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

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1answer
33 views

Restriction of an equivalence relation on a subset.

If we have an equivalence relation defined on a set E and S its subset. Is the relation defined on S is also an equivalence one? Thank you for your answers.
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1answer
117 views

determining reflective, symmetric, transitive, anti-symmetric properties and describing equivalence classes

The question: determine if p is reflective, symmetric, transitive and/or anti-symmetric, if p is an equivalence relation, describe the equivalence classes A = Z , and $apb$ if and only if $5 | (2 a + ...
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1answer
73 views

Let A be a non-empty set, and p an equivalence relation on A . Let a , b be an element of A . Prove that [ a ] = [ b ] is equivalent to apb

the question: a) Let A be a non-empty set, and p an equivalence relation on A . Let a , b be an element of A . Prove that [ a ] = [ b ] is equivalent to $apb$ b) If p is both an equivalence relation ...
1
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1answer
87 views

$A = \mathbb{R}$ , and $a\mathrel{p} b$ if and only if $\sin a = \sin b$

My question is: For the relation $p$ described below, determine if $p$ is reflexive, symmetric, transitive, anti-symmetric. In each case, if $p$ is an equivalence relation, describe the equivalence ...
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3answers
34 views

Given $n \sim r \iff n \equiv r \pmod d$, prove $\sim$ is an equivalence relation. [duplicate]

It is given that n belongs to Z and d belongs to N. How do I prove that n=r mod d defines equivalence relation? I know I have to prove it is reflexive, symmetric and transitive. But how do I do that? ...
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2answers
37 views

Can a relation be a partial order and an equivalence at the same time?

Can a relation be a partial order AND an equivalence at the same time? For instance, if we have a set A = {1, 2, 3, 4, 5} and a relation R on A defined as R = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5)}: ...
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0answers
21 views

Union and intersection of reflexive relations

If p and q are 2 reflexive relations, Are (p union q) and (p intersection q) reflexive? Similarly, check for symmetric, antisymmetric and transitive properties.
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2answers
66 views

Does R^2 has the same property as R?

If R is a relation on set A, define $R^2$ by $aR^2b$ if and only if there exists c with aRc and cRb. If R is reflexive/symmetric/transitive does $R^2$ have the same property ? I'm not sure how to do ...
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2answers
34 views

prove the equivalence of the following statements: 2x-1 is irrational; x/3 is irrational

I am stumped. I really have no idea how to solve this problem. Can someone please help me through this? THE TWO EQUATIONS ARE SEPERATE
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1answer
49 views

Proof for the equivalence classes given equivalence relation? [closed]

Let $A$ be a non-empty set, and $M$ an equivalence relation on $A$. Let $a, b \in A$. Prove that $a = b \Leftrightarrow (a,b) \in M$. If $M$ is both an equivalence relation and (simultaneously) a ...
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1answer
46 views

An equivalence relation iff G≈H, where G and H are groups [duplicate]

Problem : Let $S$ be the relation G~H iff G is isomorphic to H. Show reflexive, transitivity and symmetric. First show G is automorphism, which will imply G~G. So the identity mapping gives us ...
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1answer
21 views

Possible Equivalence Relation Question

Consider $\langle\Bbb{Z}_6, +_6\rangle$. Let $a\sim b$ if and only if $\{a,b\}$ generates $\langle\Bbb{Z}_6, +_6\rangle$. $a,b \in \Bbb{Z}_6$. Is $\sim$ an equivalence relation? I know an ...
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1answer
42 views

Find all equivalence classes

Let R by a relation defined on pairs $(m,n)$ of integers $m$ and natural numbers $n$ by $(i,j) R (k,l)$ if $il=jk$. Prove that this is an equivalence relation and give the equivalence cases. Show ...
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2answers
62 views

Identifying laws in a discrete math example

I'm studying for my upcoming discrete math test and I'm having trouble understanding some equivalences I found in a book on the subject. I guess I'm not really familiar with these rules and I would ...
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1answer
28 views

How can i do this algebra question?

The question is show that the relation $a\sim b$ defined by $a\equiv b \bmod 7$ is an equivalence relation on $\mathbb{Z}$. How many equivalence classes are there? Let us call them $[0]$, $[1]$, ..., ...
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1answer
58 views

Double check $G\sim H$ iff $G≈H$

Let $S$ be the collection of all groups. Define a relation on $S$ by $G \sim H$ iff $G ≈ H$. Prove that this is an equivalence relation. So $S$ is partitioned into isomorphism classes. Proof: Let ...
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1answer
27 views

Proving an equivalence relation(specifically transitivity)

I'm currently learning about equivalence relations. I understand that an equivalence relation is a relation that is reflexive, symmetric, and transitive. But I'm having trouble proving the transitive ...
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1answer
24 views

equivalence Relation problem with some conditions

If A be a set with $|A|=n$. if R be a equivalence Relation on A and $|R|=r$, why $r-n$ always be even ?
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2answers
44 views

Equivalence Class Question

On the set $N\times N$ define $(m,n)\simeq(k,l)$ if $m+l=n+k$. Draw a sketch of $N\times N$ that shows several equivalence classes. (hint: sketch points on graph paper). I'm not quite sure how to ...
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0answers
25 views

What is the intersection of thses equivalence relations?

Let $S$ be the following subset of the plane: $$ S \colon= \{ \ (x,y) \ | \ y=x+1, \ 0 < x < 2 \ \}.$$ Then how to describe the equivalence relation $T$ on the real line that is the ...
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2answers
50 views

Equivalence relations for $\mathbb{N} \times \mathbb{N}$ question

On the set $\mathbb{N} \times \mathbb{N}$ define $(m, n) \sim (k, l)$ if $m + l = n + k$. Show that $\sim$ is an equivalence relation on $\mathbb{N} \times \mathbb{N}$. Draw a sketch of $\mathbb{N} ...
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1answer
90 views

Absolute Value Equivalence relation inequality Question

I'm having trouble understanding what exactly to do to see if the following relation is symmetric and transitive. I've already determined that it is reflexive. Could someone please help me? For $a, b ...
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0answers
24 views

Question about logical statement regarding the following ∃x∈ℝ,(x²=2)

Is the following statement logically equivalent to ∃x∈ℝ,(x²=2): "There is at least one real number whose square is 2."
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1answer
25 views

equiv. class if aRb means a+b is a+b even

let s be set of integers. and say that aRb=a+b only if a+b is even. i've already shown that this is indeed a equivalance relation, but how to show its equivalance classes?
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1answer
39 views

Let F be a partition of A. Prove there exists unique equivalence relation R such that F=A|R?

Let F be a partition of A. Prove there exists unique equivalence relation R such that F=A|R? I don't even know how to start. I know to be a equivalence relation R must be reflexive, symmetric and ...
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2answers
59 views

Prove that $\mathbb{Q} \times \mathbb{Q}$ is countable.

Knowing that $\mathbb{Q}$ is countable, I must prove that $\mathbb{Q} \times \mathbb{Q}$ is countable. Teacher's proof: For each $a \in \mathbb{Q}$, let $A_a = \{(a,q) : q \in \mathbb{Q}\}$ so that ...
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2answers
46 views

Show that $R = \{ (x,y) \in \mathbb{Z} \times \mathbb{Z} \; : \; 4 \mid(5x+3y)\}$ is an equivalence relation.

Let $R$ be a relation on $\mathbb{Z}$ defined by $$ R = \{ (x,y) \in \mathbb{Z} \times \mathbb{Z} \; : \; 4 \mid (5x+3y)\}$$ show that R is an equivalence relation. i'm having a bit of trouble ...
0
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1answer
49 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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3answers
42 views

Prove Equivalence Relation in G

Hei, guys! I'm having some trouble with the next problem: Let $A$ and $B$ be subgroups of $G$. Show that $\sim$ is an equivalence relation when it is defined as follows: $g\sim g'\Leftrightarrow g' ...
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1answer
35 views

Check: Let G be a group and H be a subgroup of G. Define R by $xRy \iff xy^{-1} \in H$. Show R is an equivalence relation.

Let G be a group and H be a subgroup of G. Define R by $xRy \iff xy^{-1} \in H$. Show R is an equivalence relation. $\textbf{Definition:}$ R is a relation on X. R is an equivalence relation of X if R ...
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1answer
17 views

Verify Equivalence relation.

Question: Find an example of three relations $R_{1}$, $R_{2}$ , $R_{3}$ on the set S=$\{1,2,3,4,5\}$ such that $R_{1}$ is reflexive but not transitive, $R_{2}$ is transitive but neither symmetric ...
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4answers
49 views

Easy question about an equivalence relation

I was told the following in class: If we define an equivalence relation on $[0,1)$ by declaring that $x \sim y$ iff $x-y$ is rational, then there are uncountably many equivalences classes. Why is ...
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1answer
54 views

$\sin(x)$ is asymptotically equal to $x+5x^3$

Here is my question: I've never seen before this kind of fact underlined about asymptotic equalities (and why we keep only one term in these equalities) and I'm looking for reference. Here is an ...
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3answers
46 views

Does finite equivalence classes implies that the set itself is finite.

My Assignment Question: If $R$ is an equivalence relation on a set $S$ and it has only finitely many equivalence classes altogether, then $S$ itself is a finite set. From the theorem for ...
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2answers
56 views

Doubt pertaining to this Equivalence Relation.

$1$. True or false? If $R$ is an equivalence relation on a set $S$ and it has only finitely many equivalence classes altogether, then $S$ itself is a finite set. I think the answer is true ...
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2answers
27 views

What is the cardinality of $\left|C_s\right|$?

Let $$C_s = \left\{ f\in \mathbb{N}/S \to \mathbb{N} : \forall M\in \mathbb{N} / S. f(M)\in M \right\}$$ Where $S$ is an equivalence class. I need to prove $$\left|C_s\right| > \aleph_0 \implies ...
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1answer
28 views

Transitive closure of $H=\{(a,b) \in \mathbb{R}^2: |a-b| \leq 0.1\}$

$$H = \{(a, b) \in \mathbb{R}^2: |a − b| \leq 0.1\}$$ In class today we went over this problem as an example to show transitive closure. I know that the transitive closure of $H$ is "All real ...
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1answer
49 views

What is the cardinality?

Let $A=\left\{1,2,\cdots,10\right\}$ Let $f,g:A\to A$. Consider the equivalence relation $$ fRg \iff \exists h:A\to A. f=h\circ g$$ where $h$ is invertible. Now, let $g(x)=5$: Why is $\left| ...
2
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1answer
74 views

A relation that is Reflexive & Transitive but neither an equivalence nor partial order relation

Set $A = \{0,7,1\}$ 1. So for a relation that is reflexive and transitive but neither an equivalence relation nor partial order...Can a relation be both partial order and equivalence? Attempt: ...
2
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1answer
69 views

Proving that a relation is an equivalence relation

I am having difficulties proving the relation IS an equivalence relation. Let $f: X\longrightarrow Y$ be a function from a set $X$ onto a set $Y$. Let $R$ be the subset of $X \times X$ consisting ...
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1answer
44 views

What does it mean for a binary relation to be an order on “equivalence classes” under another binary relation

I am a bit confused about this statement in "Introduction to Set Theory" by Hrbackek and Jech. The statement is as follows: "|A| <= |B| behaves like an ordering on the "equvivalence classes" under ...
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0answers
21 views

prove the following equivalence $\ln(-\frac{l-x+\sqrt{r^2+(l-x)}}{l+x-\sqrt{r^2+(l+x)}})={}$ArcSinh$(\frac{l+x}{r})+{}$ArcSinh$(\frac{l-x}{r})$

Hi guys i'm trying to prove the following equivalence but i'm having some problems: $$\ln\left(-\frac{\ell-x+\sqrt{r^2+(\ell-x)^{2}}}{\ell+x-\sqrt{r^2+(\ell+x)^{2}}}\right) = \operatorname{ArcSinh} ...
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3answers
98 views

Equivalence class for the relation of having the same set of prime divisors

For an integer $n\in \mathbb{N}$ define $P(n) = \{p : p \mid n \text{, where $p$ is a prime} \}$. For example $P(12)=\{2,3\} $ and $P(1)=\emptyset$. Question: Consider the relation $R$ on ...
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5answers
96 views

How would I show that R is an equivalence relation?

If I were to consider the relation R on ℤ defined by n R m if and only if P(n)=P(m). How would I show that R is an equivalence relation? Any help is appreciated.
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1answer
63 views

Is the relation $a\mathrel R b \iff f(a) \equiv f(b)$ an equivalence relation?

Suppose that I have a relation $R$ of the form $a\mathrel R b \iff f(a) \equiv f(b)$, where $\equiv$ is an equivalence relation. In general, is $R$ also an equivalence relation? If not, what are the ...
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1answer
21 views

Comparing two circularly shifted matrices

I am looking for a way to compare two matrices A and B where B is the result of circularly shifting rows of A i.e. A = [1 2 3;4 5 6], B = [4 5 6;1 2 3] Is there an operator or metric that would ...
1
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1answer
41 views

Let R be the relation on ℤ+→ℤ+ defined by (a,b)R(c,d) if and only if a-2d=c-2b. List all the elements of the equivalence class [(3,3)].

I'm confused on how to find all the elements. I know how to find some but not all, wouldn't they be infinite? This is affecting me with the other questions as well. Thanks in advance!
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1answer
45 views

Geometric/visual interpretation of transitivity for equivalence relations on $\mathbb{R}$

If we graph equivalence relations on $\mathbb{R}$ on the plane $\mathbb{R} \times \mathbb{R}$, the properties of reflexivity and symmetry give rise to certain geometric properties--i.e. reflexivity ...
2
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2answers
41 views

Is $R=\left \{ (a,a),(a,b),(b,a),(b,b),(c,c),(c,d),(d,c),(d,d) \right \}$ an equivalence relation on $X$?

Let $X= \left \{ a,b,c,d \right \}$ and $R=\left \{ (a,a),(a,b),(b,a),((b,b),(c,c),(c,d),(d,c),(d,d) \right \}$. I want to show that $R$ is an equivalence relation on $X$. My work: $R$ is reflexive: ...
0
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1answer
28 views

suppose $R$ is an equivalence relation on $A$ such that there are only finitely many distinct equivalence classes $A_1,A_2,\ldots,A_k$ w.r.t $R$

Suppose $R$ is an equivalence relation on $A$ such that there are only finitely many distinct equivalence classes $A_1,A_2,\ldots,A_k$ w.r.t $R$. Show that $$A=\bigcup_{i=1}^k A_i$$ Since $A_i\subset ...