Questions tagged [relations]

For questions concerning partial orders, equivalence relations, properties of relations (transitive, symmetric, etc), a composition of relations, or anything else concerning a relation on a set.

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Equivalence relation question, R is the relation defined in $\mathbb{Z}$ $xRy$ ('$x$' is in relation with '$y$') if and only if $xy$ > $0$.

The exercise is the following: $R$ is the relation defined in $\mathbb{Z}$, $xRy$ ('$x$' is in relation with '$y$') if and only if $xy > 0$. Analize reflexivity, symmetry, antisymmetry, ...
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What is meant by relations in set theory? [closed]

Could you explain to me what is meant by relations in the theory of Sets? I understand that relations are sets, therefore the representation of sets can be used to represent relations.tambien que By ...
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Show that $\sim$ is an equivalence relation.

Suppose $A,B$ sets and nonempty $X\subseteq A$. For $f\in B^A$ define $\hat f = f_{|X} \in B^X$. Define a relation $\sim $ on $B^A$ by $f\sim g \iff \hat f = \hat g$. Show that $\sim$ is an ...
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1answer
45 views

Working out if a given relation is reflexive, symmetric or transitive (or all 3?) [duplicate]

On the set of integers, let 𝑥 be related to 𝑦 precisely when x ≠ y Is this Reflexive? Is this Symmetric? Is this Transitive? I'm also wondering if it can be multiple? I assume it can maybe be two ...
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Who do I do this matrix relation problem? [closed]

enter image description here Please explain how you got the answer!
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3answers
353 views

Why isn't $\{(1,1),(2,2),(3,3),(1,2),(2,1),(1,3),(3,1)\}$ an equivalence relation? [closed]

For $A={1,2,3}$, the equivalence relations are: $\{(1,1),(2,2),(3,3)\}$ $\{(1,1),(2,2),(3,3),(1,2),(2,1)\}$ $\{(1,1),(2,2),(3,3),(1,3),(3,1)\}$ $\{(1,1),(2,2),(3,3),(2,3),(3,2)\}$ $\{(1,1),(2,2),(3,3)...
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2answers
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Is a reflexive and NOT symmetric binary relation automatically antisymmetric? [closed]

If so can I just state that fact and reach a conclusion that the relation is antisymmetric?
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Is xy > 0 relation reflexive?

A relation (x,y) connects all ordered pairs from the set of real numbers where xy > 0 I wonder if it's reflexive? Because 0 is not included in the relations but every pair in the relations will ...
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1answer
19 views

Criteria a function has to follow in order for there to exist an n such that the function's nth derivative is itself

Is there some property that all functions that meet the following condition satisfy? $\exists n$ s.t $\frac{d^nf}{{dx}^n} = f(x)$ So far I have tried using the Taylor series, which yields a rather ...
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3answers
117 views

Proof $U\sim V \iff \mu(U \Delta V)=0$ [closed]

Let $(X,Y,\mu)$ be a measure space and $U,V\in Y$. Show that, and define the equivalence relation, $U\sim V \iff \mu(U \Delta V)=0$ note: the measure is of the symmetric difference. I know that it ...
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27 views

Reflexive, symmetric, nonsymmetric and antisymmetric relations examples from real life.

I have a task to make up examples of relations from real life, which follow these requirements (three separate examples): 1. reflexive, symmetric, nontransitive; 2. reflexive, nonsymmetric, transitive;...
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Prove that $t_n>n(n+1)t_{n-1}, \forall n\in \mathbb{N}$.

Despite several attempts, I have been unable to show that $$t_n>n(n+1)t_{n-1}, \forall n\in \mathbb{N}$$ where $t_n$ denotes the number of transitive relations on a set with $n$ elements. Would you ...
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1answer
42 views

Is the relation $\sim$ an equivalence relation, where $(x_1,y_1)\sim(x_2,y_2)$ means that $(x_1, y_1)=(\lambda x_1,\lambda y_1)$ for some $\lambda$? [duplicate]

Define a relation on $\mathbb R^2 \setminus (0, 0)$ by letting $(x_1, y_1) \sim (x_2, y_2)$ mean that there exists a nonzero real number $\lambda$ such that $(x_1, y_1) = (λx_2, λy_2)$. Prove that $\...
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40 views

How to determine the maximal, minimal, greatest, and smallest elements for the subset relation $\subset$?

If I have a set $S=\{\{a, b\}:a, b\in X\}$ and consider the subset relation $\subset$, how can I determine the maximal, minimal, greatest, and smallest elements? Wouldn't all the possible elements ...
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1answer
29 views

Show g is not surjective

$\begin{align} f: A \rightarrow B, \space g : B \rightarrow A, \end{align} $ and $\begin{align} f \circ g : B \rightarrow B \end{align}$ is bijective I am not sure how to show that $g$ doesnt have to ...
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Biconditional Operator in The Logical Expression for Symmetric Relations

"The logical expression for the symmetric relation should contain the biconditional operator <-> instead of implication -> . Because, if you keep ->, then for the case when F -> T ...
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2answers
144 views

What is the number of transitive relation containing exactly three ordered pairs?

On a set $A=\{1,2,3,\cdots,n\}$ , what is the number of transitive relations, $t_{n,3}$ that contain exactly $3$ ordered pairs? An example is the relation $\{(1,2),(2,3),(1,3)\}$ I calculated the same ...
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1answer
54 views

What is the largest transitive relation that is not the universal relation?

Let $A=\{1,2,3,\cdots, n\}$. Let $T$ denote a transitive relation on $A$ such that $T\neq A\times A$. What is the possibility for the maximum size for $T$? I considered the case $n=3$. I found that ...
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772 views

Is this relation symmetric? I think it is but my professor claims it isn't.

Is the relation $R = \{(a,b),(a,c),(b,a),(b,c),(c,a),(c,b),(d,d)\}$ symmetric? My professor claims that if $(d,d)$ was not included, it would be symmetric, but the inclusion of $(d,d)$ ruins it ...
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1answer
51 views

On the definition of the reflexive-transitive hull

The reflexive-transitive closure of a binary relation 𝑅 is the smallest reflexive and transitive binary relation containing 𝑅. Sentences like the above one are simultaneously widespread and ...
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1answer
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Writing down the relation in the figure?

So for the first question, I have written down the following: R = {(Tom, Tom)} R = {(Mia, Mia, Bob, Liv)} R = {(Liv, Liv, Tom, Noa)} R = {(Noa, Noa, Tom)} R = {(Gus, Gus, Tom, Liv)} R= {(Kim, Kim, ...
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1answer
38 views

Is there some notation for "picking up the $n$-th domain/column" of a relation?

Is there a good/standard notation for the "$n$-th domain/column" of an $m$-ary relation $R$ where $1\leq n\leq m$? If I denote it currently by $\mathrm{dom}_n(R)$, then that means $$ \mathrm{...
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1answer
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Is this equivalence relation: $\{(1,2),(2,1),(1,1)\}$?

Let $S=\{1,2,3\}$ and $R=\{(1,2),(2,1),(1,1)\}$ Over $S, R$ does not have pairs like $(2,2),(3,3).$ Is it reflexive even though those elements in set $S$ are not used? But it is symmetric and ...
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3answers
62 views

Prove that the product of two relations is the identity relation if both relations are bijective maps

So the question is: Suppose $R_1$ and $R_2$ are relations on a set $S$ with $R_1\circ R_2 = \operatorname{I}$ and $R_2\circ R_1 = \operatorname{I}$. Prove that both $R_1$ and $R_2$ are bijective maps. ...
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1answer
34 views

If $\: f:A\to B\:\text{ with }A⊇\{x\},\{y\}\quad$then is $\:f(x)\equiv f(y\mapsto x)\:\text{ & }\:f(y)\equiv f(x\mapsto y)$?

If two variables $x$ and $y$ are both known to be in the domain of a unary function $f$ (but not necessarily defined over the whole domain nor be equal to the other; i.e., $A∋x,y$ while $◇\:\{x\}≠\{y\}...
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1answer
77 views

Number of vacuously transitive relations

A relation $R$ on a set $A$ is vacuously transitive if it is transitive but there do not exist ordered pairs such as $(x,y)$,$(y,z)$,$(x,z)$ . If $A$ has $n$ elements, what is the number of such ...
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Sketch each of the following binary relations in the Cartesian plane and state if it is totally defined, and if it is well-defined :

Sketch each of the following binary relations in the Cartesian plane and state if it is totally defined, and if it is well-defined: (a) the binary relation $H = \{(x,y) \in \mathbb{R}× \mathbb{R}: xy=...
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Transitive relation on set [1,2,3] (exercise question)

Why $$P=[(1,1),(2,2),(3,3),(1,2),(2,1),(1,3),(3,1)]$$ Is not a transitive relation even though it meets all requirements for it. ** a relation R on a set X is transitive if, for all elements a, b, c ...
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1answer
48 views

POSET as a disjoint union of a well ordered set and a set with no least element

Let X be a POSET. Show that one can write X as a union of two disjoint sets A and B such that A is well ordered (with respect to the ordering in X) and B has no least element My approach Intuitively, ...
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1answer
53 views

Prove for Symmetry

Let $A = \{1,2,3,4\}$ and let $f:\mathcal{P}(A)$ be the function defined by saying that $f(X)$ is the sum of the elements of X, for each $X \in \mathcal{P}(A)$. (If $X = \emptyset$, then by convention ...
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Applications of transitive relations

Are there any nice applications of transitive relations or applications of counting transitive relations on a set? I tried googling it. I found that there is something in genetics related to it. Any ...
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77 views

Symmetric and Reflexive relation

Let R be a relation on the set of ordered pairs of positive integers such that ((p,q),(r,s))∈R if and only if p−s=q−r. Which one of the following is true about R? Both reflexive and symmetric ...
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Prove for Transitivity

Let $A = \{1,2,3,4\}$ and let $f:\mathcal{P}(A) \to \mathbb{N}$ be the function defined by saying that $f(X)$ is the sum of the element of $X$, for each $X \in \mathcal{P}(A)$. (If $X = \emptyset$, ...
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Write down the Equivalence classes

Let $V = \{0,1,2\} \times \{0,1,2,3\}.$ We define an equivalence relation $R$ on $V$ by saying that $(a,b)R(c,d)$ if and only if $2a-b = 2c-d$. Write down the equivalence classes for $R.$ I do not get ...
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1answer
353 views

Preservation Proof (Pierce exercise 2.2.8)

In Benjamin C. Pierce's Types and Programming Languages, there's an exercise 2.2.8 that is as follows: Suppose that $R$ is a binary relation on a set $S$, and $P$ is a predicate on $S$ that is ...
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How to find binary relations with $\rho_{1}$ and $\rho_{2}$?

With respect to $\rho$, find binary relations $\rho^{−1}, \rho^{2}, \rho\rho^{−1}, \rho^{−1}\rho$ such that $\rho_{1} = \{(x,y)\in\mathbb{R}^{2} \mid 2x\leq 2y\}$ $\rho_{2} = \{(x,y)\in\mathbb{N}^{2} ...
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59 views

Relation problem

Let $A = \{1,2,3,4\}$ and let $f: \mathcal{P}(A)\to\mathbb{N}$ be the function defined by saying that $f(X)$ is the sum of elements of $X$, for each $X\in \mathcal{P}(A)$. (If $X = \emptyset$, then by ...
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1answer
32 views

Can someone explain me this statement. "Antisymmetric relation is an equivalence relation".

"Antisymmetric relation is an equivalence relation" Whenever I used to prove this statement I come with a counterexample. Let take A= {1,2,3,4} If R= {(1,1),(2,3),(3,4)} then R is ...
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Real life example of S ≺ R if and only if Inv(S) ≺ R, here, S and R are roles, relations, or properties

The first requirement of a generalized role hierarchy to be regular is as follows: $S \prec R$ if and only if $S^{-} \prec R$. The above is used in the context of $\mathcal{SROIQ}$ RBoxes. (For ...
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1answer
60 views

Show that the set of nonnegative numbers partially ordered by divisibility has a unique maximal element.

I came across the following question while studying partial orders: Consider the nonnegative numbers partially ordered by divisibility. Show that this partial order has a unique maximal element. ...
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24 views

Guessing functional form from an equality

Suppose that it is known that two functions $f(n)$ and $g(n)$, where $n$ is a non-negative integer, satisfy $$ f(n) = g(n)-2\,g(n+1)+g(n+2) $$ From this information, can anything be said about the ...
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31 views

How to figure out what order elements go in a matrix from a relation?

Consider the following matrix: \begin{bmatrix}1&0&0&1\\0&1&0&0\\0&0&1&0\\1&0&0&1\end{bmatrix} Does this matrix represent any of the relations in 2.9(a)–(...
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1answer
51 views

Solution to equation $\psi(x)-2\psi(\frac{x}{2})+ \psi (\frac{x}{4}) = x^2$ with $\psi(0)=1$.

I had a test the day before yesterday and in that I got this question $Let \; \psi : R \; \rightarrow \; R$ be a continuous function satisfying the relation $$\psi(x)-2\psi(\frac{x}{2})+ \psi (\frac{...
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1answer
75 views

Understanding reflexive, symmetric, and antisymmetric relation with an example

Let A be a finite, non-empty, non-singleton set. Let R(A) denote the set of all reflexive relations on A , S(A) denote the set of all symmetric relations on A and T(A) denote the set of all anti-...
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2answers
61 views

Is $x^2=y^2$ a symmetric relation?

$R=\{(x,y):x^2=y^2\}$ and I have to determine whether its an equivalence relation. I found that it's reflexive but for the symmetry part I got confused as $x=y$ is sometimes said to be symmetric ...
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29 views

References for Szpilrajn extension theorem

I am looking for a reference for Szpilrajn extension theorem. The reference given in Wikipedia is not in English. It will be helpful if you suggest some textbooks or articles containing the proof of ...
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22 views

Is my proof that the relation ~ between pairs of integers is an equivalence relation correct?

since I havent written that many proofs before and because I am self studying, perhaps you could check if my proof works? Let $\mathbb{Z}$* be $\mathbb{Z}-\{0\}$ and $(x,y),(z,w),(u,v) \in \mathbb{Z} \...
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2answers
95 views

Is $f(x) = x^3 - x$ a mapping or function?

I came across this question in the exam and from my knowledge, I chose that it's a function or a function. But apparently it's not a mapping or function, which I don't understand why. From what I know:...
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6 views

ERD - ternary relationship minimum/maximum row count?

I'm trying to figure out this relationship. Given a ternary relationship of a "Shop" - A shopper can buy an item where a Seller sells it to him in a shop. No other assumptions are given ...
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1answer
38 views

Relations exercise Charles C. Pinter

let $ f: A \to B $ a surjective function. Suppose that $R$ is a relation of equivalence in $A$ such that $ R_f \subset R $ and that $S$ is a relation of equivalence in $B$ Definition: $\...

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