For reflexive, symmetric and transitive relations. Use it with the tag (relations).
3
votes
1answer
43 views
Determining if $R=\left\{(f,g)\mid \exists k\in\Bbb Z,\forall x\in\Bbb Z, \ f(x)g(x)\lt k\right\}$ is an equivalence
I need help with proving whether or not the relation $R=\left\{(f,g)\mid \exists k\in\Bbb Z,\forall x\in\Bbb Z, \ f(x)g(x)\lt k\right\}$ is an equivalence relation on $\mathbb{Z}\times\mathbb{Z}$
I ...
1
vote
1answer
44 views
Determining if this relation is an equivalence relation
$$R=\left\{(f,g)\,\Bigg|\, \exists c\in\Bbb Z,\forall x\in\Bbb Z, \frac{f(x)}{g(c)}\le 2\right\}$$
I can show that this relation is reflexive by showing that $(f,f)$ is in $R$
and so $f(x)/f(c) \le ...
2
votes
2answers
23 views
Equivalence Relations and functions on partitions of Sets
let $f$ be a function on $A$ onto $B$. Define an equivalence relation $E$ in $A$ by: $aEb$ if and only if $f(a)=f(b)$.
Define a function $\phi$ on $A/E$ by $\phi([a]_{E})=f(a)$.
Hint: Verify that ...
7
votes
1answer
82 views
Is there a name for relations with this property?
Is there a name for relations $\rho : X \rightarrow Y$ such that for all $x,x' \in X$ and all $y,y' \in Y$ we have that the following conditions $$xy \in \rho$$ $$x'y \in \rho$$ $$xy' \in \rho$$ imply ...
1
vote
1answer
40 views
Homeomorphism on Identification Space
Let $\sim$ be and equivalence relation on the unit line $X=[0,1]$ defined by $x\sim y$ if either $x=y$ or $\textbf{both}$ $x$ and $y$ $\in$ {${0,\frac{1}{2},1}$}.
Construct a homeomorphism ...
2
votes
0answers
39 views
Semi-orbital equivalence relation
Edit: I was in kind of a hurry when writing this post and made a mistake in the formula defining $G_E$. What I had written said that $G_E$ preserves the set of classes of $E$, while I meant actually ...
2
votes
2answers
36 views
Let $A$ be a set, $R$ an empty relation on $A$, what is $A/R$?
Let $A=\{0,1,2\}$ be a set and $R=\{\}$. I know that $R$ is not an equivalence relation, but does it have to be? What is $A/R$ if $R$ is empty?
Examples:
$R_1=\{(0,0),(1,1),(2,2)\}$, $A/R_1=\{[0], ...
0
votes
1answer
44 views
Equivalence relation classifying the slopes of the tangent lines to the curve?
We apply the Mean value theorem to a real analytic function $f$ (defined on $\mathbb R$) in the interval $(u,a)$ such that $f(u)=0$ and $f(a)\neq 0$ to find a $c\in(u,a)$ such that: the expression ...
3
votes
2answers
40 views
Uncountability of the equivalence classes of $\mathbb{R}/\mathbb{Q}$
Let $a,b\in[0,1]$ and define the equivalence relation $\sim$ by $a\sim b\iff a-b\in\mathbb{Q}$. This relation partitions $[0,1]$ into equivalence classes where every class consists of a set of numbers ...
2
votes
1answer
44 views
Proving this realtion is not a transitive relation
I have trouble proving how the following statement is false:
The relation $g = \{\,(x,y)\in \Bbb R\times\Bbb R\mid y = x^2\,\}$ is transitive.
I know you have to use $yRx$, $zRy$, and $xRz$, but I'm ...
1
vote
3answers
100 views
If a relation is reflexive is it symmetric and transitive?
If a relation is reflexive is it symmetric and transitive ?
let ~ means " in relation with "
if A is a set , ~ is a relation on $A$, prove that:
if $a$~$a$ for any $a$ $\in$ A then
1- $x$~$y$ ...
1
vote
0answers
82 views
Is $ℜ$ an equivalence relation?
The mean value theorem says that there exists a $c∈(u,v)$ such that $$f(v)-f(u)=f′(c)(v-u).$$ Here is my question.
Assume that $u$ is a root of $f$, hence we obtain $$f(v)=f′(c)(v-u);$$ assume that ...
1
vote
2answers
35 views
Integer proof equivalence class
I've been searching online but I couldn't find help on this matter.
How can I prove that $[(a,b)]+[(c,d)]=[(a+c,b+d)]$ is independent of the choice I make of representatives of the equivalence ...
4
votes
1answer
102 views
Mean values theorem and countable sets
The mean values theorem says that there exists a $c∈(u,v)$ such that $$f(v)-f(u)=f′(c)(v-u)$$ My question is: Assume that $u$ is a root of $f$, hence we obtain $$f(v)=f′(c)(v-u)$$ Assume that $f$ is a ...
3
votes
2answers
36 views
Functional relations : Trouble seeing transitivity
I've been given the following domain: $\;\{1,2,3,4\}$
And the following relation:
$$\{(1,1),(1,3),(1,5),(2,2),(2,4),(3,1), (3,3),(3,5),(4,2),(4,4),(5,1),(5,3),(5,5)\}$$
It states that this is an ...
3
votes
2answers
50 views
“Tricky” wording on Congruence Modulo Question?
I'm asked for all possible values, but I can only see one. The question on my practice exam reads:
Consider the equivalence class [3] for the equivalence relation "congruence modulo 7" on $\Bbb Z$. ...
0
votes
2answers
47 views
Need help counting equivalence classes.
I am having trouble wrapping my head around the concept of equivalence classes. Here is the question:
Let $X$ be the set of all nonempty subsets of the set $\{1,2,3,...,10\}$. Define the
relation $R$ ...
2
votes
1answer
81 views
What is the standard notation for a set of equivalence classes?
What is the standard notation for a set of equivalence relations? Specifically, I have a pair of objects, call them $x$ and $y$ and I denote the ordered pair as $\left(x,y\right)$. I have a set of ...
1
vote
2answers
34 views
Equivalence relation proofs: general or specific?
I'm confused about whether a specific example must exist to prove an aspect of an equivalence relation.
For example: if a set, $A$, only contains one element, $A = \{1\}$, and a relation, $R$, on ...
1
vote
0answers
29 views
Simple counting question related to equivalence classes
Let $S = \{1,2,3,...,10\}.$ Define the relation $\mathscr R$ on the power set $\mathscr P(S)$ of all subsets of $S$ by: for all $A,B \in \mathscr P(S),A\mathscr RB$ if and only if $N(A) = N(B)$.
...
1
vote
3answers
78 views
Equivalence Relations: Equivalence Classes
From my basic understanding $R$ is an equivalence relation on the set $A$, which is
a relation between elements of a set that is reflexive, symmetric, and transitive.
I am not sure how to find the ...
3
votes
2answers
53 views
Proving if a relation is an equivalence relation
I have been able to figure out the the distinct equivalence classes. Now I am having difficulties proving the relation IS an equivalence relation.
$F$ is the relation defined on $\Bbb Z$ as follows: ...
1
vote
1answer
20 views
Numerical equivalence between sets
I need some help on homework. Here is the problem I am stuck on:
Prove that every closed interval [a,b] is numerically equivalent to [0,1]
I believe that I need to find an injection between the two ...
0
votes
1answer
58 views
Rational Numbers and Equivalence Classes
Describe the rational numbers as the equivalence classes for an equivalence relation on certain pairs of integers.
0
votes
0answers
35 views
Help on equivalence relation [duplicate]
Hi guys I have a exam in 2 days and the teacher gave us practice problems to do to prepare for the exam.
I have some problems on equivalence classes that I really dont get. I know I have to show ...
4
votes
2answers
131 views
Counting the number of functions
Let $A$ and $B$ be subsets of the set $\Bbb Z$ for all integers, and let $\mathscr F$ denote the set of all functions $f:A\rightarrow B.$ Assume $A = \{1,2,3\}$ and $B=\{1,2,...,n\}$ where $n\ge 2$ is ...
0
votes
1answer
62 views
Proving $\;x-y \in2\pi \mathbb Z\;$ defines an Equivalence Relation
Prove that $\;x-y \in2\pi \mathbb Z\;$ defines an equivalence relation on $\;\mathbb R.$
2
votes
0answers
56 views
Determine and prove if the following are equivalence relations, partial ordering relations, or neither.
{$(a, b) | a∈Z^{+}, b∈Z^{+}, a = 3^{n}b, where n∈N$} (N is the set of natural numbers)
{$(a, b) | a∈Z^{+}, b∈Z^{+}$, a >2b or b >2a}
{$(a, b) | a∈Z^{+}, b∈Z^{+}$, a ≡ 0 (mod b) or b ≡ 0 (mod a)}
1
vote
0answers
46 views
Prove that the following relation is an equivlance relation.
Prove that the following relation is an equivalence relation and determine how many equivalence classes R partitions the set $Z^{+}$ into.
R = {$(a,b) | a∈Z^{+} ∧ b∈Z^{+} ∧ 10 | (a^{2}- b^{2})$}
Any ...
0
votes
1answer
21 views
defining a relation on the set of reals. proving that it is an equivalence relation and compelling description
Define a relation $\sim$ on the set $\textbf{R}$ of the real numbers by setting $a\sim b \iff b-a \in \textbf{Z}$. Prove that this is an equivalence relation, and find a compelling description for ...
5
votes
2answers
64 views
Sets, functions and relations problem
Yes this is a homework problem but I have attempted to solve it and my work is below, also this is my first question here so I'm sorry for any mistakes:
Question:
Context:
Let $A$ and $B$ be subsets ...
1
vote
1answer
53 views
Quick equivalence class clarification question
A quick clarification question, what is an equivalence class of a function? For example if you have an identity function on all integers $I_{Z}$, what would $[I_{Z}]$ = ? I know that when you have a ...
0
votes
2answers
90 views
Function and equivalence relations question
Let A and B be subsets of the set Z of all integers, and let F denote the set of all functions
f : A to B. Define a relation R on F by: for any f,g element of F, fRg if and only if f - g is a
...
2
votes
2answers
95 views
Proving two functions are equal
Let $A$ and $B$ be subsets of the set $\Bbb Z$ for all integers, and let $\mathscr F$ denote the set of all functions $f:A\rightarrow B.$ Assume $A = \{1,2,3\}$ and $B=\{1,2,...,n\}$ where $n\ge 2$ is ...
2
votes
2answers
59 views
Given a relation $R$, is it reflexive? Symmetric? Transitive?
Define the relation $R$ on the set $\mathbf Z^+$ of all positive integers by: for all $a,b \in \mathbf Z^+,aRb$ if and only if $gcd(a,b)\gt 1$.
(a) is $R$ reflexive? Symmetric? Transitive?
so here ...
2
votes
1answer
118 views
find the number of equivalence classes of $\mathbb R$.
Let $\mathscr X$ be the set of all nonempty sub sets of the set $\{1,2,3,...,10\}. $Define the relation $\mathscr R$ on $\mathscr X$ by: for all $A,B \in \mathscr X, A\mathscr RB$ if and only if the ...
1
vote
2answers
237 views
Find the number of distinct equivalence classes $[f]$ of $R$.
Let $A$ and $B$ be subsets of the set $\Bbb Z$ for all integers, and let $\mathscr F$ denote the set of all functions $f:A\rightarrow B.$
Define a relation $R$ on $\mathscr F$ by : for any $f,g ...
0
votes
1answer
30 views
Number of Equivalence Classes
Let $M=\{1,2,\ldots,20\}$ and define a function $f:M\to \mathbf{Z}$ by $f(x)=\min(x,3)$. Define an equivalence relation on M by letting two element $m$ and $n$ be equivalent if $f(m)=f(n)$. 1) How ...
4
votes
5answers
74 views
How do I work with a relation that is a set of 4-tuples?
Define the relation $\sim$ on $\mathbb{Z}\times\mathbb{Z}$ by $(a,b)\sim(c,d)$ if $a-c=b-d$. Show that $\sim$ is an equivalence relation. What is the equivalence class of $(1,2)$?
I'm not sure ...
3
votes
7answers
116 views
Define a quotient structure for $(a,b)\sim(c,d)$ iff $a+d=b+c$ in terms of $+$ and $\times$ operators
This is a past paper exam question, I don't really understand how to define this quotient structure.
The relation is on $\Bbb{N}$
In the preceding question I proved that this relation is an ...
0
votes
0answers
26 views
Equivalence relation question [duplicate]
Let $A$ be the set of all bit strings of length 12. Let $R$ be the relation define on $A$ where two bit strings are
related if the first 2 bits, the 4th bit and the 7th bit are the same. Show that $R$ ...
0
votes
1answer
58 views
Equivalence relation and its equivalence classes
Let $X$ be the set $\{1,2,3,4\}$ and also that
$$R = \{(1,1),(1,2),(1,3),(1,4),(2,2),(2,3),(2,4),(3,3),(3,4),(4,4)\}$$
How do I how that $R$ is an equivalence relation; and also its equivalence ...
0
votes
1answer
46 views
Logic: Equivalence Relation
Let $\Gamma$ be a maximally consistent set of formulas of $\mathcal{L}$. For any two terms $\tau_1$ and $\tau_2$ of $\mathcal{L}$, define that $\tau_1 \cong_\Gamma \tau_2$ if and only if ...
0
votes
2answers
65 views
Is $R$ an equivalence relation on $\mathbb Z^+$: $\;x\,R\,y \;\text{ if and only if}\;\gcd(x,y)\neq 1.$
Let $\,R\,$ be a relation $R: \{(x, y) \mid \gcd(x, y) \neq 1\}$ defined on the set of positive integers. So $$\;x\,R\,y \;\;\text{ if and only if}\;\;\gcd(x,y)\neq 1.$$
I need to figure out if is ...
3
votes
4answers
293 views
Proving an equivalence relation on $\mathbb{Z}\times\mathbb{Z}$
I'm working on some discrete mathematics problems, and have run into an issue involving proving an equivalence relation.
The relation I'm tasked with proving is the relation $R$ defined on ...
0
votes
2answers
172 views
How do you show that $R$ is an equivalence relation and enumerate one bit string from each of the different equivalence classes of $R$?
If $A$ is the set of all bit strings of length $12$. Let $R$ be the relation define on $A$ where two bit strings are related if the first $2$ bits, the $4^{\text{th}}$ bit and the $7^{\text{th}}$ bit ...
4
votes
0answers
42 views
Equiv Relation of Orbits - Group Action [duplicate]
Let $G$ be a group that acts on $X$. I want to show that the orbits of $G$ partition $X$. I am given the relation $x\sim y \iff x\in Orb(y)$. Now:
$x\sim y\iff x\in Orb(y) \iff x=gy$ for some $g\in G ...
0
votes
1answer
32 views
Proving equivalence relations with binary operations
I am having trouble with this homework problem:
Let $A$ be a set and $*$ be an associative binary operation on $A$ with the identity element $e$. Let $R$ be the relation on $A$ defined as ...
3
votes
2answers
65 views
Equivalence classes of “$x \sim y \Longleftrightarrow x -y $ is rational”.
Given the equivalence relation $x \sim y \Longleftrightarrow x -y $ is rational on the interval $[0,1)$.
How do we reason* that there are uncountably infinite number of equivalence classes?
*A ...
0
votes
1answer
37 views
For a group $G$ and subgroup $H$, is $a \sim b \iff a^{-1}b\in H$ an equivalence relation even when $H$ is not normal?
Is it true or false that defining a relation on the group $G$ based on the definition $a \sim b$ if and only if $a^{-1}b\in H$ defines an equivalence relation regardless of whether $H$ is a normal ...
