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

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3
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1answer
46 views

What is the common preimage (in $Z$) and the equivalence relation for Pushouts

Here it says: Suppose that $X$, $Y$, and $Z$ as above are sets, and that $f : Z → X$ and $g : Z → Y$ are set functions. The pushout of $f$ and $g$ is the disjoint union of $X$ and $Y$, where elements ...
2
votes
1answer
27 views

Example of an equivlance relation that is transitive

I am just tying to figure our this example but am having difficulty understand the math being used. The example state: Let R be a relation on the set $\mathbb{Z}$ defined as $(m,n)\in$ R if and only ...
0
votes
2answers
39 views

Trying to understand an example of an equivlance relation that is symmetric

I am just tying to figure our this example but am having difficulty understand the math being used. The example state: Let R be a relation on the set $\mathbb{Z}$ defined as (m,n)$\in$ R if and only ...
0
votes
1answer
49 views

Understanding an equivalence relation

Let $$R = \{ \left\langle {x,y} \right\rangle \in \wp (\mathbb{Z}) \times \wp (\mathbb{Z})|\exists t \in \mathbb{Z}.y = x + t\} $$ This is the equivalence class for $\{0\}$ $$\begin{array}{l} ...
1
vote
1answer
42 views

Proving the transitive property of an equivalence relation

I have to prove an equivalence relation.. $x$ is related to $y$ in the reals if $|x-y|\le3$ Reflexivity was easy. Symmetry was just a matter of breaking up the +ve and -ve case and it worked out. ...
0
votes
1answer
56 views

Help visualize set of all equivalence relations

I want to prove that the poset $Eq(A)$ with $\subseteq$ as the partial ordering is a complete lattice. But before even beginning to prove it, I have trouble visualizing the poset of $Eq(A)$. Kindly ...
0
votes
1answer
35 views

Determining number of different equivalence relations in a set with 4 elements

I have a set with 4 elements. Let A be $A=\{a,b,c,d\}$ How would I find number of different equivalence relations in this set? Should I use Bell's number theorem in which n would be 4? Should I ...
0
votes
2answers
28 views

Equivalence relation- Equivalence Classes And Partitions

I had the following question A is a finite set and $R \subseteq A \times A$ is a equivalence relation. Prove that $|A|$ is odd iff $|R|$ is odd. I am trying to find a general formula for this ...
0
votes
1answer
596 views

Proving equivalence relations

I just started my abstract algebra class and I am struggling with the concept of equivalence relations. I know that in order to prove equivalence relations, I have to prove the reflexive, symmetric, ...
4
votes
2answers
49 views

Using Logical Equivalences to prove $(((\neg r) \lor q) \lor ((q \lor (\neg p)) \land ((\neg p) \lor q)))$ is equivalent to $(\neg(r \land p) \lor q)$

I have been trying to solve the following proof: $$(((\neg r) \lor q) \lor ((q \lor (\neg p)) \land ((\neg p) \lor q)))\text{ is equivalent to } (\neg(r \land p) \lor q)$$ I am new to proofs and ...
0
votes
1answer
33 views

Is a relation$ R=\{(a,a):a\in I\}$ said to be reflexive, symmetric and transitive?

Consider a relation $R=\{(a,a): a\in A\}$ where $A=\{1,2,3\}$. i.e $R=\{(1,1),(2,2),(3,3)\}$ This clearly reflexive but is it necessary that all such relations are necessary to be symmetric and ...
0
votes
2answers
37 views

Logical Equivalence with →

I am given the problem of proving: $p → (q\land r) \equiv (p→q) \land(p→r)$ Using known logical equivalences. I'm not well practiced in transforming logical statements that contain →'s in them into ...
-2
votes
1answer
46 views

For a group $G$, show the relation $x\sim y$ defined by $\exists a(y=axa^{-1})$ is an equivalence relation on $G$.

Let G be a group. For $x,y\in G$, define $x\sim y$ if there exists some element $a\in G$ such that $y=axa^{-1}$. Show that ~ defines an equivalence relation on $G$.
1
vote
2answers
116 views

Conformal Equivalence of two Riemann metrics

I'm reading a paper and encountered a concept of conformal equivalence between two Riemannian metrics on a differentiable $2$-manifold $M$ : Two Riemannian metric $g$ and $f$ are conformally ...
1
vote
2answers
65 views

equivalence classes and cardinality

I need to prove that every equivalence class created by the equivalnce relation $\sim$ on $\mathbb{R}$, that is defined by: $a\sim b \Leftrightarrow (a-b) \in \mathbb{Q}$, is $\aleph_0$. Furthermore, ...
1
vote
0answers
34 views

The null set as the underlying set of a relation structure

Can the null-set underly a relation structure? Can it underly a well-order? If so, would such a well-order have order-type $0$? (Can one even have an order-type of $0$?) My motivation for this ...
0
votes
1answer
71 views

Transitive relations on a set of n elements

How to find out the total number of transitive relations in a set of n elements? I am facing a problem in finding all the possible cases, is it not possible to find all cases? If not possible, why?
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
25 views

Show that f is a symmetric relation on A

I am learning about relations and I come across this exercise. And I don't understand the problem. Let me first state the problem here: Let $f: A \rightarrow A$ be a function for which $f(f(x))=x$ ...
2
votes
2answers
46 views

Equivalence and Order Relations

I have the following problem: Provide an example of a set $S$ and a relation $\sim$ on $S$ such that $\sim$ is both an equivalence relation and an order relation. Conjecture for which sets and this ...
0
votes
3answers
43 views

Equivalence Relation Statements Proof

Let $\sim$ be an equivalence relation on a class $X$. The following are equivalent for $x,y \in X$. 1) $[x]=[y]$ 2) $x \sim y$ 3) $x \in [y]$ 4) $y \in [x]$ 5) $[x] \bigcap [y] \neq \emptyset$ ...
0
votes
1answer
70 views

Show that homeomorphism is an equivalence relation in metric spaces

It needs to be shown that homeomorphism is reflexive, symmetric and transitive in all metric spaces. Reflexivity seems to be easy to show, but I'm not sure how to do the rest. Any help?
0
votes
0answers
34 views

A question on Partitioning regarding equivalence relations

Let $S$ be the Cartesian coordinate place $\mathbb R \times\mathbb R$ and define the equivalence relation $R$ on $S$ by $(a,b) R (c,d)$ iff $b-3a = d-3c$ Find the partition $D$ determined by $R$ by ...
0
votes
3answers
122 views

Can a premise imply contradictory statements?

Can a premise imply contradictory statements? Can two contradictory premises imply the same conclusion? Determine the answers to these questions by doing the following. Prove or disprove: the ...
-1
votes
1answer
63 views

Sets Modulo Equivalence Relations

I am stuck on this question and would greatly appreciate any help: Recall, for an arbitrary set $S$ and equivalence relation $\equiv$ on $S$, $S/\equiv$ denotes the set of equivalence classes in $S$. ...
5
votes
0answers
162 views

Colored Picture for Equivalence Classes, Relations, Partitions, ..

Origin — A Book of Abstract Algebra — Charles Pinter — p120. I'm trying to sketch a colored picture for the ideas from equivalence classes, equivalence relations, partitions, etc... underneath. ...
0
votes
2answers
108 views

What does an equivalence class look like?

Let $\mathbb{R}^2 = \{Q = (a,b) | a,b\in \mathbb{R}\}$. Prove that if $q_1 = (a_1,b_1)$ and $q_2=(a_2,b_2)$ are equivalent, meaning $a_1^2+b_1^2 = a_2^2 +b_2^2$, then this gives an equivalence ...
0
votes
1answer
73 views

Shortcut method for proving equivalence relations

Define the relation R on N*N by: (x,y)R(z,w) if and only if x-z = w-y. Check whether R is an equivalence relation. Explain your answer My teacher answer is: Using the shortcut method: ...
0
votes
1answer
47 views

Equivalence class for a relation

Consider the equivalence relation on Z ! Z given by (m, n)R(p, q) if and only if mq = np: (a) Find the equivalence class represented by (2, 5). (b) Describe the set S of the equivalence classes ...
0
votes
1answer
69 views

Equivalence Relations

I would appreciate any help available for the following problem: Let $S$ be a set. Let $T$ be the set of all relations on $S$. Construct a relation $\equiv$ on $T$ in the following way: for $\sim, ...
1
vote
0answers
32 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 ...
0
votes
1answer
44 views

Very Abstract Relation with points

So I have this question on relations, that I really cant understand. I mean, I cant understand the question to be honest. Suppose a set $X$ of points on the plane and we "stabilize" a point $O ∈ X$. ...
8
votes
3answers
170 views

Can we take images of equivalence relations?

Given a function $f : X \rightarrow Y$, it is well-known that we can take the image under $f$ of any subset $A \subseteq X$, and we can take the preimage under $f$ of any subset $A \subseteq Y$. This ...
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 ...
1
vote
1answer
28 views

Equivalence involving diagonalization of block matrix

Given $T^\top L T = diag(A, B, C)$ being a block diagonalization with $L$ symmetric $A,B$ positive definite (p.d.) $T = \left( \begin{array}{ccc}I & 0 & T_2 \\ T_1 & I & T_3 \\ 0 ...
0
votes
1answer
70 views

How to show or prove equivalence relation?

I have this relation : for all integers m and n so : m R n ⇔ m ≡ n mod(3) How can I show that R is an equivalence relation
0
votes
2answers
92 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}. ...
5
votes
1answer
105 views

How to simply show that there are “78 'strict ordinal' 2x2 game matrices”

In "Theory of Moves", Steven J. Brams analyses two-player games with two strategies per player, where each player can totally rank his payoffs, although payoffs need not be comparable among players. ...
0
votes
1answer
38 views

If $E_1$ and $E_2$ are equivalence relations, is $E_1\circ E_2$ an equivalence relation?

I'm given two equivalence relations $E_1$ and $E_2$ over a set A and need to show whether the composition $E_1 \circ E_2$ is reflexive, symmetric and transitive. I only managed to show that $E_1 ...
2
votes
1answer
100 views

Equivalence relation question with cardinality and countability $A=\mathbb R,\ aSb \iff a-b\in \mathbb Q $

Let $A=\mathbb R,\ aSb \iff a-b\in \mathbb Q $ What is the cardinality of $[\pi]_S$ ? Prove that the quotient group $\mathbb R/S$ is uncountable. Well I think that cardinality is ...
1
vote
1answer
120 views

Prove or disprove question with equivalence relation, classes and quotient group

Let $A$ be a set and $R$ an equivalence relation on $A$. Prove or disprove: If $A$ is countable then all the equivalence classes of $R$ are countable. If $A$ isn't countable then the ...
2
votes
1answer
55 views

Equivalence classes of points of R^2

Let $A$ and $B$ be two sets of points in $\Bbb R^2$. We define an equivalence relation on the powerset of $\Bbb R^2$, by saying that $R(A,B)$ iff there is a translation $f$ on $\Bbb R^2$ such that the ...
3
votes
3answers
260 views

Equivalence relation question with functions

We'll define on the set: $A=\Bbb R^{[0,1]}$ the relation $R$ by $fRg$ if $f(0)=g(0)$. Make sure it's an equivilence relation. What is $[\cos x]$ ? Describe all the equivalence classes ...
1
vote
3answers
59 views

Proof of equivalence relation on a set

Let $X = \{(a,b) \mid a,b \in \Bbb Z; b \ne 0\}$. We define $(a,b)\mathrel R (c,d)$ iff $ad = bc$. Prove that $R$ is an equivalence relation on the set $X$. Which known set do the equivalence classes ...
0
votes
2answers
16 views

Having trouble with symmetry (equivalence relation)

Define a relation of $x,y \in R$ when $x = |y|$. I know this is reflexive as $x = |x|$ holds true because the relation has to have x as positive since $x = |y|$ which makes $x$ have to be positive ...
0
votes
4answers
51 views

Bijection from set of equivalence classes to $\mathbb R$

In $\mathbb R[X]$, define an equivalence relation ~ by $P_1$~$P_2$ if $P_1-P_2$ is divisible by X. I have shown that $X$ is an equivalence relation. Let $\mathscr Q$ denote the set of equivalence ...
2
votes
2answers
75 views

Properties of the relation $R=A\times B \cup B\times A$

A is a set. Let $B\subsetneq A$. $R=A\times B \cup B\times A$ Determine if the relation is (a)reflexive, (b)symmetric, (c)transitive, (d)anti-reflexive, (e)anti-symmetric, (f)asymmetric, ...
1
vote
1answer
61 views

Properties of the relation $R=\{(x,y)\in\Bbb R^2|x-y\in \Bbb Z\}$

$A= \Bbb R \\ R=\{(x,y)\in\Bbb R^2|x-y\in \Bbb Z\}$ Determine if the relation is (a)reflexive, (b)symmetric, (c)transitive, (d)anti-reflexive, (e)anti-symmetric, (f)asymmetric, (g)equivalence ...
1
vote
3answers
154 views

Define a relation on the set of all real numbers $x,y \in \mathbb{R} $ as follows:

Define a relation on the set of all real numbers $x,y \in \Bbb{R} $ as follows: $x \sim y$ if and only if $x - y \in \Bbb{Z}$ Prove this is an equivalence relation and find the equivalence class of ...
0
votes
1answer
65 views

Equivalence relation $g\sim h :\Longleftrightarrow h \in \{g,g^{-1}\}$

Let $(G, \cdot)$ be a group with an Identity element $e$. (i) A relation on $G$ is defined through $g\sim h :\Longleftrightarrow h \in \{g,g^{-1}\}$. Show that $\sim$ is a equivalence relation and ...