2
votes
2answers
50 views

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 ...
0
votes
2answers
37 views

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 ...
0
votes
1answer
36 views

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!
0
votes
1answer
36 views

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 ...
0
votes
2answers
32 views

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 ...
1
vote
1answer
38 views

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 ...
1
vote
0answers
69 views

Power Set, Bijection Function, Equivalence Relation

Let $S$ be a set and $P(S)$ the power set of $S$. For sets $A,B⊆P(S)$, we say that $A \sim B$ if there exists a bijective function $f: A \rightarrow B$. Show that $\sim $ is an equivalence relation.
0
votes
1answer
36 views

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 ...
0
votes
1answer
69 views

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 ...
0
votes
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. ...
2
votes
1answer
58 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 ...
-3
votes
2answers
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.
0
votes
2answers
60 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 $\gcd(b,a) > 1$. ...
-2
votes
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 ...
0
votes
2answers
168 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 ...
0
votes
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 $a \equiv b \pmod 5$. ...
2
votes
1answer
25 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 ...
0
votes
1answer
63 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 ...
0
votes
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 ...
0
votes
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 ...
1
vote
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 ...
2
votes
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 ...
0
votes
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 $R_2$. ...
1
vote
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 ...
1
vote
1answer
54 views

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 $b$. ...
1
vote
1answer
35 views

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 ...
1
vote
1answer
31 views

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 ...
0
votes
2answers
90 views

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. • R1={(v,w):the shortest path from v to w has an odd number of edges}. ...
0
votes
1answer
65 views

Equinumerosity between equivalence classes set and power set

I´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 ...
0
votes
2answers
85 views

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}?
0
votes
2answers
89 views

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) ...
0
votes
1answer
40 views

Equivalence classes

I'm posting this question and answers to see if I am on the right track here, just want to be sure I understand or don't understand. Bellow I will list some equivalence relations over the set $ S= ...
1
vote
0answers
33 views

Congruence induced by a subset.

Consider the additive monoid of natural numbers and calculate the congruence generated by $\{(2,3)\}$. I know the answer is that the congruence has 3 classes which are $\{0\}$, $\{1\}$, and the rest. ...
0
votes
2answers
27 views

Why are both of these not equivalence relations?

Can anyone tell me why the first set is an equivalence relation, and not the second? As far as I can see, both are reflexive, symmetric and transitive, but my books says only the second one is an ...
1
vote
1answer
44 views

Equivalence relation necklaces problem

Consider the question of making necklaces by arranging n distinguishable beads on a string. Assume that once all $n$ beads are placed on the string its ends are carefully knotted so the knot cannot be ...
0
votes
1answer
42 views

Show that $s_1(\text{equivalence class}) = s_2(\text{equivalence class})$ iff $s_1\mathrel{R}s_2$.

Let $R$ be an equivalence relation on $S$. Show that for all $s_1, s_2$ elements of $S$ we have $s_1(\text{equivalence class}) = s_2(\text{equivalence class})$ iff $s_1\mathrel{R}s_2$. I understand ...
1
vote
1answer
37 views

Equivalence relations, Cosets

Let G be a group and for elements a,b (elements of)G let a R b mean that there exists an element x(element of)G such that a=xbx^(-1). Show that R is an equivalence relation on G. Not really sure how ...
0
votes
0answers
44 views

Equivalence relations, cosets.

Let R be an equivalence relation on S. Show that for all s1, s2 elements of S we have s1(equivalence class) = s2(equivalence class iff s1Rs2. Not really sure how to go about this problem.
2
votes
1answer
94 views

Finding the equivalence relations determined by functions

I have a homework question: Consider the maps: $$f : \mathbb{R}^2 \to \mathbb{R},\; (x,y) \mapsto x^2 +y^2$$ $$f : \mathbb{R} \to \mathbb{R},\; x \mapsto x^3$$ $$f : \mathbb{C} \to \mathbb{C},\; z ...
1
vote
2answers
417 views

Reflexive but not Transitive relation

What is an example of a relation $\mathscr{R}$ on a set $S$ such that $\mathscr{R}$ is reflexive but not transitive? Here is what I have come up with. Let $S = \mathbb{Z}$. Then let $\mathscr{R} = ...
1
vote
3answers
223 views

How do I prove the following is an equivalence relation?

Define the relation $R$ on the set of all ordered pairs of real numbers as follows: $(x, y)R(s, t)$ iff $2(x - s) = (y - t)$. Prove that $R$ is an equivalence relation. Find the ...
0
votes
1answer
38 views

Number of (equivalence) relations fulfilling some additional conditions

let say I have $A=\{1,\dots,8\}$ I want to know the following things: what the number of relations on $A$? what the number of reflexivity relations on $A$? what the number of equivalence relations ...
-1
votes
1answer
121 views

equivalence class being finite [on hold]

If $R$ is an equivalence relation on a set $S$,for all $s \in S$ the equivalence class $[s]$ is finite,and there are finitely many equivalence classes altogether,then $S$ itself is finite. I know ...
0
votes
1answer
41 views

Proving convergence of a series

I need to show that the series of general term $$\tanh \frac{1}{n}+ \ln \frac{n^2-n}{n^2+1}$$ converges. I was thinking to use an equivalence as $n \rightarrow \infty$ We know that $ \tanh ...
0
votes
2answers
303 views

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 $aRb$.
0
votes
2answers
107 views

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 ...
1
vote
2answers
49 views

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 ...
2
votes
2answers
72 views

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 ...
2
votes
1answer
223 views

Numerical equivalence between sets

I need some help on homework. Here is the problem I am stuck on: Prove that every closed interval [a,b] is numerically equivalent to [0,1] I believe that I need to find an injection between the two ...
5
votes
2answers
426 views

Sets, functions and relations problem

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 ...