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

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1answer
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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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0answers
10 views

equivalence classes partition [on hold]

Let R be an equivalence relation on A. Then show that the equivalence classes A/bar/ under this equivalence relation partitions A. Conversely, if C partitions A, define ∼ on A×A by a∼b if a,b belong ...
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0answers
46 views

Is this relation an equivalence relation? If so, identify the equivalence classes. [on hold]

Determine if $ρ$ is reflexive, symmetric, transitive, anti-symmetric. In each case, if $ρ$ is an equivalence relation, describe the equivalence classes. $$A = \mathbb R \,\text{ and }\, aρ b \;\text{ ...
1
vote
1answer
58 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
vote
1answer
82 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 ...
0
votes
3answers
26 views

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

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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3answers
1k views

Why is the empty set finite?

On page 25 of Principles of Mathematical Analysis (ed. 3) by Rudin, there is the definition (excluding the irrelevant parts for this question): Definition 2.4: For any positive integer $n$, let ...
1
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2answers
31 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)}: ...
2
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1answer
43 views

Proof for the equivalence classes given equivalence relation? [on hold]

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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0answers
25 views

Equivalence relations or not? [on hold]

In both the cases I need to show whether these are equivalence relations (plus also tell if they are antisymmetric or not) and describe their equivalence classes. Hey guys, I have an exam coming ...
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0answers
4 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
65 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
32 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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votes
1answer
41 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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0answers
20 views

Equivalence Relations: Prove x E ~x [closed]

Using only the fact that congruency m is an equivalence relation on Z (integers). Prove that for all x in Z: x element of ~x (equivalence class x)
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2answers
59 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 ...
1
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1answer
33 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 ...
0
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1answer
19 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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votes
1answer
27 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
32 views

How to determine a equivalence relation?

I have a problem to understand the following output: Determine "representative system" or a "system of representatives" :).....for the following equivalence relation ...
0
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1answer
54 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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5answers
2k views

Equivalence relation $(a,b) R (c,d) \Leftrightarrow a + d = b + c$

Suppose $A$ is the set composed of all ordered pairs of positive integers. Let $R$ be the relation defined on $A$ where $(a,b) R (c,d)$ means that $a + d = b + c$. (a) Prove that $R$ is an ...
1
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1answer
38 views

Prove that $[x]_{R}=[y]_{R} \Rightarrow g(f(x))=g(f(y))$

Let $f:\mathbb{Z} \to \mathbb{N}$ and $g:\mathbb{N} \to \mathbb{N}$ be functions. And let $R$ be a equivalence relation on $\mathbb{Z}$, defined by: $$xRy \Leftrightarrow f(x)=f(y)$$ For any ...
1
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1answer
21 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 ...
2
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1answer
23 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
24 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 ...
0
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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 ...
0
votes
2answers
41 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} ...
0
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1answer
38 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 ...
0
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0answers
21 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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2answers
292 views

Difference between Reflexive and Symmetric in Discrete Maths

Difference between Reflexive and Symmetric in Discrete Maths? This is what I understand: Reflexive -> <a,a=a>, <b,b=b> uses ...
0
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1answer
22 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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0answers
32 views

Is $^\mathbb{N}\mathbb{R}$ $\sim$ $^\mathbb{R}\mathbb{N}$? [duplicate]

Is $^\mathbb{N}\mathbb{R}$ $\sim$ $^\mathbb{R}\mathbb{N}$? I know you have to use Cantor-Bernstein, and prove both directions, but i don't know how to start the proof
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1answer
36 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
56 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 ...
6
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1answer
139 views

Solve for ? - undetermined inequality symbol

So I was solving a problem in Rudin (chapter 3 #16, to be specific) and I realized how convenient it would be to have a symbol that represented an undetermined equivalence relationship. As an example ...
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2answers
43 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
42 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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1answer
311 views

What are some concrete examples of kinds of relations in math?

I'm writing an undergrad philosophy paper. My take on the issue is that the conceptual problem I'm addressing is only a problem because the word 'is' and 'relation' are too slippery. By more precisely ...
2
votes
3answers
41 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
34 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
15 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
48 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
52 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
45 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
24 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 ...
2
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1answer
45 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: ...
1
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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
46 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| ...