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

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1answer
138 views

I don't understand equivalence relations

I'm trying to show that the left coset with a subgroup is an equivalence relation. So taking some element $g \in G$ and a subgroup $H$, the left coset is defined as $gH = \{gh : h \in H\}$. That means ...
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0answers
33 views

Equivalence Set, Subsets

Just a quick question: If a = [A] and a belongs to N (set of all natural numbers) doesn't that mean that A is a subset of N? The reason I'm asking this is because I'm trying to prove the theorem ...
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1answer
81 views

Determine if R is an equivalence relation

I've got this question: Consider the relationship $R$ between ordered pairs of natural numbers such that $(a, b)$ is related to $(c, d)$ (denoted by $(a, b) R (c, d)$) if and only if $ad = bc$. ...
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1answer
45 views

Determining Equivalence Relation on $\Bbb{Z}$

Alright, I have a homework problem which I have researched, read up on and I (think) solved. I just need someone to either confirm my answer (and re-affirm my knowledge) or explain why I am wrong. ...
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0answers
28 views

Prove congruence relation.

Let $\sim$ be a relation where $x \sim y \Leftrightarrow x = y \lor (x \in I \land y \in I)$. Show $\sim$ is a congruence relation on $S$ where $S= \{a,b, c, d, e\}$ and $I = \{a, d\}$. One of the ...
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1answer
44 views

Prove transitivity of relation.

Let $\sim$ be a relation where $x \sim y \Leftrightarrow x = y \lor (x \in I \land y \in I)$. Show $\sim$ is a congruence relation on $S$ where $S= \{a,b, c, d, e\}$ and $I = \{a, d\}$. One of the ...
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4answers
48 views

Is $x \sim y \Leftrightarrow x,y$ are even an Equivalence relation?

The following instruction defineds an Equivalence relation on the set of natural numbers. $x \sim y \Leftrightarrow x,y$ are even My idea: Reflexivity: $x \sim x \Leftrightarrow x,x$ is even ...
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2answers
635 views

Equivalence Relations On A Set of All Functions

The question is, "Which of these relations on the set of all functions from $Z$ to $Z$ are equivalence relations." The first relation to consider is, $\{(f,g)|f(0)=g(0)\vee f(1)=g(1)\}$ For this one, ...
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2answers
121 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 $\Bbb Z$. ...
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1answer
63 views

Number of Equivalence relations of $\{1,2,3\}$

Let $M$ be the set $\{1,2,3\}$. How many Equivalence relations $R \subset M \times M$ exists? My idea is to count the disjoint partitions of M: $K_1= ...
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1answer
57 views

$R$ is transitive if and only if $ R \circ R \subseteq R$

Question: Let $R$ be a relation on a set $S$. Prove the following. $R$ is transitive if and only if $ R \circ R \subseteq R$. Definition 6.3.9 states that we let $R_1$ and $R_2$ be relations on a ...
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53 views

Finding an equivalence relation on $\{1, 2, 3\}$ with two equivalence classes [closed]

I need some help on a particular question, this one: Describe an equivalence relation on $\{1, 2, 3\}$ that has exactly two equivalence classes. Regards.
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2answers
36 views

Let $x\in \mathbb{R}$ and $z\in\mathbb{C}$. And define equality ($x=z$) iff $x=(0,x)$. Is this equality well defined?

Let $x\in \mathbb{R}$ and $z\in\mathbb{C}$. And define equality ($x=z$) iff $x=(0,x)$. Is this equality well defined ? Okay. It is easily shown that something goes wrong if you define equality in ...
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2answers
59 views

Is $a \sim b$ such that $\gcd(a,b) > 1$ an equivalence relation?

Is $a \sim b$ such that $\gcd(a,b) > 1$ an equivalence relation? I know that it's reflexive, since $\gcd(a,a) > 1$. It's also symmetric since $\gcd(a,b) > 1$ iff $\gcd(b,a) > 1$. ...
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3answers
46 views

Equivalence Relation on $\mathbb{R}$ and its partition $\mathbb{R}/\sim$

Define the equivalence relation $\sim$ on $\mathbb{R}$ as follows: $$\forall a,b\in\mathbb{R},\ a\sim b\ \Leftrightarrow\ b-a\in\mathbb{Z}$$ I can prove that this is an equivalence relation, but I ...
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0answers
21 views

Question about Equivalence relation and partition

This is my first time of Abstract algebra, and I don't know how to solve this problem. Although I have an idea to solve this problem, I can't assure whather it is correct or not. Please show me how ...
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2answers
53 views

Determining whether relations are equivalence classes, and finding the equivalence classes

Determine if each of the following relations is an equivalence relation. If so, determine the equivalence classes. $S = \Bbb Z$, $a \sim b \iff a \equiv b \pmod 3$ or $a \equiv b \pmod 5$. ...
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1answer
53 views

Let ~ be an equivalence relation on a set S. Show that b is an element of cl(a) <=> cl(a) = cl(b) (Where all a,b are elements of S)

This was a question on my last equivalence relations quiz and I'm not yet comfortable with the whole "class" idea. I understand that I must show transitivity, reflexivity and symmetry however I'm not ...
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2answers
167 views

Determine the number of equivalence relations on the set {1, 2, 3, 4}

Hi this was a question listed on my last proofs and conjectures midterm. It is similar to my previous post however this asks a different question which is throwing me off.. Do I simply list all ...
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1answer
42 views

Equivalence relation and quotient set

I'm studying for a test and got stuck in one question regarding equivalence relations and quotient set. Here is the question: Let $F=\mathbb{R}\to \mathbb{R}$ be the set of functions from ...
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1answer
24 views

Checking the equivalence relations of sets

$S=\{0,1,2,3\}, R:SxS, (m,n)\in R \text{ if } m+n=4$. From the condition of $R$, I found that $R=\{(2,2),(1,3),(3,1)\}$. Now I have to see if $R$ is reflexive, symmetric, antisymmetric, and ...
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1answer
62 views

Let $H=\{2^m: m ∈ Z\}$ Where $m$ is any integer, and $ a\sim b\Leftrightarrow a/b $ is an element of $H$.

Show that is an equivalence relation and describe the elements in the equivalence class $\operatorname{cl}(3)$. We're studying sets and equivalence in my mathematical proofs class. As this is a ...
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1answer
39 views

Counting Ordered Pairs

Let $A$ be a finite set with $n \geq 4$ elements and let $\rho$ be an equivalence relation on $A$. Suppose that there are exactly $4$ equivalence classes, $C_1$, $C_2$, $C_3$, $C_4$. Moreover we know ...
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1answer
45 views

Let H be a (non-empty) subset of integers. Suppose a-b <=> a~b ∈ H is an equivalence relation.

Show that $0 \in H$. Show that if $a \in H$ then $-a \in H$, and lastly, if $a$ and $b \in H$ then $a + b ∈ H$. Last bonus question on our last unit test on sets, equivalence relations and proofs. I ...
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1answer
35 views

Equivalence Relation determined by $f(x)=x^2$

I have an exercise from my professor; For the function $f(x)=x^2$, for all $x\in \mathbb{R}$, describe the equivalence relation determined by $f$. So we are working in the set $\mathbb{R}$, so ...
3
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2answers
27 views

Equivalence relation confusion

Why is $\{(x,y)\mid x-y\text{ is a rational number}\}$ an equivalence relation and and why are $\{(x,y)\mid x-y\text{ is a irrational number}\}$ and $\{(x,y)\mid x+y\text{ is an integer}\}$ not?
0
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1answer
78 views

If $X = \{1, 2, 3, 4\}$ show there are just two equivalence relations on $X$ with $1\sim 2$ and $2 \sim3$

If $X = \{1, 2, 3, 4\}$ and $\sim$ is an equivalence relation on $X$, then if $1 \sim 2$ and $2 \sim 3$ show that there are just two possibilities for the relation $\sim$ and describe both ...
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1answer
17 views

An equivalence relation on filters

Let $F$ be a filter. We say $ X \sim_{f} Y $ iff $X \leftrightarrow Y$ $\in$ $F$. I am able to prove reflexivity and associativity of the relation but not the transitivity. Need help with that. Use ...
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1answer
116 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 ...
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3answers
28 views

Relation symmetric confusion

So Symmetric = (a,b), (b,a) Set = {<1, 1>, <1, 2>, <1, 4>, <2, 1>, <2, 2>, <3, 3>, <4,1 >, <4, 4>} I understand ...
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1answer
36 views

Equivalence relation example

On the Wikipedia page about Equivalence Relations, there is a simple example: Let the set $\{a,b,c\}$ have the equivalence relation $\{(a,a),(b,b),(c,c),(b,c),(c,b)\}$. The following sets are ...
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1answer
37 views

Proving that $(a_1, b_1)\sim(a_2, b_2)\Leftrightarrow\ a_1=a_2\land b_1=b_2$

This assingment is preparation for exam. I need to prove with $(a_1, b_1)\sim(a_2, b_2)\Leftrightarrow\ a_1=a_2\land b_1=b_2$ that $\sim$ is equivalence relatio. Can you tell me how to do this. ...
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3answers
64 views

Pigeonhole Principle and Equivalence Classes

Let $A$ be a finite set with $n \geq 4$ elements and let $\rho$ be an equivalence relation on $A$. Suppose that there are exactly four equivalence classes: $C_1, C_2, C_3, C_4$. Moreover we know that ...
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0answers
18 views

Bijection between partial order and $<$

How to show the correspondence between a less than relation and partial orders ? here A less than relation $<$ on a set $S$ is a relation that satisfies If $a < b$ , then $a \neq b$. ...
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3answers
38 views

What are Equivalence Classes

In most cases I can prove whether a relation is an equivalence relation or not but have no idea what "distinct equivalent classes" are. I tried to read some examples but couldn't figure out how to ...
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4answers
86 views

Different equivalence relations of the set $\{a,b\}$

In the book of Richard Hammack, I come accross with the following question: There are two different equivalence relations on the set $A = \{a,b\}$. Describe them. OK, I found that the solution ...
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3answers
50 views

Relations and Combinatorics exercise

Be $A=\{1,2,3,\ldots,10\}$ Determine how many equivalence relations can be defined in $A$ with exactly two equivalence classes. Determine how many equivalence relations can be defined in $A$ with ...
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1answer
16 views

Little equivalence-relation problem

If $U=\{1,2,\ldots,1000\}$ and $A = \mathbb P(U) - \{ \emptyset \}$, the following relation $R$ is defined in $A$ $$XRY \Leftrightarrow (\min X = \min Y) \wedge (\max X = \max Y)$$ Calculate ...
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1answer
16 views

Verify an equivalence relation.

For a set $E $ of real numbers, define two points in $E $to be rationally equivalent if their difference belongs to $Q $. Prove that this defines an equivalence relation. (i) is trivial as 0 is a ...
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0answers
35 views

Prove that $R_1$ refines $R_2$ if and only if the partition with respect to $R_1$ is a refinement of the partition with respect to $R_2$.

Let $R_1$ and $R_2$ be equivalence relations on a set $S$. Prove that $R_1$ refines $R_2$ if and only if the partition with respect to $R_1$ is a refinement of the partition with respect to $R_2$. ...
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1answer
32 views

Define a $T$ on $ A \times B$ by $((a_1,b_1),(a_2,b_2)) \in T \leftrightarrow (a_1,a_2) \in R$. Prove that $T$ is an equivalence relation.

Let $R$ be an equivalence relation on $A$ and let $S$ be an equivalence relation on $B$. Define a $T$ on $ A \times B$ by $((a_1,b_1),(a_2,b_2)) \in T \leftrightarrow (a_1,a_2) \in R$ and ...
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2answers
38 views

Inference Proof with Quantifiers

I am trying to figure out this implication proof. Can any of you guys tell me how to prove this? Prove ∀x((¬P(x) ∧ Q(x)) → R(x)) Implies ∀x(¬R(x) → P(x))
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24 views

Is an equivalence class same as residue class?

So my teacher's notes say this: An equivalence class consists of those integers which have the same remainder on division by n. They are also known as "congruence classes modulo n" Since ...
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1answer
48 views

Equivalence class of $T$ on $\mathbb{R} \times \mathbb{R}$ given by $(x,y) T (a,b)$ iff $x^{2}+y^{2}=a^{2}+b^{2}$

What is the equivalence class of $T$ on $\mathbb{R} \times \mathbb{R}$ given by $(x,y) T (a,b)$ iff $x^{2}+y^{2}=a^{2}+b^{2}$ I can see that the equivalence class cannot be negative, as the square of ...
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2answers
413 views

Attempting to find the equivalence class of 5.

For $a,b \in \mathbb{R}$ define $a \sim b$ if $a - b \in \mathbb{Z}$ How would you find the equivalence class of 5. In other words what I'm trying to describe is the set $[5]$ = {$y : 5 \sim y$}. And ...
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1answer
58 views

Equivalence Relation Quotient

Let $S$, $R$ be two equivalence relations in $A$, with $S \subset R$. Let $1^* : A/S \to A/R$ be the map induced by relation preserving map $1_A$. Define $(S a) R/S (Sb)$ if $1^*(S a)=1^*(Sb)$. ...
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1answer
102 views

Propositional Logic with rules of inference problem.

$$ \begin{array}{l} 1.\>\>\>\> (r ∧ ¬s) ∨ (q ∧ ¬s)\\ 2.\>\>\>\> ¬s → ((p ∧ r) → u)\\ 3.\>\>\>\> u → (s ∧ ¬t)\\ ...
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3answers
706 views

Prove that this is an equivalence relation and give all the different equivalence classes [closed]

Let $R$ be a relation defined on real numbers by letting $a\mathrel R b$ iff $\cos (a) = \cos (-b)$. Prove that this is an equivalence relation and give all the different equivalence classes. Also ...
2
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4answers
88 views

Equivalence relation and subgroup

I am taking abstract algebra now, and there's a lemma: Let $H$ be a subgroup of group $G$, for $a,b \in G$,define $a\sim b$ if $ab^{-1}\in H$, then it is an equivalence. I know how to prove it and how ...
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3answers
48 views

Partition induced by the Equivalence Relation

I'm not sure I understand this concept. Let's say we have a "Is parallel to" relation from the set of all lines in the Cartesian plane. What would be the partition induced by this relation? Thank ...