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

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6
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1answer
142 views

Confusion between an element and its preimage

Let $X$ be a set and $\sim$ is an equivalence relation on $X$, so that the quotient set $X/_\sim=\bigcup_{x\in X}{[x]}$ with $[x]=[y]$ if and only if $x\sim y$. Consider the quotient map $f:X\to ...
0
votes
2answers
23 views

How to proof equivalence relation?

I need help with this problem: Let $S=\left\{\left[\begin{matrix} a & b \\ c & d \end{matrix}\right] : a,b,c,d \in \mathbb{C}\right \}$ and $M=\left\{\left[\begin{matrix} a & b \\ ...
0
votes
2answers
19 views

What is an algorithm for describing the partition of this equivalence relation?

Let $\mathbb{R}$ be the set of real numbers,$ f : \mathbb{R} \to \mathbb{R}$ a map, and $E$ the equivalence relation on $ ℝ $ defined by $E = \{(x,y) \in \mathbb{R} \times \mathbb{R} \mid ...
3
votes
2answers
23 views

Test for symmetric property of this ordered pair

Suppose the set $$S={1,2,3}.$$ I must show that the equivalence relation $$R=\{(1,1),(1,3),(2,2),(3,1),(3,3)\}$$ is on the set. The reflexive property states that: $$(a,a) \in R \;\forall a \in ...
2
votes
1answer
55 views

Can a relation be both anti-reflexive and anti-symmetric?

Is it possible for a relation to be both anti-reflexive and anti-symmetric?
2
votes
0answers
78 views

Relative Identity vs Set Theory

The last complete, but unpublished, paper by the late Tom Etter titled "Three-Place Identity" purports to prove that all of mathematics can be expressed in terms of relative identity. In his own ...
0
votes
1answer
13 views

Anti-symmetric or asymmetric for a relation between pairs in the set of Z x Z?

Is this anti-symmetric or asymmetric? I at first thought asymmetric because anti-symmetric would mean a = c and b = d which would not be true. But because the domain is the Cartesian product of ...
0
votes
1answer
30 views

Equivalence relations regarding binary relations

Let $R \subseteq X \times X$ be a binary relation for $X = \{a, b, c, d\}$. $R = \{(a, a), (b, c), (c, d), (b, d)\}$. Is the relation an equivalence relation? I don't know if I am proving it correctly ...
0
votes
1answer
21 views

A question on eqivalence relation

Please explain what the examiner means by asking: Also find [3,6] in Q 1 (a)(i)??
2
votes
1answer
33 views

Does this notion of “weak” isomorphism exist in literature?

Let $(M,\circ)$ and $(N,\ast)$ be two magmas. I'd like to relax the notion of isomorphism by defining a notion of "weak" isomorphism in the following way: $M$ and $N$ are "weakly" isomorphic if there ...
7
votes
1answer
57 views

When is the closure of an equivalence relation automatically an equivalence relation?

Let $X$ be a topological space, not necessarily nice, let $R$ be a subspace of $X \times X$, and let $\overline{R}$ be the closure of $R$ in $X \times X$. Then: If $R$ is a reflexive binary relation ...
-1
votes
1answer
24 views

Equivalence relation with functions [closed]

I've the following questions from asked: Let $A$ and $B$ sets, let $R$ be a relation on $A$, and let $f:A\rightarrow B$. The function f respects the relation $R$ if $xRy \implies f(x)=f(y)$ for all ...
2
votes
3answers
84 views

About the group of units

I'm stuck at this section of the following problem: Let $R$ be a commutative ring and $S$ be a subset of $R$, with $S$ multiplicatively closed. Find the group of units of $S^{-1}R$. My try: ...
0
votes
2answers
38 views

Well-defined and Equivalence relations

I am wondering why the following is well-defined... The definition of well-defined is given as; $g:(X/\sim) \to Z$ is well-defined if a mapping $f:X \to Z$ can be found where $f$ has the property $x ...
1
vote
2answers
19 views

Equivalence classes in $\mathbb{Z}_n$

I've the following exercise: Solve each of the following equations in the given set $\mathbb{Z}_n$: 1) $[5]+x=[1]$ in $\mathbb{Z}_9$ 2) $[2]\cdot x=[7]$ in $\mathbb{Z}_{11}$ For 1), is $x=5$ ...
0
votes
1answer
54 views

Proof of an equivalence relation

Let S be the relation on R defined by $xSy \Leftrightarrow x=|y|$, $\forall x,y\in\Re$ Is the relation reflexive, symmetric and/or transitive? By my proof that 1) $x=|y| \Rightarrow |y|=x$ ...
-1
votes
0answers
30 views

geometric description of equivalence classes [on hold]

For each of the following relations on $\mathbb{R}^{2}$, give a geometric description of the relation classes $[(0,0)]$ and $[(3,4)]$ 1) Let $S$ be the relation defined by $(x,y)S(z,w)$ iff ...
0
votes
1answer
25 views

Equivalence classes of $\Bbb Z$ with the operation $\mod n$

So I came across this phrase in my abstract algebra textbook: The integers $\mod n$ also partition $\Bbb Z$ into $n$ different equivalence classes; we will denote the set of these equivalence ...
0
votes
2answers
23 views

simple clarification of equivalence relation and order relation notation meaning

So by definition an equivalence relation is a binary relation on a subset of the cartesian product of our set A, i.e $A\times A$ satisfying the 3 conditions. But I'm a little confused about ...
0
votes
1answer
16 views

More clarification on an equivalence relation problem already answered

So this problem already has a solution: Problem with Equivalence Relations I'm good for the majority of it except for part c), I wasn't able to figure that out on my own or by looking at the ...
1
vote
1answer
39 views

Isn't reflexivity and symmetry implied in equivalence relations?

It looks like for all "nice" sets, the set $S\times S$ will have symmetry and reflexivity by default. The tough part is usually showing transitivity. However, are there any non-empty sets such that ...
1
vote
1answer
37 views

How to understand this definition of equivalence relations

I often see this type of definitions of equivalence: Suppose that $f$ and $g$ are differentiable on $\mathbb R$. We can define an equivalence relation on such functions by letting $f(x) \sim g(x)$ ...
4
votes
1answer
32 views

Defining Equivalence relations

So I am not really comfortable with equivalence relations, so this example from Wikipedia gives me trouble. Here is what it says: Let the set $\{a,b,c\}$ have the equivalence relation ...
2
votes
1answer
24 views

Intersection of two sets which are equivalence on set A is always equivalence?

If $R_{1}$ and $R_{2}$ are equivalence relations on set A ,then$ R_{1}\bigcap R_{2}$ must be equivalence relation. firstly, I am not understanding the function of R,I think that, this is only a ...
3
votes
2answers
30 views

why are equivalence relations called so?

"an equivalence relation is the relation that holds between two elements if and only if they are members of the same cell within a set that has been partitioned into cells such that every element of ...
0
votes
1answer
15 views

Transitive Relations Problem,

Let S be the set of all three-digit numbers, and define x~y to mean that x and y have the same first and last digit. (i) Show that the relations ~ is transitive. (ii) List two numbers in the ...
2
votes
2answers
25 views

Equivalence Classes Output

I understand that an equivalence relation is a set that is reflexsive, symmetric, and transitive. I don't quite understand equivalence classes though. For example: What would the equivalence class be ...
0
votes
2answers
36 views

Why is this an equivalence relation, and what does the equivalence classes contain?

I'm doing some discrete mathematics exercises, but I can't seem to wrap my head around this relation: $$R(x, y) \text{ if } \exists z(\text{LiesInPart}\circ\text{LiesInCountry}(x,z) \wedge ...
1
vote
1answer
30 views

How to determine an equivalence class?

Let $A$ be a nonempty set and let $B$ be a fixed subset of $A$. A relations $R$ is defined on the power set $\mathcal{P}(A)$ by $X\mathrel{R}Y$ if $X \cap B = Y \cap B$. Let $A=\{1,2,3,4,5\}$ and ...
1
vote
2answers
22 views

Relations and Equivalence - numbers are related if they have the same floor

$S$ is defined on $\mathbb{Q}$ by $xSy$ if and only if $⌊x⌋=⌊y⌋$ (Note that$⌊q⌋$is defined to be the largest integer less than or equal to q. You can think of it as “$q$ rounded down”.) We've been ...
0
votes
1answer
19 views

Equivalence relations and classes

$T$ is defined on $\mathbb{N} \times \mathbb{N}$ by $(a,b)\mathrel{T}(c,d)$ if and only if $a \leq c$ and $b \leq d$. I know this is a partial order relation as it is Transitive, Anti Symmetric and ...
0
votes
1answer
15 views

Describing equivalence classes over the set of natural numbers

So for this problem we are supposed to describe the equivalence class for the following relation. $x \thicksim y$ iff $x$ mod 2 = $y$ mod 2 and $x$ mod 4 = $y$ mod 4. I am confused on what is meant ...
1
vote
1answer
16 views

verifying properties of relations to test equivalence

We are doing some more with relations and this time we are given a relation and told that it is not equivalent. We need to find out which property it does not fulfill. So we at least know there is ...
0
votes
2answers
37 views

Can someone explain how to find the number of equivalence classes and elements?

I am struggling so much with this topic. Trying to do some practice questions but I don't seem to get it. What I'm working on is Let $A = \{ 1, 2, 3, \dots, 2014 \} = \{ x \mid 1 \le x \le ...
0
votes
1answer
23 views

Proving that two equivalence classes are disjoint?

I am having trouble with the following proof: Define the relation $R$ on $\mathbb{Z}$ by $nRm$ if $n-m$ is divisible by $2$. Prove that the equivalence class for $0^{(\bar{0})}$ and the equivalence ...
0
votes
1answer
37 views

Which equivalence class represents the zero element $0_{\mathbb Q}$ in $\mathbb Q$?

The Statement of the Problem: We identify $\mathbb Q$ with the set of equivalence classes $[a,b]$, where $(a,b) \in \mathbb Z \times \mathbb N^+$ and $(a,b) \sim (a'b')$ iff $ab'=ba'$. We define ...
1
vote
1answer
23 views

Union of equivalence relations

As part of a HW assignment in the course elementary set theory, I was given the following question: Let $A$ be a set and let $T$ and $S$ be two equivalence relations on $A$. Prove: $S\circ ...
1
vote
1answer
19 views

Equivalence Relation on the set of ordered pairs of positive integers

Have a homework question, but how can I show that the given relation R is reflexive, symmetric and transitive, so that it is an equivalence relation. Appreciate assistance from anyone. "Let R be the ...
1
vote
1answer
28 views

Composition of equivalence relations

As part of a HW assignment in the course elementary set theory, I was given the following question: Let $A$ be a set and let $T$ and $S$ be two equivalence relations on $A$. Prove: $S\circ ...
0
votes
1answer
31 views

Help with equivalence classes for $x\sim$y iff $x-y\in\mathbb{Q}$.

Help with equivalence classes for $x\sim$y iff $x-y\in\mathbb{Q}$. I need to show the equivalence classes for $[0]_{\sim}$ and $[\sqrt{2}]_{\sim}$. Here is what I did: $[0]_{\sim}$ = ...
0
votes
0answers
5 views

Is bisimilarity an equivalence relation

I want to know if bisimilarity is an equivalence relation. I need to make a proof showing that this is true but I have searched and I can only find for branching bisimilarity.
0
votes
1answer
28 views

Determining a relation if reflexive, symmetric, and transitive

I just get stuck in this relation and need to find if this relation is Reflexive/ Irreflexive or Neither, Symmetric/ Antisymmetric or Neither, Transitive or Not. $$W_1 = \{(a , b) \in \mathbb ...
2
votes
1answer
23 views

The equivalence relation generated by a relation

Let $X$ be a non-empty set and let $r\subseteq X\times X$ be a relation on $X$. Let $R$ be the intersection of all equivalence relations on $X$ that contain $r$. Prove that if $xRy$, then one of the ...
0
votes
1answer
35 views

Partition and equivalence relation

Consider the equivalence relation between non-empty subsets $A , B$ of $\{ 1,2,3, 4,\dots,100\}$ defined by the condition: the greatest element of $A$ is the same as the greatest element of $B .$ ...
0
votes
1answer
23 views

Make the set $R$ transitive

Let $R$ be a relation on a set $A=\{w,x,y,z\}$ defined by $R=\{(w,x),(y,x),(x,y),(z,z)\}$. Using the original relation, $R$, make the necessary minimal additions to make $R$ transitive. I thought ...
1
vote
0answers
33 views

quick question about a definition of homeomorphism set classes

Take $S$ a surface of general type. I want to define $Q$ the set of homeomorphism classes determined by the surface $S$. How can i define $Q$? I think that $Q$ is determined by all surfaces $S^{'}$ ...
0
votes
3answers
36 views

Finding the equivalence classes of a relation R

Let A = {0,1,2,3,4} and define a relation R on A as follows: R = {{0,0},{0,4},{1,1},{1,3},{2,2},{3,1},{3,3},{4,0},{4,4}}. Find the distinct equivalence classes of R. How do I solve this problem? ...
0
votes
1answer
33 views

Proving Equivalence Relations On A Set

Let X be the set of all nonempty subsets of {1,2,3}. Then X = {{1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}} Define a relation R on X as follows: for all S and T in X, SRT if, and only if, the least ...
3
votes
1answer
27 views

Describe the equivalence classes for each equivalence relation

Let ~ be an equivalence relation on $\mathbb R^2$ such that $\left( x_1, y_1 \right)$ ~ $\left(x_2, y_2 \right)$ iff $y_1=y_2$. Let ~ be an equivalence relation on $\mathbb R^2$ such that $\left(x_1, ...
0
votes
0answers
28 views

Show that same cardinal number defines and equivalence relation

Same cardinal number must satisfy all three properties : symmetry, reflexivity, transitivity Symmetry Suppose bijection $f:A→A$, Then by definition, |A| = |A| Reflexivity Suppose bijections ...