2
votes
1answer
26 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
43 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. ...
0
votes
2answers
82 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
26 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
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
51 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
36 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
38 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
24 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
23 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
42 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
29 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
13 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
23 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
26 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
23 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
27 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 ...
0
votes
1answer
58 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
27 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
31 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
27 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
16 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
44 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
59 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
47 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 ...
3
votes
1answer
47 views

Discrete math functions help?

I'm doing a review for my discrete math test on functions and I'm having troubles with a few questions. Can I get some guidance in how to do these questions so I can be more prepared for the test? ...
1
vote
2answers
37 views

transitive property in a binary relation

I'm looking at a True or False question in my book and it is very close to identical to the definition of the transitive property in the book, though this answer is False. If someone could explain to ...
1
vote
2answers
63 views

understanding reflexive transitive closure

Suppose I have the following relation $$R = \{(1,1), (2,3), (3,1)\}$$ To make it reflexive we add the following missing pairs: $$ \{(2,2), (3,3)\}$$ Now I wonder how to find the reflexive transitive ...
1
vote
0answers
33 views

Composition relation of P∘P

Consider the following relation P on the set B = {a, b, {a, b}}: P = {(a, a), (a, b), (b, {a, b}), ({a, b}, a)}. Answer questions 6 to 8 by using the given relation P. Question 6 Which one of ...
6
votes
2answers
291 views

Is there a relation that is irreflexive, anti-symmetric and not transitive?

from the set $\{a, b, c, d\}$? Of the one's I have tried, it at best is two of the three, but never all.
1
vote
0answers
64 views

General (Set Builder) definition for a relation composed with itself n times

Questions What does the set builder notation for $S\circ R$ look like? I'm having the most trouble knowing when there is too much information or not enough information on either side of the 'such ...
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
25 views

Are all relations that are symmetric and anti-symmetric a subset of the reflexive relation?

Simple question, but I can't seem to find a guaranteed answer. A symmetric set contains (a, b) if it contains (b, a), but an ...
0
votes
1answer
59 views

Help understanding a theorem on transitivity of a relation

The theorem states this: The relation R on a set A is transitive if and only if $R^n \subseteq R$ for n = 1, 2, 3,... What I'm reading is that the nth power of that set is transitive if the set ...
0
votes
0answers
17 views

Transitive closure of Relation

We know that $$R^* = R \cup R^1 \cup R^2 \cup \cdots \cup R^n$$ $\text{Where R is a relation from set A with n elements}$ My problem is, why we had limited to $R^n$ ? There can be more paths of ...
0
votes
1answer
56 views

Discrete Maths: I'm not familiar with this notation

I've the following relation: $(x,y) \in A \times B, x S y ↔ 2|(x-y)$ What does $2|(x-y)$ mean? Thank you.
0
votes
1answer
45 views

Reflexive relation on set of $n$ elements

How many reflexive relations are there on a set of $n$ elements? I did the problem and I got the answer $2 ^ {n ^ 2}$. Is it correct? Thanks for the help..!!
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
2answers
117 views

How do I prove if a relations is symmetric,transitive or reflexive?

I have no idea how to start this problem. It is asking to prove if the following relation R on the set of all integers where $(x,y) \in R$ is reflexive, symmetric and/or transitive. 1) $(x, y)\in R ...
1
vote
1answer
835 views

Prove that between every two rational numbers a/b and c/d that there is a rational number and there are an infinite number of rational numbers [duplicate]

So the full problem is Prove that between every two rational numbers $a/b$ and $c/d$ that: There is a rational number There are an infinite number of rational numbers I am having ...
-2
votes
1answer
62 views

Determining if $R$ is reflexive, transitive or symmetric on $S$ [closed]

Determine if $R$ is reflexive on $S.$ $S = \mathbb{R}$. Define $R$ by $(a, b)\in R$ if $|a|=|b|.$ $S = \mathbb{R}$. Define $R$ by $(a, b)\in R$ if $a = b + 1.$ $S = \mathbb{R}$. Define $R$ by $(a, ...
0
votes
1answer
46 views

Determining why this is transitive

Why $R_3 = \lbrace (1,2),(3,4)\rbrace$ is transitive? It's like, transitive is said because there's $\{a,b\}$,$\{b,c\}$ then there will be $\{a,c\}$ right? But then, why is that one is said to be ...
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 $O ∈ X$. ...
0
votes
2answers
43 views

To find $R\circ R^{-1}$ in Discrete mathematics

Today I came across a question in DMS which says: If $R$ is the relation “Less Than” from $A = \{1, 2, 3, 4\}$ to $B = \{1,3,5\}$ then find $R\circ R^{-1}$. Now what is $R\circ R^{-1}$? I know ...
1
vote
1answer
29 views

Number of Relations

I was stacked in one question . It was about number of reflexive relations on set with N elements. I know the solution but i don't know the logic behind it . I know we construct nxn matrix and number ...
0
votes
1answer
63 views

Calculating a union of 2 relations

I have 2 relations: $$ xSy \Leftrightarrow y = 2x$$ and $$ xTy \Leftrightarrow y = 3x$$ The problem I have is calculating $$x(T \cup S)y$$ and $$xS^+y $$ Could you please help me?
2
votes
1answer
97 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 ...