is an equivalence relation on the set of rational numbers. I'm kind of confused with this question and what it is asking me to do. In order to solve ...
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2answers
56 views

Is this relation reflexive?

Suppose $R$ is a relation on $N_4=\{1,2,3,4\}$ such that $R\circ R = R$. How would I prove that $R$ is reflexive? Could you give me some tips about how to start off?
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1answer
31 views

Relation Reflexive? [duplicate]

Suppose $R$ is a relation on $N_4=\{1,2,3,4\}$ such that $R\circ R=R$. How would I prove that $R$ is reflexive? I am geting this statement as false, Please Let me know , How to prove this ?
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0answers
64 views

How to construct a transitive relation?

Given a binary relation $\mathcal{R}$ on a set $A$, you can be assured that the relation $\mathcal{R} \cup \mathcal{R}^{-1}$ is symmetric. (I can give a proof of this, if you would like.) I wanted to ...
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1answer
27 views

Disc quotient that is homeomorphic to the pinched torus

I apologize for my previous post. There was a mistake. I want to write a quotient of the disc $D^2:={\{z\in\mathbb R^2;\parallel z\parallel \leq 1 }\}$ by an equivalence relation which is ...
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2answers
26 views

Relation antisymmetry check

Hello the question I am having trouble with is Describe a binary relation on 1, 2, 3 that is reflexive and transitive, but not symmetric nor antisymmetric. I Have the answer ...
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2answers
44 views

Matrix Similarity Question

Show that similarity is an equivalence relation. More specifically, recall that we say $A, B \in M_{n \times n}(F)$ (set of $n\times n$ matrices) are similar if there exists an invertible $Q$ such ...
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1answer
59 views

Example of equivalence relation where reflexivity/symmetry is not trivial

For basically any equivalence relation that I've encountered in my mathematical studies, the only problematic part (if at all) was proving transitivity. Is there a "real-world" (i.e. appearing in some ...
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1answer
103 views

How to find union and intersection of these relations?

Problem: Let $R_1$ and $R_2$ be the "divides" and "is the multiple of " relations on the set of all positive integers respectively. That is, $R_1 = \{(a,b) | a \text{ divides }b\}$ and $R_2 = \{(a,b) ...
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2answers
57 views

How to mathematically show that the relation is transitive?

Problem: Show that the relation $x R y$ iff $x \leq y$ is a poset over the set of integers $\mathbb{Z}$ My work: I know that to show the relation is a poset or a post order, I have to show the ...
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1answer
94 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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2answers
55 views

How to prove the relation is transitive?

Problem: Consider the relation R on $N$ defined by $x$R$y$ iff $2$ divides $x + y$. Prove that R is an equivalent relation My work: I know that to prove that a relation is an equivalent relation, ...
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2answers
54 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 ...
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4answers
123 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 ...
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1answer
52 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), ...
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1answer
87 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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0answers
145 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 ...
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1answer
23 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 ...
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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 ...
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2answers
43 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 ...
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1answer
57 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 ...
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3answers
56 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: ...
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1answer
33 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
50 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
42 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
105 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
23 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
51 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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2answers
135 views

Some equivalence relation from flipping binary trees

I know almost nothing in combinatorics, so this question might be very easy, or well-known. Fix a number $n$. We will consider rooted planar binary trees with $n$ leaves. We will distinguish between ...
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1answer
28 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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49 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
108 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
59 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
45 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
90 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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4answers
395 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
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 $A$.
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1answer
24 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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2answers
1k views

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
60 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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1answer
32 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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2answers
34 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
84 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
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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
82 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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1answer
36 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
35 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
53 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 $|X|$ ...