-1
votes
1answer
41 views

determining reflective, symmetric, transitive, anti-symmetric properties and describing equivalence classes

The question: determine if p is reflective, symmetric, transitive and/or anti-symmetric, if p is an equivalence relation, describe the equivalence classes A = Z , and $apb$ if and only if $5 | (2 a + ...
0
votes
1answer
22 views

Transitive Relations on a set

I am trying to study binary relations (for myself, it's not an assignment!) I have the set $\{1,2,3,4\}$, and one of the relations in the exercise is $\{(1,3),(1,4),(2,3),(2,4),(3,1),(3,4)\}$. A ...
1
vote
1answer
20 views

How does one find/list equivalence classes?

Can someone explain how I would find/list the equivalence classes (And number of equivalence classes) of these two examples? Example 1: A is the set of all possible strings of 3 or 4 letters in ...
0
votes
1answer
27 views

How would I draw the diagram for this relation?

The question I am trying to solve is below. I have proven it is an order but am unsure how to draw the diagram for it. Can someone point me in the right direction? Let A = {1, 2, 3, 4}, and let R be ...
0
votes
1answer
28 views

How to prove this equivalence relation?

How would one go about proving this is an equivalence relation? I have no idea where to start. $\cal R$ is the relation on $\Bbb Z \times \Bbb Z$, such that $((a, b),(c, d)) \in \cal R$ if and only ...
1
vote
1answer
21 views

Proving an equivalence relation(specifically transitivity)

I'm currently learning about equivalence relations. I understand that an equivalence relation is a relation that is reflexive, symmetric, and transitive. But I'm having trouble proving the transitive ...
0
votes
1answer
17 views

Composition of relations: Incomplete proof.

Let $R$ be a relation from $A$ to $B$, and $S$ be a relation from $B$ to $C$, and $T$ be a relation from $C$ to $D$. I want to prove that $T\circ (S\circ R)=(T\circ S)\circ R$. This is how I proved ...
1
vote
1answer
39 views

Partial order relation (Antisymmetric property), given a relation $xRy \iff x-y\le 4$

Given the set: $A=\{1,2,3,\dots,19,20\}$. The relation $R$ is defined on $A$ as: $xRy\Leftrightarrow x-y\leq4$ Is $R$ a partial order relation? I know that for a relation to be partial order it has ...
1
vote
1answer
50 views

The composition of the $<$ relation with itself

I am struggle with answering this question. I do not understand how to approach this question. 1.Let <􏰈 denote the less than relation on the set of integers. Describe the squared relation <^2 ...
0
votes
3answers
22 views

Why is this Relation R (graph) not transitive?

Let the arrow graph of R be the following: If we get the ordered pairs we have that R = { (a,a), (a,b), (a,c), (b,b), (b,a), (b,c), (c,a), (c,b), (d,d) } If we analyze this: *Reflexive - NOT ...
0
votes
1answer
17 views

Set of ordered pairs of the transitive closure R* of R

I pretty much know how to get the ordered pairs by doing the arrow graph method since the matrix method is much more complex. let R be: R = { (a,b), (b,a), (a,c), (c,d), (c,e), (e,c) } (I am ...
0
votes
2answers
30 views

How do you determine if a relation is transitive?

Suppose I have the relation P such that $$ x P y $$ iff $$ x = y^2 $$ How do I determine whether or not the relation is transitive?
1
vote
1answer
27 views

Steps to determine if a relation of a set is reflexive,symmetric or transitive?

I am having problem understanding these concepts. For example, let $A = \{2,3,4,5,6,7,8\}$. The definition I found says that $x R y \iff 3 | (x-y)$. How do I know if the relation $R$ on $A$ is ...
2
votes
1answer
27 views

I have two symmetric relations on a set. How can I prove that the symmetric difference is irreflexive?

I have this problem. Let R and S be symmetric relations on a set A. Prove or disprove: $R \oplus S$ is irreflexive. Now I'm assuming it's not true, because $(x,x)$ can be an element of $R$ ...
0
votes
1answer
46 views

What is the cardinality?

Let $A=\left\{1,2,\cdots,10\right\}$ Let $f,g:A\to A$. Consider the equivalence relation $$ fRg \iff \exists h:A\to A. f=h\circ g$$ where $h$ is invertible. Now, let $g(x)=5$: Why is $\left| ...
3
votes
3answers
519 views

How many functions are transitive?

Let the set of all functions defined as: $\left\{a,b,c,d\right\} \rightarrow \{a,b,c,d\}$ How many functions are transitive? I've been told to use the fact that a function is transitive iff "it's ...
2
votes
1answer
57 views

Proving that a relation is an equivalence relation

I am having difficulties proving the relation IS an equivalence relation. Let $f: X\longrightarrow Y$ be a function from a set $X$ onto a set $Y$. Let $R$ be the subset of $X \times X$ consisting ...
2
votes
1answer
30 views

Is this relation symmetric

$R = \{(X, Y) \in \mathscr{P}(A)^2| X \subset Y \text{ and }X \neq Y \}$ I know that $(X,Y) \in R$ holds true since $X \subset Y$. However I'm unsure if $(Y,X) \in R$ since if $Y \subset X$ then ...
1
vote
1answer
72 views

Is the subset relation on the powerset of a set, with qualification, reflexive?

I was wondering if the subset relation is reflexive? $R = \{(X, Y ) \in P(A)^2\mid X\subseteq Y \text{ and } X \neq Y \}$ I assumed they it was reflexive since for all $X \in P(A), X \subseteq X$ is ...
0
votes
0answers
32 views

Properties of R, R^n, R*

I was talking to a friend who mentioned that eventually, R^n and R* are equivalent. This confuses me because I don't see how it's necessarily the case. But it does seem to hold, for instance: R = ...
1
vote
1answer
30 views

Let R be the relation on ℤ+→ℤ+ defined by (a,b)R(c,d) if and only if a-2d=c-2b. List all the elements of the equivalence class [(3,3)].

I'm confused on how to find all the elements. I know how to find some but not all, wouldn't they be infinite? This is affecting me with the other questions as well. Thanks in advance!
3
votes
1answer
36 views

Is there a specific name for a directed graph that is composed of only loops?

Recently I have been doing practice questions for my Final exam tomorrow and this one question appeared that was interesting, but I couldn't seem to find the other half of the answer to it. Q: Given ...
2
votes
1answer
82 views

Relations examples (reflexivity, symmetry, transitivity)

I've found the two textbooks I'm using to to be particularly unhelpful in explaining these concepts, especially as they relate to English examples (non-existent). The first few following questions ...
0
votes
1answer
50 views

Properties of a relation on matrices: $(m_1,m_2)\in R$ iff $m_1\cdot m_2$ is defined

Let $M$ be a set of matrices of integers. Let $R$ be the relation on $M$ defined as follows: For any two $m_1, m_2 \in M, (m_1, m_2) \in R$ iff the matrix multiplication $m_1 \cdot m_2$ is defined. ...
1
vote
2answers
94 views

Why relations are defined as the smallest

Often relations are defined as follows: The xxxxx relation is the smallest relation satisfying... My question is why relations are defined as the smallest ...
1
vote
2answers
41 views

If $R$ is a transitive realation, then $R\circ R\subseteq R$

Here's the question I'm struggling with: Let R be a transitive relation on a set A. Prove the R composed with R is a subset of R. I'm kind of lost on how to prove this. I've started with saying: ...
2
votes
2answers
112 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
53 views

Sets and Relations in Math

I have not knowledge about relations, could you help me to solve this excercise step by step, to use in futures excercices? Thanks for your time. Given the set $A = \{1, 2, 3, 4\}$ and $B = \{1, 3, ...
2
votes
3answers
37 views

let A be a set of 6 distinct postive integers each <= 12, show that the sum of non empty subests of A cannot all be distinct [closed]

let A be a set of 6 distinct postive integers each <= 12, show that the sum of non empty subsets of A cannot all be distinct. for when does this not continue to hold up ( ie instead of 12 , its ...
0
votes
3answers
53 views

Giving an equivalence relation that corresponds to set partitions

My question is: Give equivalence relation that corresponds to the partitions A1 = {1,3,5} A2 = {2} A3 = {4,6} of the set A = {1,2,3,4,5,6} I don't know what the format of the relation should be, in ...
0
votes
1answer
28 views

Define a numeric relation that is reflexive, but not symmetric or transitive.

Define a numeric relation that is reflexive, but not symmetric or transitive. I've googled on this one quite a bit and am stuck.
1
vote
2answers
30 views

Relations on a set, check my answers?

I've been struggling with identifying relations on a set, and was hoping someone could check my answers and make sure I'm on the right track. Let A = {1,2,3,4} and R be a relation on the set A ...
0
votes
1answer
46 views

Blood relation - How A is related to B

This is a data sufficiency question - Q - How is A related to B? ...
1
vote
1answer
58 views

Is the property reflexive, symmetric, anti-symmetric, transitive, equivalence relation, partially ordered given the relation below?

I'm working on this and I'm supposed to figure out if the following properties apply to the below relations. Properties are: ...
0
votes
1answer
19 views

How do I derive a contradiction from an assumption that is “not asymmetric”

Let $R$ be a transitive relation on a set $A$. Define another relation, $S$, such that, for any $x,y \in A$, $Sxy$ iff $Ryx$. Moreover, let $S$ be irreflexive. Prove: $S$ is asymmetric on $A$. ...
0
votes
2answers
27 views

I need help proving that a relationship is not anti-symmetric

Basically this is what the problem says: R and S are two are anti-symmetric, prove that $R\:∪\:S$ is anti-symmetric as well. I know it isn't because when I try to simplify it, it ends up getting ...
1
vote
2answers
25 views

Not sure how to do Non-Homogeneous Recurrence Relations

I have a sample exam paper, and the answer is given, but I can't work out the answer from the question: Find the solution of: $a_n = \frac{1}{3}a_{n-1} + 2$ using $a_0 = 4$ Given Answer: $a_n = ...
2
votes
1answer
17 views

Need help proving that $ fRg \Leftrightarrow fg = f $ on $ B^{n} $ to $ B $ if and only if $ f(b_1,…,b_n) \leq g(b_1,…,b_n) $

I'm trying to gather my thoughts for proving the following claim: For $ fRg \Leftrightarrow fg = f$ on $B^{n}$ to $B$, show that $ fRg $ if and only if for any input values $ b_1,...,b_n $, we ...
0
votes
1answer
32 views

Equivalence Relation ~

let S = {1,2,3,4} Explain why each of the below are not equivalence relation. { (1,1), (1,2), (2,1), (2,2), (3,3) } { (1,1), (1,2), (2,3), (1,3), (2,2), (3,3), (4,4) } { (1,1), (2,2), (3,3), ...
0
votes
1answer
32 views

Partition induced by a Relation

Here's the problem: Let $A=\{1,2,3,4,5,6,7,8,9\}$. Define a relation $R$ on set $A$ by $xRy$ if and only if $2\mid(x+y)$ Assuming that $R$ is an equivalence relation, determine the partition of set ...
0
votes
1answer
19 views

Need help clarifying relation properties

So I am facing some issues determining the right properties for: $ xRy\;if\,\sin^2(x) + \cos^2(y) = 1 $. (On real numbers) Obviously this one is reflexive as $\sin^2(x) + \cos^2(x) = 1 $ is a basic ...
0
votes
2answers
28 views

Bijection of a function.

Define the function f: $(2,\infty) -> (-\infty,-1)$ by $f(x)= \frac{-x}{x-2}$. Show that f is bijective. I know i need to prove both injective and surjective, and I was able to solve the equation ...
1
vote
1answer
95 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
1answer
28 views

Symmetric relation , why are these symmetric?

$R_1 = \{(a,b)$ such that $a \leq b \}$ $R_2 = \{(a,b)$ such that $a>b \}$ $R_3 = \{(a,b)$ such that $a=b$ or $a=-b \}$ $R_4 = \{(a,b)$ such that $a=b \}$ $R_5 = \{(a,b)$ such that $a=1+b \}$ ...
0
votes
1answer
34 views

Is a relation induced by a partition always an equivalence relation?

Is a relation induced by a partition always an equivalence relation? I'm having some serious trouble understanding this concept and I was wondering if this is true.
1
vote
1answer
31 views

Proving a relation is a total order relation

Consider question #21 part a: Here is the solution: However, consider the definition of a total order relation: The solution didn't prove that the relation is a partial order relation. This ...
0
votes
1answer
23 views

How to count the number of distinct equivalence classes for a relation involving truth tables?

I am having trouble with question 22 part (2): Here is the solution: How did the author know that there are 256 distinct equivalence classes? Where did they get $2^8$ from?
0
votes
1answer
73 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. ...
1
vote
1answer
63 views

Let A = {1,2,3,4} Let F be the set of all functions from A to A. (check the parts)

Let $\operatorname{S}$ be a relation on $F$ defined by: $\forall f, g \in F, f\,\operatorname{S}\,g \iff f(i) = g(i), \exists i \in A$. (a) Recall that the identity function $I_A : A \mapsto A$ is ...
1
vote
1answer
52 views

$M_{R^n}$; how to derive $n$ for transitive closure?

When finding the transitive closure of a relation $R$, I convert $R$ into a boolean matrix $M_R$, and find the union between $M_R$ and its powers up to $n$. $$M_{R^*} = M_{R^1} \lor M_{R^2} \lor ...