Tagged Questions
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], ...
2
votes
1answer
43 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 ...
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 $\Bbb Z$. ...
3
votes
2answers
52 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
3answers
71 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 ...
0
votes
2answers
87 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
...
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
60 views
Proving that $R$ is an equivalence relation.
Let $A$ be the set of all statement forms in three variables $p$, $q$, and $r$. Let $R$ be the relation defined on $A$ as follows: For all $P$ and $Q$ in $A$,
$$P\; R\; Q \longleftrightarrow P\; ...
2
votes
3answers
103 views
Is $f:\mathbb{Z}_{30}\longrightarrow\mathbb{Z}_{30}$ defined by $f([a])=[7a]$ well defined?
To tell the truth, I'm not even sure what this means.
The professor gave an example saying that $\mathbb{Z}_m=\{[0],[1],[2],\dots,[m-1]\}$, and I sort of understand that.. but I have no idea what ...
4
votes
5answers
74 views
How do I work with a relation that is a set of 4-tuples?
Define the relation $\sim$ on $\mathbb{Z}\times\mathbb{Z}$ by $(a,b)\sim(c,d)$ if $a-c=b-d$. Show that $\sim$ is an equivalence relation. What is the equivalence class of $(1,2)$?
I'm not sure ...
1
vote
1answer
82 views
Show that an equivalence relation is equal to the union of its equivalence classes
Given an equivalence relation $\sim$ with equivalence classes $C_1,\dots,C_n$, show that $$\mathbin{\sim} = \bigcup_{i=1}^n(C_n\times C_n)\;.$$
I could use a hint on where to start approaching this ...
0
votes
1answer
131 views
Equivalence Relation? Column-equivalence on the set of all $m\times n$ matrices.
How do I show that column equivalence on the set of all $m\times n$ matrices is an equivalence relation?
I know that to show it is an equivalence relation, I need to show that column equivalence is ...
1
vote
2answers
129 views
Equivalence Relation problem
Let $S$ be the relation on $\mathbb{N} \times \mathbb{N}$ defined by $(a,b)S(c,d)$ if and only if $ad=bc$. Prove this is an equivalence relation on $\mathbb{N} \times \mathbb{N}$.
I think I've found ...
3
votes
4answers
292 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
1answer
87 views
Is the relation $\{ (a,b) \in\Bbb Z^2 : |a-b| \le 10 \}$ an equivalence relation or not ??
Is the relation ↖ an equivalence relation or not ??
I know we are supposed to prove that it is reflexive , transitive and symmetric . I found that it is an equivalence relation but i am not sure , so ...
2
votes
3answers
49 views
Disjoint Equivalence
Why do equivalence classes, on a particular set, have to be disjoint? What's the intuition behind it? I'd appreciate your help
Thank you!
1
vote
3answers
112 views
Describing A Congruence Class
The question is, "Give a description of each of the congruence classes modulo 6."
Well, I began saying that we have a relation, $R$, on the set $Z$, or, $R \subset Z \times Z$, where $x,y \in Z$. The ...
1
vote
1answer
30 views
The Importance Of Equivalences
Okay, I asked a question earlier today, Congruence Class, pertaining to finding equivalence classes. I already know how to solve such problems, now my question is, what is the importance of ...
1
vote
2answers
114 views
Congruence Class
The question is, "What is the congruence class$[n]_5$ (that is, the equivalence class of $n$ with respect to congruence modulo 5) when n is
a) 2?
b)3?
c) 6?
d)−3?"
I know this is more work ...
1
vote
3answers
417 views
Equivalence Relation On A Set Of Ordered-Pairs
The question is, "Let R be the relation on the set of ordered pairs of positive integers such that $((a, b), (c, d)) ∈ R$ and only if $a+d=b+c$. Show that R is an equivalence relation."
There are two ...
1
vote
3answers
221 views
Finding The Equivalence Class
Okay, so the question I am working on is, "Suppose that A is a nonempty set, and $f$ is a function that has A as its domain. Let R be the relation on A consisting of all ordered pairs $(x, y)$ such ...
3
votes
2answers
258 views
Equivalence Relations On A Set of All Functions
The question is, "Which of these relations on the set of all functions from $Z$ to $Z$ are equivalence relations."
The first relation to consider is, $\{(f,g)|f(0)=g(0)\vee f(1)=g(1)\}$
For this one, ...
0
votes
1answer
89 views
Equivalence Relations On A Set of All Functions From Z to Z
The question is, "Which of these relations on the set of all functions from $Z$ to $Z$ are equivalence relations.
$\{(f,g)|f(1)=g(1)\}$
I just want to make certain that I am interpreting this ...
1
vote
1answer
43 views
Deciding If A Relation On A Set Is An Equivalence Relation
The relation I am looking at is $\{(0,0),(1,1),(1,3),(2,2),(2,3),(3,1),(3,2),(3,3)\}$, and is on the set $\{0,1,2,3\}$
Apparently, the only thing that does not qualify this as an equivalence relation ...
1
vote
3answers
110 views
Equivalence Class Definition
I am currently reading about the subject given in the title of this thread. The definition they give for equivalence classes in my textbook is a rather ostentatious in its wording, so I just want to ...
3
votes
3answers
106 views
$x,y$ related if and only if $x\cap\{{1,3,5\}}=y\cap\{{1,3,5\}}$.
Let A be the power set of $\{1,2,3,4,5\}$, let $z= \{1,2,3\}$, and let $(x,y) \in R$ if and only if
$$x \cap \{1,3,5\} = y \cap \{1,3,5\}$$
I'm supposed to find the equivalence class, number of ...
0
votes
1answer
94 views
Equivalence Classes and such
Let A be the set of all possible strings of 3 or 4 letters in alphabet ${A,B,C,D}$, let z = $BCAD$, and let $(x,y)\in R$ if and only if $x$ and $y$ have the same first letter and the same third letter
...
1
vote
2answers
251 views
Equivalence Relations and Partitions
My university "textbook" for discrete math is Schaum's Outline. In this outline he goes over Equivalence Relations and Partitions, and I got confused at a particular theorem.
From the book:
Theorem ...
0
votes
1answer
97 views
Equivalence Relations with inverses?
I have no Ida how to approach this problem:
Suppose S is a relation on a set X which is reflexive and transitive. Then S intersection S inverse is an equivalence relation on X.
Any idea on how I ...
2
votes
2answers
127 views
Equivalence Relation problem?
I need help with this problem:
Suppose $\sim$ is a relation on a set $S$ which is both symmetric and transitive. Let $A = \{x∈S\ \vert \text{ for some }y∈S, x\sim y\}$. Prove that $\sim$ is an ...
