-1
votes
1answer
10 views

Equivalence relation and equivalence class question

Show that the relation $\sim$ defined on the set $X = \mathbb{N} \times \mathbb{N} = \{(a, b) : a \in \mathbb{N}; b \in \mathbb{N}\}$ as $(a,b) \sim (c,d)$ if and only if $a + d = c + b$ is an ...
0
votes
1answer
24 views

Give an example of a relation R on $A^2$ which is reflexive, symmetric, and not transitive

I am just looking for some clarification on this exercise: Let $A = \{a,b,c,d\}$. Give an example of a relation $R$ on $A^2$ which is reflexive, symmetric, and not transitive. I understand that if I ...
1
vote
2answers
40 views

How to find the Equivalence class for a given set?

I'm really having trouble understanding these equivalence classes. Could someone please guide me through the following problem step by step, and help explain this. I have a final exam next week, and ...
-1
votes
1answer
25 views

list all the equivalence relation [duplicate]

list all the equivanlance relations in the set A={1,2,3,4) so there should be 15 right? so what I got so far (1 1) (22) (33) (44) (12) (13) (14) (21) (23) (24) (31) (32) (34) (41) (42) (43) these ...
0
votes
1answer
31 views

Equivalence Relation on R (real numbers)

Let R be the relation on R(real numbers) defined by: For all x, y (that belong) to R(real numbers), x relates y <=> x-y (that belongs) to Z. (a) Is R an equivalence relation? Prove your answer. ...
0
votes
3answers
27 views

Describing Equivalence Classes using set builder notation

How would you describe all the equivalence classes for the relation: $congruence$ $modulo$ $5$ over $Z$, using set builder notation?
2
votes
1answer
19 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
2answers
37 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))
0
votes
1answer
64 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)\\ ...
0
votes
3answers
45 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 ...
0
votes
2answers
38 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 ...
1
vote
1answer
31 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. ...
4
votes
2answers
35 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
3answers
98 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 ...
0
votes
1answer
40 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 $O ∈ X$. ...
0
votes
1answer
67 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
79 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}. ...
1
vote
1answer
80 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
86 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 ...
3
votes
3answers
201 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 ...
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 ...
1
vote
3answers
87 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 ...
2
votes
3answers
119 views

Is this relation reflexive, symmetric and transitive?

Define a relation $R$ on the set of functions from $\mathbb{R}$ to $\mathbb{R}$ as follows: $(f, g) \in R $ if and only if $f(x) - g(x) \geq 0$ for all $x \in \mathbb{R}$ Is this relation ...
0
votes
3answers
26 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 ...
0
votes
0answers
53 views

What is this equivalence relation explicitly?

Let $S \colon = \{ \ (x,y) \in \mathbf{R}^2 \ | \ \ y = x +1, \ \ 0 < x < 1 \ \}$, and let $T$ be the intersection of all the equivalence relation on the plane that contain $S$. Then how ...
0
votes
2answers
113 views

Reflexive, Symmetric, Anti-Symmetric relations

Let $A = \mathbb Z \times ( \mathbb Z\setminus {0} )$. A binary relation $R$ on $A$ is defined as follows: For all $(a,b),(c,d) \in A$ $$(a,b) \,R\,(c,d) \iff ad = bc$$ now how do I find if $R$ is ...
0
votes
2answers
77 views

Discrete Math - Equivalence Classes

I'm trying to understand a problem that my textbook gives me. Here is the problem: The relation $R$ is an equivalence relation on the set $A$. Find the distinct equivalence classes of $R$. $A = \{0, ...
1
vote
1answer
42 views

how many elements does Ia have?

Let $A=\{1,2,3,4\}$. Let $F$ be the set of all functions from $A$ to $A$. Let $R$ be the relation on $F$ defined by $f,g \in F$ $f R g \Leftrightarrow |f(A)|=|g(A)|$ $f(A)=\{f(x): x\in A\}$ ...
1
vote
1answer
35 views

Need help to understand equivalence class

This is in my note Let S={1,2,3,4} Let R be the relation on P(s) defined by xRy <=>|x|=|y| how many equivalence classes are there ? 5 [∅]={∅} [{2}]={{1},{2},{3},{4}} [{2,3}]={{1,2},.......... ...
1
vote
0answers
25 views

Finding equivalence relations containing specific equivalences

"Find the number of equivalence relations on the set $\{1,2,3,\ldots,7\}$ such that: a) $1\sim2$ and $3\sim4$. b) $1\not\sim2$, $1\not\sim3$ and $3\not\sim2$." Solving this problem is equivalent to ...
-2
votes
1answer
51 views

Basic Equivalent Relations Question [duplicate]

For $x,y\in \mathbb R$ $x\sim y$ if and only if $x-y \in \mathbb Q$. I need help with the following questions: If $a \in \mathbb Q$, what is the equivalence class of $a$? If $a \in \mathbb Q$, prove ...
2
votes
1answer
19 views

What is the structure of a directed graph with vertex set A which has a relation R

I am studying for a test and found this question in the book: Let $R$ be an equivalence relation on the set A (Non-empty). Let $D_R$ be the directed graph with vertex set $A$ and an arc from $x$ to ...
0
votes
0answers
34 views

How does Dilworth’s Theorem apply to the set {0, 2, 6, 7}?

I'm having some serious problems with Dilworth's Theorem. My question is 'how does Dilworth’s Theorem apply to the set {0, 2, 6, 7}?'. Any help is appreciated.
0
votes
1answer
35 views

Binary relations: transitivity and symmetry

I've been looking at some examples for transitivity and symmetry. Suppose $A=\{0,1,2 \} $ and the relation $R=\{ (0,0),(1,1),(2,2),(1,2),(2,1) \}$ Well for starters this is clearly reflixe since ...
0
votes
1answer
50 views

Binary relations, closures and equivalences

Let $R$ be the relation on $Z$ such that $xRy \iff x-y=c$. Well, what I have so far is $R=\{ 0,-1,1,0,-1,1,0 \cdots\}$ Is $R^* $ and equivalence relation? Why not? This is where problems start: I ...
1
vote
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 ...
0
votes
1answer
161 views

Find the equivalence class of 0

R is a relation defined on the integers by $(a,b) \in R$ is $a^2-b^2$ and is divisible by 3. I set a or b to zero to get all the negative and positive values in the equivalence class. Although I want ...
0
votes
2answers
65 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
5answers
83 views

Number of equivalence relations with a fixed size

How can I find the number of equivalence relations R on a set of size 7 such that |R|=29? Any advice would be greatly appreciated! :D
1
vote
3answers
87 views

Determine if these are equivalence relations

I would appreciate if someone could go through the task and the answers I've got and check if I've done it correct, if not please correct me. Here is the task: Below we have listed some ...
-2
votes
1answer
57 views

Show that $∼$ satisfies the three properties of an equivalence relation. [closed]

Given sets $A$ and $B$, say that $A\sim B$ (the sets are equicardinal) if $\lvert A\rvert\sim\lvert B\rvert$ (that is, there exists a bijection $f$ from $A$ to $B$.) Show that $\sim$ satisfies the ...
0
votes
1answer
40 views

Proving that the relation $(x,y)S(x',y') \iff x - x' \in \mathbb{Z} \land y = y'$ is of equivalence.

The relation $S$ is of equivalence. I have to prove it. I managed to prove reflexibility and transitivity, but I'm having problems with symmetry. How can I prove it? The relation $S$ is defined ...
2
votes
1answer
320 views

How to determine the equivalence classes of a relation?

I don't fully understand how to find the equivalence classes of a relation. Over $\mathcal P(E)$, where $E = \{1,2,3,4,5,6\}$, $ARB \iff |A\cap\{1,2\}| = |B\cap\{1,2\}|$ From what I've seen, ...
1
vote
1answer
51 views

Determining equivalence classes of certain pairs for the relation $(a,b)R(c,d) \iff a^2 + 7b^2 = c^2 +7d^2$

This is an equivalence relations exercise. It has two parts. The first is about proving that the relation is of equivalence, which seems to be fine to me, but I'll put it there anyway. With the second ...
2
votes
1answer
60 views

Equivalence relation class $\bar{0}$

In the set $\mathbb{Z}$ we define the following relation: $$a\Re b \iff a\equiv \bmod2\text{ and }a\equiv \bmod3$$ 1)Prove that $\Re$ is an equivalence relation. (Done) 2) Describe the equivalence ...
0
votes
1answer
28 views

What is the equivalence class of a relation's element?

I'm studying about equivalence relations. My book has the following definition for an equivalence class: If $R=(G,A,A)$ is a relation of equivalence over the set $A$, the equivalence class of ...
3
votes
2answers
351 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 ...
4
votes
2answers
102 views

Prove that $\sim$ defines an equivalence relation on $\mathbb{Z}$.

For $a,b \in \mathbb{R}$ define $a \sim b$ if $a - b \in \mathbb{Z}$ I don't understand how I'm suppose to prove this: Prove that $\sim$ defines an equivalence relation on $\mathbb{Z}$ Also can you ...
1
vote
2answers
173 views

Trouble determining whether relations are reflexive, symmetric and transitive.

I'm having trouble understanding whether or not relations are reflexive, symmetric and transitive. I know that for a relation to be any of those it must satisfy the conditions: reflexive: for every ...
0
votes
2answers
32 views

Is this an Equivalence Relation and why?

if $I$ is a set of positive integers and relation $\def\R{\mathrel R}\R$ is defined over the set $I$ by $x\R y$ iff $x^y = y^x$. Is this an Equivalence Relation and why?