2
votes
2answers
49 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 ...
1
vote
1answer
54 views

Showing $ (R ∪ R^{-1})^∗ = R^∗ ∪ R^{−1∗} $ is false by giving a counterexample.

Show that $$ (R ∪ R^{-1})^∗ = R^∗ ∪ R^{−1∗} $$ is false by giving a counterexample. I tried the following, but every time it keeps coming out as true (instead of false): If $R = \{(a,b), ...
2
votes
3answers
66 views

Finding an area of a triangle inside of a triangle, given certain areas of other triangles, and area ratios.

I'm studying for the Waterloo Math Contest (Galois, Gr. 10) taking place in April of 2015 and I am preparing by looking at previous problems and solving them. This is question 4(c) on the 2010 Galois ...
0
votes
0answers
10 views

For which sets, $X$ the relation is a partial function

Given $T=\left\{\ \left<A,B\right> \in (P(X))^2 | A\subseteq B \right\}$ For which sets, $X$, the relation $(P(X))^2-T \cap (P(X))^2-T^{-1}$ is a partial function? Here's my solution: ...
1
vote
2answers
49 views

Find the $f(x)$ from the given information

So tomorrow I tackled a maths test where I faced a question which was saying, Question: Let $f:R-\{0,1\}\rightarrow R$ be a function satisfying the relation ...
0
votes
2answers
37 views

Finding the equivalence classes of a trigonometric relation

I have been asked to respond to the following: Define a binary relation R on $\mathbb{R}$ as ${\{(x, y) \in \mathbb{R} \times \mathbb{R} \mid \sin(x) = \sin(y)\}}$. Prove that R is an ...
1
vote
1answer
28 views

Domain of definition of the function

I was going through some questions of Relations and Functions and now I am stuck to one. Question says Question: Domain of definition of the function $$f(x)=\frac{9}{9-x^2}+\log_{10}(x^3-x)$$ ...
0
votes
1answer
35 views

relation r reflexivity, transitivity, symmetry

check if relation r is reflexivity, transitivity, symmetry. r is a binary relation in the set of natural numbers such that x r y (x mod 3) = (y+1 mod 3). x-y-1≡(3 mod) <=>x-y-1=3k, for some k ∈ ...
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
35 views

Equivalence relations and power sets.

Let $\mathcal{A}$ be the class of all sets and define the relation $R$ on $\mathcal{A}$ as: $A\space R\space B$ iff there is a bijective function $f:A \to B$. Prove that $R$ is an equivalence relation ...
0
votes
1answer
54 views

Proving a relation is transitive

I am trying to understand transitive relations. I understand given that a set may have $\{(a,b)(b,c)\}$ it must contain $(a,c)$ for it to be transitive. But for longer sets I am getting confused in ...
0
votes
2answers
10 views

Find a relation over $P(${$1,2,3$}$)$ such that $|R|=12$ and the transitive closure of $R$ is the proper subset relation

I'm having trouble finding relation $R$ over $P(${$1,2,3$}$)$ such that $|R|=12$ and the transitive closure of $R$ is $T$, the proper subset relation over $P(${$1,2,3$}$)$. My thoughts: a pair of ...
0
votes
1answer
69 views

Show that if $R$ is a strict partial order on $X$, and $R$ is not linear, then there exists a strict partial order $R'$ and $R' \supsetneqq R$.

Question: Show that if $R$ is a strict partial order on $X$, and $R$ is not linear, then there exists a strict partial order $R'$ and $R' \supsetneqq R$. My attempt: By definition 6.23,6.3.1, and ...
0
votes
1answer
45 views

Determining Equivalence Relation on $\Bbb{Z}$

Alright, I have a homework problem which I have researched, read up on and I (think) solved. I just need someone to either confirm my answer (and re-affirm my knowledge) or explain why I am wrong. ...
0
votes
2answers
33 views

Use of “for all” in definition of reflexive and symmetric relations.

My book says that a relation R on A is reflexive, if $\ (a,a) \in R, \ for\ every \ a \in A$ symmetric, if $\ (a_1,a_2) \in R \implies (a_2,a_1) \in R,\ for\ all\ a_1,a_2 \in A$ Although I ...
1
vote
1answer
63 views

Proving isomorphisms from posets.

An isomorphism from a poset $(S_1,R_1)$ to a poset $(S_2,R_2)$ is a bijection $f: S_1 \rightarrow S_2$ such that, for all $x,y \in S_1$ $(x,y) \in R_1 \leftrightarrow (f(x), f(y)) \in R_2$ When ...
2
votes
1answer
55 views

$R$ is transitive if and only if $ R \circ R \subseteq R$

Question: Let $R$ be a relation on a set $S$. Prove the following. $R$ is transitive if and only if $ R \circ R \subseteq R$. Definition 6.3.9 states that we let $R_1$ and $R_2$ be relations on a ...
2
votes
1answer
363 views

Composite Relations - Give Examples of relations $R_1$ and $R_2$ such that $R_2 \circ R_1 = R_1 \circ R_2$ and $R_2 \circ R_1 \neq R_1 \circ R_2$

Let $S =[a,b,c]$. Give examples of a. relations $R_1$ and $R_2$ on $S$ such that $R_2 \circ R_1 = R_1 \circ R_2$ b. relations $R_1$ and $R_2$ on $S$ such that $R_2 \circ R_1 \neq R_1 \circ R_2$ My ...
0
votes
1answer
55 views

Relation $R$ on $V$ is given by $x+y$ is even [closed]

A relation $R$ on $V$ is given by $x+y$ is even. How can we show that if integers $x$ and $y$ are $R$-related then either $x$ and $y$ are both even or $x$ and $y$ are both odd? I've been looking ...
2
votes
1answer
24 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
194 views

Count number of binary relations between sets

He, I have following questions: We have sets $A$ and $B$, $\left | A \right | = m,\left | B \right | = n$. 1) How many binary relations are there from $A$ to $B$? 2) How many binary relations are ...
0
votes
1answer
28 views

Prove the following $R \subseteq A\times B$ and $S\subseteq B\times C \rightarrow $ $ S \circ R $ is symetric

I want to prove the following $ S \circ R $ is symetric, A,B, C are sets $R \subseteq A\times B$ is Symetric $S\subseteq B\times C$ is Symetric Any Suggestions? Thanks!
1
vote
0answers
35 views

Prove the following $(R\cap S)^n=R^n \cap S^n$

I would like to prove the following without induction. $$(R\cap S)^n=R^n \cap S^n$$ We can start by take $(a,b)\in (R\cap S)^n$ its represent a path from $a$ to $b$ right? Any hints? Thanks.
0
votes
3answers
55 views

Check the following Relation $R=\{(x,y) |\exists k\in \mathbb{Z} \cdot x*y=3k \}$

I would like to check the following relation: $$R=\{(x,y) |\exists k\in \mathbb{Z} \cdot x*y=3k \},R\subseteq \mathbb{Z} \times \mathbb{Z}$$ Reflexivity Symmetric Transitivity Asymmetric Can I ...
0
votes
2answers
43 views

Binary relations: The Lion quiz.

This is my third question today and I think I'm abusing the platform a bit. In any case, here's the question: Let $L$ be the total number of lions that live in Africa today. A binary relation $R$ ...
1
vote
1answer
37 views

What is meant by a|a notation?

Is the "divides" relation on set of positive integer reflexive? In solution of above question I found following. ...
0
votes
2answers
85 views

Reduction Transitive Relation Problem

I have this problem on my homework, it's my last one left but I'm having trouble with it. Any help would be appreciated.
0
votes
2answers
61 views

Unknown maths topic, does the numbers hold

Let $A=\{a,b,c,d,e,f\}$ and let $R\subseteq A\times A$ be a relation which is symmetric and transitive. You have been given some partial information about the relation which is that the following ...
0
votes
1answer
146 views

Maximal and minimal elements of the partial order relation

Let $A=\{1,2,3,4\}$ and $H$ is the set of antisymmetric relations on $A$. I think that $H = \{(1,2),(2,3),(3,4),(1,3),(1,4),(2,4)\}$. How would I find the minimum/maximum and min/max elements?
0
votes
4answers
119 views

How to find $f(2013)$ if $f(5)=45$ and $f(m)+f(n)= f(m+n)$ for all $m,n\in\mathbb N$?

$f: \mathbb N\to\mathbb N$, $f(m)+f(n)=f(m+n)$ for all $m,n\in\mathbb N$, and $f(5)=45$. Find $f(2013)$. I messed up my original posting, its fixed now. I changed $m+ n$ to $f(m+n)$.
0
votes
1answer
271 views

Relation squared of $xRy$ iff $x-y=c$

Let $R$ be the relation on $\Bbb{Z}$ such that $xRy$ if and only if $x-y=c$. (a) Define $R^2$. Can anyone help me with $R^2$? I am not sure where to start. From similar questions, I saw that it ...
0
votes
1answer
61 views

Show that there are no squares included in the sequences

Show that there are no squares included in in the sequences (11, 111, 1111, 11111, ....) (22, 222, 2222, 22222, ....) (33, 333, 3333, 33333, ....) and so on and so forth for all numbers $1 ...
0
votes
2answers
52 views

Describing relations

(a). Describe all relations $R$ on $A$ which are simultaneously symmetric and antisymmetric. (b). Describe all relations $R$ on $A$ which are reflexive, symmetric, and antisymmetric. I have no ...
0
votes
2answers
27 views

Why are both of these not equivalence relations?

Can anyone tell me why the first set is an equivalence relation, and not the second? As far as I can see, both are reflexive, symmetric and transitive, but my books says only the second one is an ...
0
votes
1answer
54 views

Question on increasing/decreasing subsequences?

Here's the question: Describe a sequence consisting from 1 to 10,000 in some order so that there is no increasing or decreasing subsequence of size 101. I'm not quite sure how to do this. My first ...
0
votes
1answer
85 views

Is this poset a lattice?

Given the set $\Bbb Z^+\times\Bbb Z^+$ and the relation $$\begin{align*} (x_1,x_2)\,R\,(y_1,y_2)\iff &(x_1+x_2 < y_1 + y_2)\\ &\text{ OR }(x_1 + x_2 = y_1 + y_2\text{ AND }x_1 \le y_1)\;: ...
1
vote
2answers
121 views

Symmetric relations

Let $A=\{ 1, 2, 3, 4 \}$ is $ B = \{ (1, 2), (2, 1), (1, 3), (3, 1) \} $ Is $B$ a symmetric relation on $A$? I said no because not all $x, y \in A$ are in $B$ Is this correct?
1
vote
1answer
135 views

Prove the relation to be a Linear Order.

Let (a, b),(x, y) ∈ R × R and define ≺ as follows: (a, b) ≺ (c, d) iff a < c or a = c and b < d: Define (a, b) ≼ (c, d) if and only if (a, b) = (c, d) or (a, b) ≺ (c, d). Show that ≼ is a ...
2
votes
3answers
450 views

How to solve recurrence relation: f(n) = f(n-1) + 2(n-1) when f(1) = 1?

I am just learning about recurrence relations, and this is an absolute beginner's question. I understand what's going on in the formula, but I have no clue how to write it's solution? This probably ...
2
votes
2answers
78 views

Set theory relation: irreflexive and transitive

Which of the following relations on $T = \{1, 2, 3\}$ is irreflexive and transitive. $\{(2, 1), (2, 3)\}$ $\{(1, 1), (2, 1), (3, 2)\}$ $\{(2, 1), (1, 2), (3, 2), (2, 3)\}$ $\{(1, 1), (2, 2), (3, 3), ...
0
votes
1answer
290 views

Antisymmetric Relation

Determine whether the relation R on the set of all people is antisymmetric. (a) a is taller than b. (b) a and b are born on the same day. (c) a has the same first name as b. ...
0
votes
0answers
22 views

Prove that T comp (S comp R)=(T comp S) comp R

W, X, Y, Z are sets. R is the relation from W to X, S is the relation from X to Y, and T is the relation from Y to Z. Prove that T composition (S composition R)=(T composition S) composition R If ...
4
votes
1answer
97 views

$F : \mathbb{Z} \to \mathbb{Z}$, $F(n) = 2 -3n$. Is $F$ one-to-one? Onto?

Define $F : \mathbb{Z} \to \mathbb{Z}$ by the rule $F(n) = 2 -3n$, for all $n \in \mathbb{Z}$. Is $F$ one-to-one? Onto? Now, I understand that one-to-one means that nothing in the co-domain is ...
0
votes
1answer
39 views

Discrete and combinatorial mathematics

Suppose we have a relation on a set $A$, i.e. $A \times A$, where $|A| = n; \;n$ a positive integer. How can we count the number of relations on set $A$ which are reflexive, symmetric, transitive and ...
1
vote
2answers
412 views

Reflexive but not Transitive relation

What is an example of a relation $\mathscr{R}$ on a set $S$ such that $\mathscr{R}$ is reflexive but not transitive? Here is what I have come up with. Let $S = \mathbb{Z}$. Then let $\mathscr{R} = ...
3
votes
1answer
75 views

$\beta$ as the relation “is a brother of”

So I have a question about relations. In particular, here is the formal question: Let $\beta$ be the relation "is a brother of" and let $\sigma$ be the relation "is a sister of". Describe ...
0
votes
2answers
66 views

Abstract Algebra topic: Equivalence relations [duplicate]

If R1 is reflective and not transitive, R2 is transitive but not symmetric and R3 is symmetric but not reflexive. We need to find an example of a set S and the three relations R1 R2 R3.
1
vote
1answer
109 views

relations - examples and counterexamples

The question is to find an example of a set $S$ and three relations $R_1$, $R_2$, and $R_3$ on it, such that $R_1$ is reflexive but not transitive, $R_2$ is transitive but not symmetric and $R_3$ is ...
0
votes
1answer
38 views

Number of (equivalence) relations fulfilling some additional conditions

let say I have $A=\{1,\dots,8\}$ I want to know the following things: what the number of relations on $A$? what the number of reflexivity relations on $A$? what the number of equivalence relations ...
1
vote
2answers
132 views

Partially ordered set Question : $A=\{1,2,3,4,5,6\}$ ,$R =\mathcal P(A) \times \mathcal P(A) $

I`m trying to prove that this relation is partially ordered set: $A=\{1,2,3,4,5,6\}$ $R =\mathcal P(A) \times \mathcal P(A) $ $(B,C)R(D,E) \Longleftrightarrow (B \subset D) \vee ((B=D)\wedge(C ...