# Tagged Questions

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### Binary relation, reflexive, symmetric and transitive

I have a question regarding an image. I'm currently studying binary relations and the following image confused me: What got me confused is that the page from which I got the link ...
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### Finding the equivalence classes of a trigonometric relation

I have been asked to respond to the following: Define a binary relation R on $\mathbb{R}$ as ${\{(x, y) \in \mathbb{R} \times \mathbb{R} \mid \sin(x) = \sin(y)\}}$. Prove that R is an ...
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### Prove a relation is a equivalence

Let $\sim$ be defined so that $a\sim b$ when $a+b$ is even. Is this an equivalence relation? Equivalence relations confuse me a lot, so any help is appreciated!
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### Equivalence relations and power sets.

Let $\mathcal{A}$ be the class of all sets and define the relation $R$ on $\mathcal{A}$ as: $A\space R\space B$ iff there is a bijective function $f:A \to B$. Prove that $R$ is an equivalence relation ...
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### Define an equivalence relation on $\Bbb{Z}$ by $x\sim y$ if $x+2y$ is divisible by $3$

How do I approach this problem? I know how to approach on the equivalence relation of triangles(finding $x\sim x$, finding if $x\sim y$ then $y \sim x$, and finding if $x \sim y$ and $y \sim z$, then ...
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### Equivalence Class.

Let R be the relation of congruence mod 4 on Z: a R b if a - b = 4k, for some k in Z. What integers are in the equivalence class of 31? How many distinct equivalence classes are there? What are ...
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### Power Set, Bijection Function, Equivalence Relation

Let $S$ be a set and $P(S)$ the power set of $S$. For sets $A,BâŠ†P(S)$, we say that $A \sim B$ if there exists a bijective function $f: A \rightarrow B$. Show that $\sim$ is an equivalence relation.
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### Show that the equivalence classes of $\sim$ are left cosets of $H$ in $G$.

Let $H \leq G$ and define a relation on $G$ by $x \sim y$ if $y^{-1}x \in H$. Show that $\sim$ is an equivalence relation on $G$ and then show that the equivalence classes of $\sim$ are left cosets ...
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### Show that if $R$ is a strict partial order on $X$, and $R$ is not linear, then there exists a strict partial order $R'$ and $R' \supsetneqq R$.

Question: Show that if $R$ is a strict partial order on $X$, and $R$ is not linear, then there exists a strict partial order $R'$ and $R' \supsetneqq R$. My attempt: By definition 6.23,6.3.1, and ...
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### 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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### $R$ 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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### 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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### 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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### 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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### 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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### 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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### 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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### Let $H=\{2^m: m âˆˆ Z\}$ 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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### Let H be a (non-empty) subset of integers. Suppose a-b <=> a~b âˆˆ 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 $a + b âˆˆ H$. Last bonus question on our last unit test on sets, equivalence relations and proofs. I ...
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### 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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### 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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### 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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### 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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### 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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### What is the proper way to format a hypothetical syllogism proof?

Problem: Show that these three statements are equivalent, where $a, b \in R:$ (i) $a < b$, (ii) the average of $a, b,$ is greater than $a,$ and (iii) the average of $a$ and $b$ is less than $b$. ...
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### Proof that a given relation is an equivalence relation

Can someone can tell me if my proof of the next propostion is correct? Define the following relation: $$a\sim b \iff a-b=km, m\in \mathbb{Z}$$ Show $\sim$ is an equivalence relation And so here's my ...
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### How can be a class of paths be open for a connected open set?

I have the following excercise: Let $A$ be an open set. If $x,y\in A$ we write $x\sim y$ when there is a path from $x$ to $y$, this is, $\exists P=\bigcup_{i=0}^{n} [r_{i-1},r_i]$ with $a=r_0$ and ...
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### 3-dimensional cube shortest path question

Let Q be the graph consisting of vertices and edges of a 3-dimensional cube. Two relations are defined on the vertices of Q. â€¢ R1={(v,w):the shortest path from v to w has an odd number of edges}. ...
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### Equinumerosity between equivalence classes set and power set

IÂ´m currently working on the following problem: "Let $\xi$ = $\{$ $\bot$ $\mid $$\bot is a equivalence relation over \mathbb{N}$$\}$ Show that $\xi$ and $2^\mathbb{N}$ (power set) are ...
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### How to calculate equivalence relations

How can I calculate how many equivalence relations can be defined on a given set? For example: How many possible equivalence relations can be defined on S = {a,b,c,d}?
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### Define a relation $\sim$ on $\mathbb{N}$ by $a\sim b$ if and only if $ab$ is a square

(a) Show that $\sim$ is an equivalence relation on $\mathbb{N}$. (b) Describe the equivalence classes [3], [9], and [99]. (c) If $a\sim b$, which attributes of $a \text{ and } b$ are equal? For (a) ...
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### Let $R$ be an equivalence relation on a set $A$, $a,b \in A$. Prove $[a] = [b]$ iff $aRb$.

Hello I need help with the proof strategy for this problem. Let $R$ be an equivalence relation on a set $A$ and let $a,b \in A$. Prove that $[a] = [b]$ if and only if $aRb$.
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### Minimum Equivalence Relation

Let $X= \{1,2,3,4\}$, and $R = \{(1,2),(3,4)\}$. Show the minimum equivalence relation on $X$ that extends $R$. How many elements does the quotient set $X/R$ have ? Can somebody give hints to solve it ...
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### easy homework: Equivalence classes, how do they look?

Let's say that I got a set = { Arnold, Harrison } and I want to display the equivalence class of [ Harrison ] The actual ...
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### Equivalence Relations and functions on partitions of Sets

let $f$ be a function on $A$ onto $B$. Define an equivalence relation $E$ in $A$ by: $aEb$ if and only if $f(a)=f(b)$. Define a function $\phi$ on $A/E$ by $\phi([a]_{E})=f(a)$. Hint: Verify that ...
Yes this is a homework problem but I have attempted to solve it and my work is below, also this is my first question here so I'm sorry for any mistakes: Question: Context: Let $A$ and $B$ be subsets ...