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

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1answer
23 views

Finding equivalence classes in a set of functions with a condition

I can't seem to work around this problem... Let $n$ and $m$ be two positive integers. $F$ is the set of functions from {1,..,n} to {1,...,m}. We define the relation $R$ as: $f R g$ if and only if ...
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0answers
26 views

Does $\lim\limits_{x\rightarrow a}(f-g)(x)=0$ then $f(x)\sim_{x\rightarrow a}g(x)$ and revert?

Why does $\lim\limits_{x\rightarrow a}(f-g)(x)=0$ then $f(x)\sim_{x\rightarrow a}g(x)$ false and $f(x)\sim_{x\rightarrow a}g(x)$ implies $\lim\limits_{x\rightarrow a}(f-g)(x)=0$ false? I was given ...
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0answers
35 views

Equivalence Relations and Cardinality

I'm looking at the question below from a past paper: What is an equivalence relation? Say that two sets $X$ and $Y$ are related via the relation $\rho$ if $X$ and $Y$ have the same cardinality. Prove ...
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1answer
14 views

Explain this step in solving this system of linear congruences.

I'm looking at this example and it doesn't make sense to me. We have to solve the following systems of linear congruences : $x\equiv 1\pmod 5$ $x\equiv 2\pmod 6$ $x\equiv 3\pmod 7$ We take ...
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4answers
57 views

Equivalence relation on $\mathbb{N}$ with infinitely many equivalence classes

Does there exist an equivalence relation, defined on $\mathbb{N}$, with infinitely many equivalence classes, all of which contain infinitely many elements? I see no reason for such a relation not to ...
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1answer
14 views

Relations and restriction of a function.

This is a homework question: "Let R be an equivalence relation on a set S. For A ⊆ S, we define RA to be the restriction of R to elements of the set A, i.e., RA is a relation on A such that for any a,...
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2answers
23 views

Equivalence relations on intersections

Let $R_1$ and $R_2$ be equivalence relations on a set $S$. Their intersection is on the relations considered as sets of ordered pairs. Identify the equivalence classes of the relation $R_1 \cap R_2$. ...
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1answer
13 views

Which language L have exactly one equivalence class

Consider the alphabet {a,b}, for which language does the equivalence relation R have exactly one equivalence class? From what i understand about equivalence class, each state is consider a class. So ...
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1answer
46 views

How to describe an equivalence class?

For example: the relation given is $x\sim y$ if $f(x)=f(y)$. What do you have to say when describing a equivalence class?
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1answer
78 views

Prove that $f : \mathbb R \smallsetminus \{−1\} \to\mathbb R \smallsetminus \{1\}$ given by $f(x) = \frac{x − 3}{x + 1}$ is bijective

I know for a function to be bijective it must be one to one and onto. Here's what I have Take by cases Case 1 (one to one) $$ \begin{align*} \frac{x-3}{x+1} &= \frac{y-3}{y+1} \\[1ex] (x-3)(y+1)...
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1answer
17 views

Cocycle condition on bundles as some equivalence relation?

The three cocycle conditions on the transition maps of a fiber bundle look alot like the three condition defining equivalence relation - refelxivity, symmetry, and transitivity. I was wondering ...
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1answer
81 views

Cardinality of equivalence relations in $\mathbb{N}$

I came across this long proof on this site: Cardinality of relations set But I would like to know whether my direction can work. Say we want to find the cardinality of all equivalence relations in $...
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2answers
71 views

Show that R is an equivalence relation and determine all distinct classes

Let R be a relation on Z define as follows: m R n <--> 3|($m^2$-$n^2$) show that R is an equivalence relation and determine all distinct equivalence classes. EDIT: I looked several places and ...
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0answers
32 views

prove that for any pair of natural, there is a power of 2 that separates the pair of natural

i need to prove that: $$ \forall i,j \in \{1, \_ ,N \} \subset \mathbb{N} \ \exists k \in \mathbb{N} / (r_{2^k}(i) \leq 2^{k-1} \wedge r_{2^k}(j) > 2^{k-1})\vee (r_{2^k}(i) > 2^{k-1} \...
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1answer
27 views

equivalence relation and sets

The question: Let $X$ and $Y$ be two sets, and let $S$ be an equivalence relation on set $X$ and $T$ be an equivalence relation on set $Y$. Define a relation $R$ on $X ×Y$ by $(a,b)R(c,d)$ if and ...
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1answer
24 views

Equivalence classes in the logical equivalence on some finite set of propositional formulas

I'm having trouble understanding the following problem: Let $S_n$ be the set of all formulas that can be built up with the atoms $\{A_1,...,A_n\}$. How many equivalence classes does the ...
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1answer
26 views

Equivalence relations and composition, intersections of them

I've been having some trouble with this one, I hope someone get's it. Let S and R be equivalence relations within X. Prove that if R∘S is an equivalence relation, then it is equal to the ...
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44 views

Lagrange's theorem intuition

I cannot grasp the intuition behind |G|/|H|=[G:H]. Starting from the equivalence relation x~y if and only if x^(-1)*y is in H, I can see a sort of division, but in my mind, the equivalence relation ...
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0answers
17 views

Prove that $P_{1}$ and $P_{2}$ have only one common refinement iff $E^{P_{1}} \cap E^{P_{2}} = =_{a}$

Let $A$ be a set. $P_{1}$,$P_{2}$ Are partitions of $A$. Let $E^P$ be the equivalence relation associated to a partition $P$ $$E^P:=\{\langle a,b\rangle \mid \exists \mathcal B\in P\ (b\in\mathcal B\...
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2answers
39 views

Prove that the union of two equivalence relations on the same set an equivalence relation iff?

Let $R$ and $E$ be equivalence relations on set $A$. Prove that $R\cup E$ is an equivalence relation on set A iff for all $a\in A$, $[a]_{R} \subseteq [a]_{E}$ OR $[a]_{E} \subseteq [a]_{R}$. ...
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1answer
30 views

What is an equivalence class of an equivalence relation?

I might be interpreting this wrong but in my book it says: If ~ defines an equivalence relation on $A$ then the set of equivalence classes of ~ form a partition of $A$. To me, this means that the set ...
0
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1answer
26 views

Proved that if $R\cup E$ equivalence relation so $a / E \subseteq a / R$ OR $a / R \subseteq a / E$

Let R, S be equivalence relations on A. Proved that if $R\cup E$ Is an equivalence relation on A So $\Rightarrow$ For all $a\in A$ , $$a / E \subseteq a / R$$ OR $$a / R \subseteq a / E$$ I ...
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1answer
42 views

Can $R=\{(1,6),(2,7),(3,8)\}$ be said transitive?

Given a relation $R=\{(1,6),(2,7),(3,8)\}$. It is clear that it is not reflexive and symmetric but can we say that it is transitive?
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1answer
21 views

Determinate the quotient topology

I was trying to find the quotient topolgy for the next example: Let R be the real numbers with the usual topology ($\tau$) and define the relationship $\mathcal{R}$ over R as follows, a $\mathcal{R}$...
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0answers
13 views

Determine the given relation is Equivalence Relation or not.

$R_{1} \oplus R_{2}$ I know that $R_{1} \oplus R_{2} = R_{1} \cup R_{2} - R_{1} \cap R_{2}$, and $R_{1} \cup R_{2}$ is not necessarily an equivalence relation but $R_{1} \cap R_{2}$ is always an ...
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2answers
16 views

How do I prove symmetry of a relation given a function?

Let G be a group. For all $g\in G$ , define the function f: G → G that sends x to $gxg^{-1}$. Define the relation ~ on G by a~b if $a = f(b)$ for some $g\in G$. Prove that ~ is an equivalence relation....
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6answers
74 views

Why $yC_1x \iff yC_2x$ implies $C_1 = C_2$? $C_i$ is a relation.

Here is the text from the book Topology by Munkres: Studying equivalence relations on a set $A$ and studying partitions of $A$ are really the same thing. Given any partition $\scr D$ of $A$, there ...
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1answer
15 views

Counting ordered pairs in quotient set

We have equivalence relation E on the set $$A = \{1,2,3,4,5\}$$ So the quotient set: $$A/E = \{\{1,2,3\}, \{4\}, \{5\}\}.$$ How much orderd pairs we can find in E? How to count the ordered pairs?...
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46 views

LEN-Model equivalency

Starting position is a principal-agent-model with incomplete information (moral hazard) and the following properties: Agent utility: $u(z)=-e^{(-r_az)}$ Principal utility: $B(z)=-e^{(-r_pz)}$ Effort ...
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2answers
31 views

What is the equivalence class of the following equivalence relation?

If we have a equivalence relation $R=_{def} \{((x_1,y_1),(x_2,y_2)) ~~| ~~x_1-y_1=x_2-y_2 \} \subseteq \mathbb{R}^2 \times \mathbb{R}^2 $ What is the equivalence class $[(0,1)]_R$? I thought it ...
0
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1answer
123 views

Show that R is an equivalence relation on X for x, y in X iff f(x) = f(y)

$f:X→Y$ $x,y ∈ X,xRy$ iff $f(x) = f(y)$ Show that R is an equivalence relation on X. Also when $X = Y = \mathbb{R}$ and $f: \mathbb{R} \to \mathbb{R} $ with $x \mapsto x^2$ for all $x∈R$ find the ...
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2answers
111 views

Prove that the relation on $X$ given by $x\sim y$ if $f(x)=f(y)$ is an equivalence relation

Let $f:X\to Y$ be a function between two sets. Prove that the relation on $X$ given by $x\sim y$ if $f(x)=f(y)$ is an equivalence relation. I know that it should have $3$ cases, reflexive (for all $x$...
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1answer
27 views

Stuck on equivalence relations

Let R be the relation on the set of ordered pairs of positive integers such that $((a, b), (c, d)) \in R$ if and only if $ad = bc$. What are the equivalence classes of this relation? I am completely ...
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0answers
21 views

equivalence class of kernel relation of floor function

By taking particular values of $k$, I found that equivalence class of $k$, $[k]=\{k,k+1,k+2,...,2k-1\}$, equivalence class of $2k$, $[2k]=\{2k,2k+1,2k+2,...,3k-1\}$ and so on, but how to present it ...
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0answers
10 views

Number of symmetric relations

Let set $A=\{1,2,3\}$ Find number of symmetric relations that can be defined on $A$ containing ordered pairs $(1,2)$ and $(2,1)$ is? Can someone give me some hint for this question?
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1answer
40 views

Show that either $[x] = [y]$ or $[x]$ and $[y]$ are disjoint?

An equivalence class $[x]_R$ is defined by $[x]_R = [y ∈ X : xRy]$. In this proof I'm supposed to start by assuming $[x]$ and $[y]$ are not disjoint. Therefore, there is some element $z$ that is in $[...
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4answers
155 views

How do you show one way equivalences in mathematics?

In the real world, it seems you can often take a set of things and create new things with them. Let's say you want to make bread, for example. Normally, you can create bread by putting together flour, ...
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5answers
27 views

Show that $x$ is an element of its own equivalence class?

If $R$ is an equivalence relation on $X$ and $x$ is an element of $X$, the equivalence class is defined as $[x]_R = [y ∈ X : xRy]$. Since $x$ is equivalent to itself, doesn't that automatically make ...
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1answer
38 views

Relations and their fallacies

I need to find a flaw in the proof of the following statement: Any relation that is symmetric and transitive is also reflexive. False Proof: aRb implies bRa,by symmetry. Then by transitivity, aRa ...
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2answers
54 views

Proving an equivalence relation on a $\mathbb Z\times \mathbb Z$

I am having some trouble understanding how to prove that a given binary relation is an equivalence relation. I understand that to prove that a relation is an equivalence relation, you must prove ...
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1answer
38 views

Which of the following equivalence classes are equal?

Let R be the relation of congruence modulo 3. Which of the following equivalence classes are equal. [7], [-4], [-6], [17], [4], [27], [19] The answer is: [7]=[4]...
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0answers
15 views

Is Congruence Class an example of Equivalence Class? (Understanding the definition through Examples)

Would you say a congruence class such as $[0]_{4}, [1]_{4}, [2]_{4},$ and $[3]_{4}$ an example of Equivalence classes since: the set of all elements of $\{..., -8, -4, 0, 4, 8,...\}$ is related to $[...
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2answers
38 views

What does it mean for f([x])=[2x] for a function mapping R/Z to R/Z?

Let X=R/Z (the circle), with a map $f : X → X$ given by $f([x]) = [2x]$. I'm a little lost on what $f([x]) = [2x]$ means. I thought the function was mapping the equivalence class [x] to the ...
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0answers
17 views

Find equivalence classes

A is set of all propositional expressions on the propositions p1 , p2 , . . . , pn. Relation R on A is defined as (P, Q) ∈ R if P ↔ Q is a tautology. I think it is an equivalence relation, what should ...
0
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1answer
30 views

Equivalence Classes of an Equivalence Relation Confusion (definition and solution included)

The Definition of Equivalence Classes of an Equivalence Relation is given as: Suppose $A$ is a set and $R$ is an equivalence relation on $A$. For each element $a$ in $A$, the equivalence class of a, ...
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1answer
26 views

Showing an equivalence relation

Let $[n]$ denote the set of all permutations. Let $R$ be the relation $$i_1 i_2\cdots i_n=j_1 j_2\cdots j_n$$ if and only if there exists a $k$ such that $j_1 j_2 \cdots j_n=i_k i_{k+1}\cdots i_n i_1 ...
0
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1answer
123 views

Equivalence classes under logical equivalence by 13 valuations

Let L be the set of 5 propositional variables. Under the equivalence relation given by logical equivalence, how many equivalence classes of propositional terms are given the value TRUE by 13 ...
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2answers
114 views

Why not just define equivalence relations on objects and morphisms for equivalent categories?

My question is regarding this blog post https://unapologetic.wordpress.com/2007/05/30/equivalence-of-categories/ which I will paraphrase below: The author gives the example of a category $\mathscr{C}$...
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1answer
48 views

definition of the directed colimit of a functor

Let $(\Lambda,\le )$ be a directed set, which we can understand as a small category: The set of all objects is $\Lambda$ and for $\lambda,\lambda '\in \Lambda$ there exists an unique morphism $i_{\...
0
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1answer
26 views

Proving Equivalence Relations by providing an example based on given subsets.

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 $A$ and $...