0
votes
2answers
19 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
0answers
9 views

How to proove these properties of compositions of relations?

From wikipedia: If R and S are injective, then S ∘ R is injective, which conversely implies only the injectivity of R. If R and S are surjective, then S ∘ R is surjective, which conversely implies ...
0
votes
1answer
28 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
23 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 $a=1+b \}$ ...
1
vote
1answer
22 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
18 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
11 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
0answers
21 views

Relations basics

I need help explaining some of the properties of sets. Suppose you're given three sets A, B, C with A = {z, y, d}, B= {a, x, z, d} and C = 0. How many elements are there in AxBxC? The answer ...
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. ...
1
vote
1answer
53 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
42 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
39 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
25 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 ...
0
votes
2answers
54 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
29 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
265 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
41 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
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
1answer
16 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
47 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
15 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
49 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
41 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
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 ...
0
votes
2answers
69 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
501 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
61 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
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
2answers
37 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
25 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
59 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?
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
87 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 ...
1
vote
1answer
50 views

Quotient group with functions and relations question

For the set $\mathbb Z/5 \mathbb Z $ (the quotient group of $\mathbb Z$ with the relation R that is defined by $xRy$ if $5|y-x$) We'll define the following operations (both are $\cdot, +$ ...
3
votes
3answers
204 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
1answer
44 views

Transitive closure of binary relation with proof of equivalence

On the set X = {1,2,3,4,5,6,7,8,9}, there is binary relation Q = {(1,9),(2,5),(3,7),(4,1),(5,8),(6,2),(7,3),(8,6),(9,4)}. Make a transitive closure T of the relation Q. Decide and prove whether the ...
0
votes
1answer
33 views

What is the number of inclusive relations?

A binary relation on a set T is inclusive if every element in T relates to at least one element. As an example we can say that {(2, 3), (3, 4)} is not inclusive since 4 does not relate to any ...
1
vote
2answers
50 views

Function and Relations

I have a small question, if anyone could shed some light I would be really grateful! we have this relation R: ∀ x,y ∈ "≈" , [(x≈y) ⇔ (| x - y| ≤ 0.5)] Also relation R belongs to real numbers ℝ. ...
2
votes
1answer
520 views

Number of relations on a set with n elements

Let $A$ be a set with $n$ elements. How many (1)symmetric, (2)anti-symmetric and (3)asymmetric relations are there on $A$ ? (4)How many linear relations are there ? Here's what I did: ...
1
vote
1answer
110 views

How to prove relation is asymmetric if it is both anti-symmetric and irreflexive

Prove a relation is asymmetric if it is both anti-symmetric and irreflexive (anti-reflexsive). I tried to go from the definitions of the relations: Anti symmetric: $\forall x,y \, (xRy \land ...
0
votes
1answer
55 views

recurrence relation related problems

I'm having some difficulties of finding the recurrence relations of; number of divisions of internal region of n sided polygon number of paths from one point to another point in an NxN grid Can ...
1
vote
3answers
89 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 ...
1
vote
3answers
48 views

Why $(a,b)\mid a+b\le3$ is not reflexive and is symmetric?

Why $(a,b)\mid a+b\le3$ is not reflexive and is symmetric? I read because a€ Z so by counter example $(5,5)$, $5+5$ is not less than or equal to $3$ So it's not reflexive But why it's symmetric ? I ...
1
vote
1answer
129 views

Why the divides relation on the set of positive integers antisymmetric

I'd like to know why the divides relation on the set of positive integers antisymmetric. The book says $a|b$ and $b|a$ then $a=b$. But I think if a|b and b not divides a for example $1|2$ but not ...
2
votes
3answers
121 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
1answer
49 views

Find the number of subsets $S$ of $X$ (of any size) that satisfy the following property

Let $X=\{1,2,\dots,10\}$ define the relation $R$ on $X$ by: for all $a,b\in X$, $a\mathrel{R}b \iff ab$ is even. 1) Find the number of subsets $S$ of $X$ (of any size) that satisfy the ...
3
votes
1answer
67 views

Is this relation an equivalence relation? Check my solution please.

Define a relation R on the set {$2, 3, 4, ... $}, as follows. $(x, y)$ ∈ R if and only if $x$ and $y$ have a common factor greater than $1$. Is this relation reflexive? Is it symmetric? Is it ...
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 ...
1
vote
3answers
35 views

Help understanding what's required for a relation to be symmetric?

Let $\,X=\{1,2,3,4,5\},\;$ does a symmetric relation on $\,X\,$ need to have all the elements of $\,X\,$ in the relation? Or can if have just a few elements of $X$ like this relation: $\,A ...