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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Difference between Inclusion and continuation

Halmos defines the order continuation as follows: We shall say that a well ordered set A is a continuation of well ordered set B if B is a subset of A, if, in fact, B is an intial segment of A and ...
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1answer
19 views

$aRb$ iff $a=b(10)^k$ for some $k \in \mathbb{Z}$ [Prove Equivalence Relation]

The question: $R$ is a relation on $\mathbb{N}$ defined by $aRb$ iff $a=b(10)^k$ for some $k \in \mathbb{Z}$. Prove that $R$ is an Equivalence relation. The problem: I can define an ...
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2answers
35 views

How to find relation between 2 numbers

I have been practicing programming for many months now and what I found difficult is not about solving problem. But it is how to find the "how to solve problem" to make computer solves that for me! ...
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1answer
19 views

Does an asymmetric relation entail an antisymmetric relation?

So if there exists an asymmetric relation within a set, does it also entail that there will be an antisymmetric relation in that same set? If so, then it is possible to find out whether a set ...
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1answer
29 views

How is a relation defined on ordered sets?

I am reading that $(\mathbb{Z}, \leq )$ is a total ordered set. I understand how it satisfies reflexivity, antisymmetry, transitivity. But it says that because for any $a,b \in \mathbb{Z}$, either ...
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0answers
10 views

Partial Order Matrix Representaion

What would be the general matrix representation of a partial ordering ? i.e. since the relation must be reflexive, mat(i,i) = 1.
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2answers
59 views

Good book for self-studying Binary Relations

I am studying economics and I frequently encounter Binary Relations. But without any good knowledge of it, I get confused. Here is some background, if it's helpful: I know calculus(single and ...
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1answer
50 views

Why is this relation recursive?

A relation $R \subset \mathbb{N}^d$ is called recursive if there exists a primitive recursive function f with $$ (x_1 ,\dots,x_d) \in R \Leftrightarrow f(x_1,\dots,x_d)=0.$$ In Kurt Gödel's article ...
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1answer
48 views

condition for transitivity

In transitive relations, $aRb$ and $bRc$ implies $aRc$. But what if there are no $bRc$, can we say that the relation is transitive? For example, are relations $R\subseteq V\times V$, corresponding to ...
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1answer
30 views

I have two symmetric relations on a set. How can I prove that the symmetric difference is irreflexive?

I have this problem. Let R and S be symmetric relations on a set A. Prove or disprove: $R \oplus S$ is irreflexive. Now I'm assuming it's not true, because $(x,x)$ can be an element of $R$ ...
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3answers
88 views

A big list of non-trivial examples of functions from outside mathematics

I will be teaching my students about functions, and want to stress that functions are not only the usual mathematical ones (linear, logs, exponential, ...), but that function is fundamentally a ...
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1answer
120 views

Where can I learn more about the Galois connection induced by a graph on its own powerset?

Given a binary relation $R \subseteq X \times Y$, we get an antitone Galois connection $(F,U) : \mathcal{P}(X) \rightarrow \mathcal{P}(Y)$ in the usual way: The function $U : \mathcal{P}(X) ...
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2answers
34 views

Determining if the relation is an equivalence one.

Determine if the relation : $$x \sim y \iff |y-x| \text{ is an integer multiple of } 3$$ is an equivalence one. Now, I think this is an equivalence relation but I am having troubles formally ...
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1answer
28 views

Why is this relation a function?

I need to determine whether or not the relation $\{ (a^2,a) | a \in \Bbb {R}, a \geq 0\}$ is a function from $\Bbb {R}$ to $\Bbb {R}$. I think that it is a function. But I don't know how to ...
2
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2answers
69 views

Is the relation a function

I'm trying to determine if the relation $\{(\frac{a}{b}, a-b) | a,b \in \Bbb {Z}, b \neq 0\}$ is a function from $\Bbb {Q}$ to $\Bbb {Z}$. I know that a relation is a function from A to B if dom(f)=A ...
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2answers
40 views

Is the relation a function?

Is the relation on $\Bbb {R}$ a function from $\Bbb {R}$ to $\Bbb {R}$? $$\{(a^2,a)\mid a \in \Bbb {R}\}$$ How do I determine whether or not the relation is a funtion? Would I treat $(a^2,a)$ as ...
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1answer
28 views

Transitive closure of $H=\{(a,b) \in \mathbb{R}^2: |a-b| \leq 0.1\}$

$$H = \{(a, b) \in \mathbb{R}^2: |a − b| \leq 0.1\}$$ In class today we went over this problem as an example to show transitive closure. I know that the transitive closure of $H$ is "All real ...
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1answer
46 views

What is the cardinality?

Let $A=\left\{1,2,\cdots,10\right\}$ Let $f,g:A\to A$. Consider the equivalence relation $$ fRg \iff \exists h:A\to A. f=h\circ g$$ where $h$ is invertible. Now, let $g(x)=5$: Why is $\left| ...
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3answers
519 views

How many functions are transitive?

Let the set of all functions defined as: $\left\{a,b,c,d\right\} \rightarrow \{a,b,c,d\}$ How many functions are transitive? I've been told to use the fact that a function is transitive iff "it's ...
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2answers
57 views

Prove that the binary relation “is a subset of” is a…

Prove that the binary relation "is a subset of" is a partial order (POSET)? Should I try to prove this in reference to the power set of a general set? When is this relation a total order?
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1answer
23 views

An accessible example of a preorder that is neither symmetric nor antisymmetric

For a project I am working on, I need an example of a preorder (reflexive and transitive relation) that is neither symmetric (like an equivalence relation) nor antisymmetric (like less than or equal ...
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1answer
68 views

Example of relation that is neither transitive nor intransitive?

I have been struggling to think of an example of a relation that is neither transitive nor intransitive, does anyone have any tips? I ended up finding one website that described this as non ...
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1answer
29 views

Is it Preference relation?

I need to check if the relation $ \succeq ( \space \succeq \space \subset X × X, \space X=VB[0,1] ) $ define as below $$ f \succeq g \Longleftrightarrow Var(f+g) \geq Varf $$ is preference ...
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1answer
45 views

A relation that is Reflexive & Transitive but neither an equivalence nor partial order relation

Set $A = \{0,7,1\}$ 1. So for a relation that is reflexive and transitive but neither an equivalence relation nor partial order...Can a relation be both partial order and equivalence? Attempt: ...
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1answer
58 views

Proving that a relation is an equivalence relation

I am having difficulties proving the relation IS an equivalence relation. Let $f: X\longrightarrow Y$ be a function from a set $X$ onto a set $Y$. Let $R$ be the subset of $X \times X$ consisting ...
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2answers
33 views

The relation on the set of functions

Let $\varphi: \mathbb{R}^{2} \to \mathbb{R}$ be a symmetric (not necessarily continuous) function (so, $\varphi(x,y)=\varphi(y,x)$ $\forall (x,y)\in \mathbb{R}^{2}$), let $\mathcal{F}$ be the set of ...
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1answer
41 views

What does it mean for a binary relation to be an order on “equivalence classes” under another binary relation

I am a bit confused about this statement in "Introduction to Set Theory" by Hrbackek and Jech. The statement is as follows: "|A| <= |B| behaves like an ordering on the "equvivalence classes" under ...
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1answer
101 views

The category with binary relations as objects

I have reconsidered my ideas and remember how I thought ones upon a time. I will make a last try and delete if it doesn't work: Set is the category where sets are objects and functions are morphisms ...
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1answer
19 views

Re-calculating Value of $100 in Each State by Specific State

I'm using this Tax Foundations graphic for data. How would I re-calculate each state based on a specific state? For example, what if I wanted to base the control state on Missouri, which is $113.51. ...
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1answer
81 views

how to show a function is bijection

I have taken two numbers $p$ and $r$ where $p,r\in A = \{0,1,\ldots,4i + 1\}$ where $i\geq 1$ and $q\in B = \{0,1,\ldots,n-1\}$. Let $X$ contains all elements obtained by cartesian product of $A$ and ...
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1answer
150 views

Some questions concerning continuity and relations

A lot of equivalent conditions for functions between topological spaces $$ X\overset f\longrightarrow Y $$ are proved on this site. Here some of them formulated from the perspective of 'relations': ...
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1answer
63 views

Is the relation $a\mathrel R b \iff f(a) \equiv f(b)$ an equivalence relation?

Suppose that I have a relation $R$ of the form $a\mathrel R b \iff f(a) \equiv f(b)$, where $\equiv$ is an equivalence relation. In general, is $R$ also an equivalence relation? If not, what are the ...
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1answer
35 views

A question about equivalence classes

Theorem: Let $P$ be a partition of a set $S$, and let $a$ and $b$ be $\in S$. Define the relation $R$ on $S$ as follows: $aRb$ iff there exists an $X \in P$ such that $a \in X$ and $b \in X$. ...
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319 views

Can only one ordered pair be a relation?

I'm sorry, but I really can't find an answer to this no matter how deep I dig. A relation is defined as any set of ordered pairs. But what about a set of only one ordered pair? Is it still a ...
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1answer
28 views

Composition of Ordered Pair

I'm doing math exercises from a Computer Science book and I am confused as to how the following result (from the solutions manual) is obtained: Given the function f={(a,b), (a,c), (c,d), (a,a), ...
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1answer
106 views

Defining a partial order on $A\times B$, given partial orders on $A$ and on $B$

Let $(A,\preceq_A)$ and $(B,\preceq_B)$ be partially ordered sets. Define $C = A \times B$ and define the relation $\preccurlyeq$ on $C$ to be $(a,b) \preccurlyeq$ $(a',b')$ if and only if ...
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0answers
15 views

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
39 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
34 views

Quotient set cardinal in $\mathbb{Z}_{12}$

In $\mathbb{Z}_{12}$ define the equivalence relation xRy if $x^2 = y^2$ Then what is the cardinal of the quotient set?
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31 views

Cardinal of the quotient set

Let $X = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}$. In $X \times X$ define the equivalence $(a,b)\:\mathcal{R}\:(c,d)$ if $a+b=c+d$. Then what is the cardinal of the quotient set? I know that $|X| = 9$, so $| X ...
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3answers
35 views

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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1answer
47 views

What are the steps to calculate the number of elements in a quotient?

Let $X = \{0,1,2,3,4,5,6,7,8,9\}$ and $ Y = \{0,2,4,6,8,9\}$. In $P(X) =$ power set of $X $ define the following relation: $$A R B \Leftrightarrow A \setminus Y = B \setminus Y $$ Then, how many ...
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43 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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38 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
30 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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1answer
73 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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2answers
33 views

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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37 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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0answers
32 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
34 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!