Tagged Questions

2
votes
2answers
21 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 ...
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
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 ...
1
vote
3answers
98 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- $x$~$y$ ...
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 $\Bbb Z$. ...
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 ...
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
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
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
2answers
92 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
56 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
117 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 ...
0
votes
1answer
57 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
49 views

Transformation or relation

Could anyone please chack my task on equivalence relation? THank you!! In this task it says: if $f\colon X \to Y$ is a transformation. We define the relation $R$ to $X$ : $x'\sim_R x'' ...
0
votes
2answers
71 views

Showing a relationship is transitive

I can't figure out why the following relationship is transitive: Consider the relation $R=\left\{(a,a),(b,b),(c, c),(d,d),(a,b),(b,a)\right\}$ on set $A =\left\{a,b, c,d\right\}$. Is $R$ ...
3
votes
2answers
69 views

Working with Equivalence Classes and Quotient Sets

I have a doubt about working with equivalence classes and quotient sets. The definition that I know, is that given an equivalence relation $\sim$ on a set $A$, the set of all elements of $A$ ...
3
votes
3answers
73 views

Confusion on understanding a proposition on equivalence classes

I am given to prove this proposition on equivalence classes. Each element of $A$ is an element of one and only one equivalent class. The part that is confusing is one and only one. It sounds ...
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 ...
-1
votes
2answers
79 views

Prove that two equivalence classes of two elements are equal iff the elements are related

Show that for equivalence relation $\sim$ on the set $A$ and $a,b\in A,[a]=[b] \iff a\sim b$. This is an advanced practice problem.
1
vote
2answers
56 views

Permutations and Equivalence Relations

Let X be a nonempty set and let $\sigma \in$ Sym(X). Define the two place relation $\sim$ on X as follows: x$\sim$y if and only if $\sigma^{k}(x)=y$ for some integer k. Prove that $\sim$ is and ...
0
votes
4answers
105 views

Equivalence Relations: Abstract Algebra

I need someone to check my proof. Question: On the set $\mathbb{R}^2$ of ordered pairs define the 2-plane relation $\sim$ as follows $(a,b)\sim(c,d)$ if and only if $a^2+b^2=c^2+d^2$. Prove that ...
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 ...
3
votes
4answers
843 views

Understanding equivalence class, equivalence relation, partition

Im having difficulty grasping a couple of set theory concepts, specifically concepts dealing with relations. Here are the ones I'm having trouble with and their definitions. 1) The collection of ...
0
votes
1answer
51 views

Am I correct? State the necessary and sufficient condition for R to be an equivalence relation on A.

Let R be a relation on a given nonempty set A. State the necessary and sufficient condition for R to be an equivalence relation on A. My attempt The conditions for any equivalence relation are ...
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!
0
votes
1answer
54 views

Equivalence-relations question.

Show that for all class $ \{A_i \}_{i\in I}$ of $A $, the relation $T$ is equivalence relation where $T$ is defined to be: $xTy$ iff exist $i\in I$ such that $\{ x,y\}\subseteq A_i$. My attempt: ...
1
vote
1answer
60 views

Union of equivalence relation

Let $S$ be a equivalence relation on $A$, let $B$ be a subset of $A$ and suppose that $T$ equivalence relation on $B$. Defining $R=S \cup T$. Its given that there is exist $x\in A$ such that $B$ is ...
1
vote
1answer
64 views

Verifying this equivalence relation proposition

$\def\class#1{\mathopen{[\![}#1\mathclose{]\!]}}$Proposition: If $\sim$ is an equivalence relation on $A$ and $a,b\in A$, then either $\class a \cap \class b = \emptyset$ or $\class a = \class b$. ...
3
votes
2answers
61 views

Having the same equivalent class imply that the elements are equivalent?

You have given that the equivalence class of $x$ and the equivalence class of $y$ is equal. $[x]=[y]$. Does this imply that $x\sim y$? If yes, how do I prove it, if no, what sort of counter-examples ...
1
vote
1answer
190 views

Equivalence Relations and Equivalence Classes

We will define the relation ~ on $\mathbb N \times \mathbb N$ by $(a,b)\sim (c,d)$ iff $a + d = b + c$. Prove that the operation given by: $[(a,b)][(c,d)] \stackrel{\text{def}}= [(ac + bd, ad + bc)]$ ...