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

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Solve the relation with congruence

On $\Bbb Z$ consider the relation $xRy \Leftrightarrow x-y \not\equiv 0 \mod 3$. Prove (with explanation), whether the relation reflexive, symmetric, antisymmetric transitive is and prove if they are ...
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1answer
24 views

Does symmetric property of Equivalence hold when the left hand side of the equation is in indeterminate form?

The Symmetric Property of Equivalence is a=b implies b=a. This property does not have any conditions on a or b. But what if through manipulation I get $a=\frac00$ where $a$ is a real number does this ...
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3answers
53 views

Find the equivalence classes for $a T b \iff \frac a b \in \Bbb Q$

Given the set $S = \{ x − \sqrt 5 y : x,y \in \Bbb Q, \ x − \sqrt 5 y \ne 0 \}$, assume the relation $T$ is defined on $S$ by $a T b \iff \frac a b \in \Bbb Q$. How can I find the distinct ...
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3answers
87 views

Equivalence relation: $aRb$ iff $2a+3b$ is divisible by $5$

Prove that $R$ is an equivalence relation: $aRb$ iff $2a+3b$ is divisible by $5$. Here $a,b\in \mathbb{Z}$ (set of integers). I can prove that $R$ is reflexive and transitive. How to prove it's ...
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3answers
56 views

$x\sim y$ if $|x-y|\le 3$, then is $\sim $ or R an equivalence relation?

Let R or $\sim$ be the relation defined on Z by $$x\sim y\text{ if } |x-y| \le 3$$ Is $\sim$ an equivalence relation? It is reflexive and symmetric if I did it correctly. However, I am having doubt ...
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3answers
64 views

Equivalence relation and Distinct equivalence classes

Given the set $S = \{x-y \sqrt5: x, y$ are rational numbers and $x-y \sqrt5 \neq 0\}$. Assume the relation $T$ is de fined on the set $S$ by $a T b$ if $a/b$ is a rational number. Question has ...
2
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1answer
94 views

Finding distinct equivalence classes

Q: Given the set $S = \{x - y\sqrt 5 : \text{x, y are rational numbers and }x - y \sqrt5 \neq 0 \}$. Assume the relation defined on the set $S$ by $a\ T\ b$ if $a/b$ is a rational number. Find ...
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0answers
85 views

Unitary Farey Sequence Matrices

Take the Farey sequence $\mathcal{F}_n$ with values $a_m\in \mathcal{F}_n$ and put them into a vector $$ \vec v_k=\frac1{\sqrt{|\mathcal{F}_n|}}\biggr(\exp(2\pi i k a_m)\biggr)_m $$ The dimension of ...
0
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0answers
34 views

E1 and E2 are equivalent then they are “almost equivalent”

Given : 2 statements E1, E2 in relational algebra are "almost equivalent" if every phase in the database D ,except finite number of D's E1(D)=E2(D). E(D) means the result of activating the statement E ...
2
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2answers
65 views

What is the symbol for “coincident” in geometry?

I am looking for a symbol to say that one geometrical figure coincides with another without writing the phrase "is coincident with." For example, the altitude $a$ of an equilateral triangle coincides ...
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1answer
30 views

Suppose $(a+i)^{a+i}\equiv a^a \mod p$ for all $a=1,2,..$ Then $i$ is divisible by $p(p-1)$

Suppose $(a+i)^{a+i}\equiv a^a \mod p$ for all $a=1,2,\dots$ Then $i$ is divisible by $p(p-1)$. Solution: Take $a=p$ then we see that $(i+p)^{p+i}\equiv p^p \equiv 0 \mod p$ Since $i+p\equiv 0 ...
2
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1answer
36 views

Can we always define a congruence category?

In Awodey's Category Theory the congruence category is defined as follows... We have a congruence ~ on a category $C$. Then $C^\tilde{}$ is defined as: $(C^\tilde{})_0=C_0$ $(C^\tilde{})_1=\{ ...
1
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1answer
38 views

The relation $a_1 \sim a_2 \iff f(a_1 ) = f(a_2 )$ is an equivalence relation

Suppose a function $f : A → B$ is given. Define a relation $\sim$ on $A$ as follows: $a_1 \sim a_2 \iff f(a_1 ) = f(a_2 )$. Prove that $\sim$ is an equivalence relation on $A$. I know that in ...
0
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1answer
29 views

Finding distinct equivalence classes.

I am going through some practice questions and am having trouble to finding distinct equivalence classes and this is my understanding so far. Let S be a nonempty subset of Z, and let R be a relation ...
0
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1answer
32 views

First-order equivalence formulas in Logic

Can someone help me understand as to why the following are equivalent when x is a bound variable that does not occur free in A? $\forall x (A \lor B) \iff A \lor \forall x B$ $\exists x (A ...
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2answers
36 views

Partition into “fibers” $f^{-1}(y) \in Y$

Consider any surjective map f from a set X onto another set Y. We can define an equivelance relation on X by $x_1Rx_2$ if $ f(x_1)=f(x_2)$. Check that this is an equivelance relation. Show that the ...
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1answer
20 views

On an exercise that asks for a homeomorphism between a quotient space and a metrizable space.

I have the solution to the exercise but have a doubt on one thing, I state the exercise: Given $$ X = \{ (x,y) \in R^2 | x = \frac{1}{n}, n \in N \}$$ and $Y = X/_{\sim}$ where the equivalence ...
4
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0answers
63 views

Given $S=\{0,1,2,3,4,5\}$, find the partition induced by the equivalence relation $R$

I am currently taking discrete math and have been given the following question to answer. Given $S=\{0,1,2,3,4,5\}$, find the partition induced by the equivalence relation $R$ where ...
2
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0answers
28 views

Cardinality of this equivalence class

I'm looking at the following equivalence relation on $\mathbb{Z}$: $a \sim b$ if and only if there exist $n,m \in \mathbb{N}_{>0}$ so that $a^n = b^m$ I'm trying to determine what the cardinality ...
2
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2answers
53 views

Is the relation $a $~$ b$ iff $ ab$ is square on $\mathbb{Z}$ transitive?

I'm trying to determine whether the relation given above is a equivalence relation. I've already proved it is reflexive and symmetric, but I'm stuck trying to prove (or disprove) its transitivity. I ...
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3answers
41 views

Construct an equivalence relation on a given set

can anyone help me on this problem? I have the set $\{0,1,3,8,9\}$ and I want to define an example of an equivalence relation. I know that to be an equivalence relation it needs to be reflexive, ...
3
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2answers
43 views

Does $R=\{(x,y) \in \mathbb{Z}\times\mathbb{Z} : 3|(x+y)\}$ define an equivalence relation?

Given $R=\{(x,y) \in \mathbb{Z}\times\mathbb{Z} : 3|(x+y)\}$, Is $R$ reflexive? Is $R$ symmetric? Is $R$ transitive? Reflexivity: Could $(1,1)$ be a counter-example because $3\nmid(1+1)$? Symmetry: ...
2
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2answers
114 views

Define a relation on the integers such that $a R b$ iff $\;3\mid (a + 2b)$?

I've seen relations defined as functions between sets and as sets of ordered sets; however, I've never seen a relation defined as $3\mid(a+2b)$. What does this mean? --Edit-- I'll try and express my ...
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0answers
29 views

Check if equivalence relation

Check if $(x,y)\rho(a,b)\Leftrightarrow sgn(y-\pi x)=sgn(b-\pi a)$ is equivalence relation on $\mathbb{R^2}$, find the set of equivalence classes and $C_{(1,\pi)}$. Give geometric representation. ...
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2answers
58 views

Is this a valid construction of the natural numbers under ZF?

First, define an equivalence relation, $\sim$, such that two sets, $A$ and $B$ are equivalent, $A\sim B$ if and only if there exists a bijection between them. Then define $$0=[\emptyset]_\sim$$ Where ...
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1answer
38 views

Let $X = \{−1,0,1\}$ and $A =\mathcal{P}(X)$, and $R$ is defined on $A$ as for all sets $S,T \in A$, $\ldots$

Let $X = \{−1,0,1\}$ and $A = \mathcal{P}(X)$, and $R$ is defined on $A$ as for all sets $S,T \in A$, $$ SRT \Longleftrightarrow \text{the sum of the elements in $S$ equals the sum of the elements in ...
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3answers
56 views

Proving equivalence relation for 7 | (3a + 4b)

I know this might be quite trivial, but I just can't seem to figure out how to prove $$R = \{(a,b) \in \mathbb{Z} \times \mathbb{Z} : 3a + 4b \text{ is divisible by } 7\}$$ is a symmetric relation, ...
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0answers
58 views

Equivalence of Group Actions, Transitivity, and Conjugate Subgroups

Some Preliminary Definitions and Properties: Actions of a group $G$ on sets $X$ and $Y$ are equivalent if the corresponding action of $G$ on maps from $X$ to $Y$ fixes some bijection. In this case, ...
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1answer
62 views

Find the distinct equivalence classes

Let $B = \{0,1,2,3,4\}$ and let $\{0\},\{1,3,4\},\{2\}$ be a partition of $B$ that induces a relation $Q$. Find the distinct equivalence classes of $Q$.
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1answer
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proof or deproof linear equivalence of X, X is an amount.

Again I am stuck at some proof. I need to proof or deproof that for all linear equivalences: R:(X,X) is R = So far I think it is correct because we get symmetry and linearity, but I have ...
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0answers
130 views

Equivalence relation for strings

Let R be the relation consisting of all pairs (x,y) such that x and y are strings of uppercase and lowercase English letters with the property that for every positive integer n, the nth characters in ...
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1answer
37 views

Connections between Posets and WQO's

Here is the question that I posted on the Mathematics Chat Room that I was unable to find an answer to: Question: Under what conditions/properties is a poset ever a wqo (well-quasi-order)? Can we ...
2
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1answer
25 views

Proof: Sum / Intersection of family of equiuvalence relations is equivalence relation

I have to check if sum and intersection of family of equivalence relations is equivalence relation. Here is the exercise: Let $\mathcal{R}$ be a family of equivalence relations defined on some set ...
3
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1answer
55 views

Is $xRy \iff x+y = 0$ an equivalence relation?

$R$ is a relation on real numbers. $xRy \iff x+y = 0 $. Is it an equivalence relation? My answer is no proof: -(Reflexive) let $x = a$ , $aRa \iff 2a=0$. Since $2a = 0$ doesn't hold for every real ...
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1answer
37 views

How to prove an equivalence relation with more than 2 variables?

Let $R$ be a relation of positive integers $$((a,b),(c,d)) \in R \iff ac = bd.$$ Prove that $R$ is an equivalence relation. So I need to prove that this relation is reflexive , transitive and ...
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3answers
18 views

Show that $(H, \circ)$ is a subgroup of the group $G$

Question: Let $G$ be a group and $H$ be a nonempty subset of $G$. A relation $\rho$ defined on $G$ by ``$a\rho b$ if and only if $a\circ b^{-1}\in H$" for $a,b\in G$, is an equivalence relation on ...
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0answers
77 views

Symmetric closure of the reflexive closure of the transitive closure of a relation

Give an example to show that when the symmetric closure of the reflexive closure of the transitive closure of a relation is formed, the result is not necessarily an equivalence relation. My attempt ...
1
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1answer
68 views

How to proof that nested intervals are an equivalent relation?

I want to show that a relation on the space of all sequences of nested intervals is an equivalence relation. Definition: Let $[a_n,b_n]_{n\in\mathbb{N}}$ and $[c_n,d_n]_{n\in\mathbb{N}}$ be two ...
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2answers
32 views

Equivalence relations on metric spaces

Let $d:X\times X \rightarrow \mathbb{R} \cup \{\infty\}$ be a metric on the set X. I should prove that $d(x,y)\neq \infty$ is an equivalence relation but I'm not sure what this expression means. ...
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1answer
25 views

What is the empty relation?

I was reading the Wikipedia article on equivalence relations and one section says that "the empty relation R on a non-empty set X is vacuosly symmetric and transitive but not reflexive." What is the ...
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1answer
94 views

Show that a relation is equivalent if it is both reflexive and cyclic.

A relation $R$ on set $X$ is called cyclic if whenever both $xRy$ and $yRz$ then $zRx$ where $x,y,z\in X$. Show that a relation on $X$ is an equivalence relation if and only if it is both reflexive ...
2
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1answer
49 views

Describing Distinct Equivalence Classes of a Relation

Suppose that $R$ is a relation on the set of complex numbers $\mathbb{C}$. The relation $R$ is defined as follows: For any two complex numbers $w,z \in \mathbb{C}$, $$w R z \Leftrightarrow ...
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2answers
70 views

Is the set $\Bbb Q$ a quotient set of $\Bbb Q^*$?

Let $\Bbb Q^*=\{\frac a b: a\in \Bbb Z, b\in \Bbb N\}$. From this definition we can see $c=\frac 2 3$ and $d=\frac 4 6$ are elements of $\Bbb Q^*$. Claim: $$\frac 2 3\neq \frac 4 6$$ Proof: ...
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1answer
62 views

Is “to be conjugate” is an equivalence relation?

Let denote $P_x$ the minimal polynomial of $x$ over a field $K$. We say that $x$ and $y$ are conjugate if $P_x(y)=0$. Is "to be conjugate" is an equivalence relation ? The question behind this ...
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0answers
58 views

How many distinct strict ordinal 2x2x2 games exist?

Consider the same type of strict ordinal games as described in How to simply show that there are "78 'strict ordinal' 2x2 game matrices" and add a third player with two strategies ...
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1answer
62 views

Equivalent classes of similar/equivalent $n\times n$ matrices

Is there a natural way to find describe all the equivalence classes of $F^{n\times n}$ under equivalence, F an arbitrary field? Here equivalence is just the normal definition: $A$ is equivalent to $B$ ...
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0answers
46 views

find the Equivalence classes of this equivalence relation

Is this correct? Let R and S be the equivalence relations on Z X Z defined by ((a,b),(c,d)) ∈ R if and only if ab=cd and ((a,b),(c,d)) ∈ S if and only if ad=bc Find the equivalence class ...
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0answers
44 views

Show that multiplication $[(x, y)] * [(n,m)] = [(xn + ym, xm + yn)]$ is also well- defned.

I'm having a bit of trouble on this proof. It's part of the construction of the integers. $R$ is the relation, $\mathbb{N}$ the natural numbers, $((x,y),(n,m)) \in ...
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2answers
28 views

Similarity transformation-proof of equivalence

I am getting stuck with following problem: Show that \begin{align} \dot{x} = f(x/t) \end{align} is equivalent to \begin{align} \dot{y} = (f(y) − y)/t \end{align} using the transformation ...
0
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1answer
28 views

$a^2 \equiv b^2$ mod 4 equivalence classes.

so we have the relation $a^2 \equiv b^2$ mod 4. And to find equivalence classes we say b or a = 0 so $a^2=4k$ so $a=+-2\sqrt{k} $ so all even numbers. But when we get to a=1 then $a^2=4k+1$ after ...