This tag is intended for questions concerning partial orders, equivalence relations, properties of relations (transitive, symmetric,...), composition of relations and similar stuff. More-or-less the things about relations taught in the first elementary set theory or discrete math course.

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similarities between two binary matrices

I want to measure the similarities between two matrices A and B. Both A and B contains the feature vectors of sounds and are in binary format. i want to see what is the similarities between these two ...
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1answer
30 views

What is the name of this property of relation?

What is the name of property of a binary relation $R$ that states that $\lnot(a\mathrel{R} b) \iff \lnot(b \mathrel{R} a)$ for all $a, b$?
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1answer
43 views

Suppose that $(X,Y,R)$ and $(Y,Z,S)$ are functions. Prove that $(X,Z,S \circ R)$ is a function [closed]

I am given this definition to help me with the proof: Suppose that X,Y and Z are sets, that R is a relation on X and Y and S is a relation on Y and Z. We define a new relation S ∘ R on X and Z as ...
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2answers
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Composite Relations

I'm new to functions and relations, and I've only just figured out that there are 16 relations on a set with 2 elements. I can't figure out what is meant by R ; R ⊆ R other than the fact it is a ...
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2answers
24 views

Anti-symmetric relations

I'm having issues wrapping my head around anti-symmetric examples in specific contexts. I understand that if BOTH $a$, $b$ belong to $\mathbb{R}$ then $a = b$ and if $a \ne b$ then they aren't ...
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2answers
41 views

how to find the equation of this set of points?

What the relation (Equation) between these numbers (X, Y, Z)? Your answer will be highly appreciated.
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1answer
283 views

What are some concrete examples of kinds of relations in math?

I'm writing an undergrad philosophy paper. My take on the issue is that the conceptual problem I'm addressing is only a problem because the word 'is' and 'relation' are too slippery. By more precisely ...
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0answers
36 views

Is a set of some $m \times n$ matrices a relation?

A relation between sets $A_i, i = 1, \dots, n$ is defined as a subset of $\prod_i A_i$. Given $m, n \in \mathbb N$, is a set of (some or all) $m \times n$ matrices over $\mathbb R$ considered a ...
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1answer
52 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 ...
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1answer
26 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 ...
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2answers
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Finding Domaing and Range

Can you please tell me how i am going to solve these? $R=\{(x,y)\in \mathbb R^2 | x^2=y^2\}$ $R^{-1}=?$ $R\circ R^{-1}=?$ $\text{dom} (R)=?$ $\text{range}(R)=?$ Thanks..
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0answers
30 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 = ...
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1answer
29 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!
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1answer
33 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 ...
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2answers
54 views

Functional relations : Trouble seeing transitivity

Given the following domain: $\;\{1,2,3,4\}$ And the following relation: $$\{(1,1),(1,3),(1,5),(2,2),(2,4),(3,1), (3,3),(3,5),(4,2),(4,4),(5,1),(5,3),(5,5)\}$$ It states that this is an equivalence ...
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1answer
24 views

Geometric/visual interpretation of transitivity for equivalence relations on $\mathbb{R}$

If we graph equivalence relations on $\mathbb{R}$ on the plane $\mathbb{R} \times \mathbb{R}$, the properties of reflexivity and symmetry give rise to certain geometric properties--i.e. reflexivity ...
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293 views

Can functions be defined by relations?

So let us say that for whatever reasons, we are not allowed to use function symbols in first-order logic. Then can we define and use a function only by relations?
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1answer
40 views

Function on the set of relations: $f(R)=RK$ where $K$ is a fixed relation

$\ M$ is the set of all relations on $\ A = \{1,2,3\}$ $\ K$ is the following relation on A $\ K=\{(1,1),(2,1),(3,1)\}$ let there be $\ f :M\rightarrow M$ $\ f(R) = RK$ prove that ...
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1answer
70 views

finding the transitivity of a given set [closed]

Let $A=\{1,2,3,4,5,6\}.$ Define the relation $R$ on $A$ as follows: $a\,R\,b$ if and only if $|a-b| \leq 2$. I need to decide whether $R$ is transitive on $A$, and if so, why. (And if not, why ...
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Examples of relations: reflexive but not transtive; transtitive but not symmetric; symmetric but not reflexive [closed]

Find example of a set $ S $ and three relations $R_1, R_2 ,R_3$ on it such that $R_1$ is reflexive but not transitive, $R_2$ is transitive but not symmetric and $R_3$ is symmetric but not ...
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1answer
32 views

Proving that this relation is transitive

I have seen this question on a book I am reading and could not figure it out fully. The question is as follows: "Suppose A is a set, and $F\subseteq P(A)$. Let $$R_F=\{ (a,b)\in AxA|\text{ for every ...
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1answer
225 views

Relation and the complementary relation: reflexivity and irreflexivity

How do I show that the relation R on a set A is reflexive if and only if the complementary relation R is irreflexive. Because of iff: I start with let R be a relation from a set A to B. The ...
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1answer
24 views

suppose $R$ is an equivalence relation on $A$ such that there are only finitely many distinct equivalence classes $A_1,A_2,\ldots,A_k$ w.r.t $R$

Suppose $R$ is an equivalence relation on $A$ such that there are only finitely many distinct equivalence classes $A_1,A_2,\ldots,A_k$ w.r.t $R$. Show that $$A=\bigcup_{i=1}^k A_i$$ Since $A_i\subset ...
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1answer
25 views

Proving Transitivity

Consider a relation defined by $Z$ where $(a,b) = 2a^2 + b^2 -3ab = 0$ Is the relation $R_1$ reflexive? symmetric? transitive? Is it an equivalence relation? I have said that it is ...
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1answer
83 views

Prove that $D_r \circ D_s = \{ (x,y) \in \mathbb{R}^2 \ | \ |x-y|<r+s \}$, where $D_a = \{ (x,y) \in \mathbb{R}^2\ | \ |x-y| < a \}$

Suppose $r$ and $s$ are two positive real numbers. Let $D_r = \{ (x,y) \in \mathbb{R}^2\ | \ |x-y| < r \}$ and $D_s = \{ (x,y) \in \mathbb{R}^2 \ | \ |x-y| < s \}$. Prove that $D_r \circ D_s = ...
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3answers
56 views

If R is $(a,b)R(c,d) \iff a+d =b+c$ show that R is an equivalence relation.

The relation R is defined n all positive integers such that, $(a,b)R(c,d) \iff a+d =b+c$ . Show that R is an equivalence relation. In order to be an equivalence relation, R has to be reflexive, ...
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Proof for a relation

Define a relationship $R$ on $\mathbb{Z}$ by declaring that $xRy$ if and only if $x^2 \equiv y^2 (mod 4)$. Prove that $R$ is reflexive, symmetric and transitive. I'm unsure of where to start with ...
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1answer
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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 ...
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1answer
35 views

I am working on basic functions, I am asked is x-5=y^2 a function,

i use the square root property and get plus or minus the sqaure root of x-5=y, then I come to my question, for any value of x greater than 5, how many values of y result? I need some insight to fully ...
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1answer
185 views

Equivalence Relation? Column-equivalence on the set of all $m\times n$ matrices.

How do I show that column equivalence on the set of all $m\times n$ matrices is an equivalence relation? I know that to show it is an equivalence relation, I need to show that column equivalence is ...
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2answers
23 views

Prove that R is an equivalence relation on F

A relation $R$ is defined on the set $F = \{f: \Bbb R \to \Bbb R\}$ $$fRg \iff f(0) = g(0).$$ My approach: This is reflexive because: $f(0) = f(0)$ is same as $f(0) = g(0)$ This is symmetric ...
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3answers
76 views

What is it called when !(a < b) and !(b < a) implies a = b?

I thought it would be some kind of symmetric equality but its impossible to do a google search on this, all I get are definitions of reflexive, symmetric and transitive. I'm not really sure which ...
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1answer
34 views

Proving theorems on relations

I came across the following three statements about relations. I understand why the statements are true, but I am not sure how to demonstrate them mathematically. In the following, $A,B$ are sets and ...
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49 views

How to figure out the solution to this equality problem? [closed]

Let $x, y, z$ be strictly positive, real numbers. If: $$\frac{x^2}{y^2} + \frac{y^2}{z^2} + \frac{z^2}{x^2} = \frac{x}{z} + \frac{y}{x} + \frac{z}{y}$$ then prove that $x = y = z$. Thanks!
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1answer
47 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. ...
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2answers
92 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 ...
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0answers
37 views

Properties of relations on Z

$$R_3 = \{(x,y) \in \mathbb{Z} \times \mathbb{Z}|\frac{x-y}{5} \in \mathbb{Z}\} $$ $$R_4 = \{(x,y) \in \mathbb{Z} \times \mathbb{Z}|x > y \} $$ I need to know which one is reflexive, symmetric, ...
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2answers
30 views

Relations on set A and conditions of existence of descending chains

I am reading through the Elements of Set theory by Herbert Enderton and even though I have passed this exercise long ago,the more I look at my solution the less I believe it. Problem goes like this: ...
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1answer
40 views

Terminology on pullbacks

I'm quite confused with the use of pullbacks, and in particular I wonder which terminology I shall use in the following examples. Let $X$ and $Y$ be arbitrary sets. Suppose that $f,g:X\to Y$ and I ...
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1answer
620 views

What is meant by “m|n”? Two letters separated by a vertical bar (|)

I am new to this subject, and not not sure what "|" symbol means on this statement. Let $R_2 \subset\Bbb N \times\Bbb N$ be defined by $(m, n) \in R_2$ if and only if $m|n$.
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1answer
70 views

Let R be a relation on set A. Prove that $R^2 \subseteq R <=>$ R is transitive $<=> R^i \subseteq R ,\forall i \geq 1$

this is my first question here. I'm still relatively new to more advanced mathematics and don't have much experience with proofs yet. I'm self-studying at the moment and therefore have no one to check ...
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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: ...
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11k views

Are there real-life relations which are symmetric and reflexive but not transitive?

Inspired by Halmos (Naive Set Theory) . . . For each of these three possible properties [reflexivity, symmetry, and transitivity], find a relation that does not have that property but does have ...
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1answer
39 views

Relations that are: reflexive but not transitive; transitive but not symmetric; symmetric but not reflexive

I have an incomplete answer to my question. Can anyone help me answer the last two parts. My question is: Find example of a set $S$ and three relations $R_1$, $R_2$, $R_3$ on it such that ...
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3answers
71 views

What would make a function reflexive, transitive, and/or symmetric?

A binary relation $R$ is a subset of the Cartesian product between two sets $X$ and $Y$, containing a set of ordered pairs $\{(x,y) : x \in X, y \in Y\}$. $R$ is a function if each element of $X$ is ...
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1answer
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Equivalence between different definitions of transitive closure

Let $W$ be an arbitrary, non-empty set and let $R$ be an arbitrary binary relation on $W$. Define the transitive closure of $R$ as $R^+ = \bigcap \{ R' \; | \; R' \text{ is a binary transitive ...
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1answer
43 views

Relations and functions with valence 0

From http://en.wikipedia.org/wiki/First-order_logic#Non-logical_symbols Relations of valence 0 can be identified with propositional variables. For example, P, which can stand for any ...
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3answers
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An example of symmetric transitive relation that is not reflexive on a set of natural numbers $\mathbb{N}$ [duplicate]

An example of symmetric transitive relation that is not reflexive on a set of natural numbers $\mathbb{N}$. My guess is that such relation does not exist, but I don't know how to prove it.
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3answers
43 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 ...
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2answers
34 views

What is the name of the Speed, Distance, Time relationship?

Really simply, I'd like to know if there is a name used to describe the speed, distance & time relationship. i.e. As this is basically the same relationship that applies to current, voltage and ...