# 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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### Find a set of minimal elements in set $\langle \mathbb N\setminus\{0\},|\rangle$

We consider a relation $|$ on set $\mathbb N\setminus\{0\}$ Find set of minimal elements in set $\langle \mathbb N\setminus\{0\},|\rangle$ Prove that in set $\langle \mathbb N\setminus\{0\},|\rangle$...
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### Consider two relations $R$ and $S$ on a set $A$ with $R$ being antisymmetric. Prove $R \cap S$ is antisymmetric.

Consider two relations $R$ and $S$ on a set $A$ with $R$ being antisymmetric. Prove $R \cap S$ is antisymmetric. Theorem The subset of an antisymmetric relation is also antisymmetric. The intersection ...
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### Find equivalence classes of a relation $x\mathcal{R}y \iff (\exists{t}\ne 0)(tx=y)$

On a set $X = \mathbb{R}^2$ is defined an equvalance relatiion $x\mathcal{R}y \iff (\exists{t}\ne 0)(tx=y)$ What are the equvalance classes? I think that equvalance classes of some x are y that by ...
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### Finding transitive closure

Answer the following, related to the relation $R$ on domain $D$, where $D = \{1,2,3,4,5\}$ and $R=\{(1,1), (2,2), (3,3), (4,4), (5,5), (4,3), (3,4), (5,4), (4,5), (5,2), (2,4)\}$: List the elements in ...
1 vote
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### Find cardinality of a set {$f \in \mathbb{N}^{\mathbb{N}}|f\le h$} where $h(n)=n+1$

On a set $\mathbb{N}$ is defined a partial order relation $f \le g \iff \forall{n\in\mathbb{N}} f(n) \le g(n)$. Also let $h: \mathbb{N}\to\mathbb{N}$ given by a formula $h(n)=n+1$. Find cardinality ...
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### What are the equivalence classes of a relation $\mathbb{R}: f \sim g \iff f′ = g′$. [closed]

What are the equivalence classes of a relation $\mathbb{R}: f \sim g \iff f′ = g′$. I can’t understand how this relation should be partitioned.
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### Check if this function injective or surjective $f(x,y)=[x]_R \cap[y]_R$

Let $R \subseteq \mathbb{N}\times\mathbb{N}$ be an equivalance relation. Function $f: \mathbb{N}\times\mathbb{N} \to \mathcal{P}(\mathbb{N})$ is describes as $f(x,y)=[x]_R \cap[y]_R$ 1)Check if it ...
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### About 2 operators such that $X*(Y+Z) = (X*Y)+(X*Z)$ , $W+(U+V)\neq (W+U)+V$

This question is a follow-up and analogue of a previous very similar question. I have to seperate questions since it is a bit of a rule ; 1 question at a time , hence this seperate one. Looking at the ...
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### Proof that order on $A \times B$ is linear only if $A$ or $B$ has no more than 1 element

$\le_A$ is linear order on $A$ and $\le_B$ is linear order on $B$, relation $\le$ on $A\times B$ denote by: $\lt a_1,b_1\gt \le \lt a_1, b_2 \gt$ $\iff$ $a_1 \le_A a_2 \land b_1 \le_B b_2$ Question is ...
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### About 2 operators and $A*(B+C) =(A+B)*(A+C)$

Consider $2$ binary operators defined for a finite set with $n$ elements. Operator $*$ behaves like a commutative latin quandle : $$x*x = x$$ $$a*b=b*a$$ $$a*(b*c)=(a*b)*(a*c)$$ And forms a latin ...
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### Is "substitution" an axiom?

SUBSTITUTION: Let $\mathbf{F}$ be any algebraic structure. For each $a$, $b\in \mathbf{F}$ if $a=b$ then $a$ can be replaced with $b$ in any mathematical statement involving $a$ and the statement will ...
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### How to write a function as a relation?

How do we write a function as a relation?: In this thread the top answer has written a function as a relation as "$aRf(a)$" where I assume we can just define $b=f(a)$? I'm sure this is ...
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### When is a function $f$ transitive?

I want to understand/unpack this definition of a transitive function: "A function is transitive iff it's restriction to it's image is the identity function on it's image" (From This post). ...
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### Proving that the composition of relations preserves countability

Suppose there exists a relation $R$ such that $R$ is countable. I would like to prove that for all $k \in \mathbb{N}_1$, $R^{k}$ is countable (assuming that this actually is a true statement, which I ...
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### Understanding deviation and squared error

I'm wondering if set of numbers 1 can have a lower standard deviation and a lower variance than an other set of numbers 2 while having a higher mean squared error comparing to the other set 2?
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### What is the mathematical relation that can best represent the following curve?

Polynomial functions do not seem to represent the curve well. I do know the curve, but can not arrive at the mathematical relation that represents it. Please help me identify the equation for the ...
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### Number of different relations that are both symmetric and reflexive on a set with 4 elements [duplicate]

Let S = {x | x is a prime number and 1 < x < 10}. Let N be the number of different relations that are both reflexive and symmetric that can be defined on S. Since there is only 4 elements in S={...
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### Is my understanding of partial functions and total functions correct?

After reading this Math.SE answer, I wanted to know if my understanding of partial vs. total functions is correct.  \newcommand{\+}{\mkern2mu} \newcommand{\viff}{\Big\Updownarrow} \begin{gather*} f \...
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### Relations, Transitive functions & Idempotent functions

I've read all the stack threads I can find on transitive functions and want to pull together the ideas/questions it leaves me with. This all started because as per this thread I wrongly thought ...
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### Discrete Mathematics - Trees and Relations Question

I am completely stuck on this multiple choice trees question. In this multiple-choice question, more than one option can be correct. I believe this is a trees question; if it is not, then I have gone ...
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### Understanding in what sense does the 'equals to' sign indicate equality in different scenarios.

I recently came to the realization that I've been taking the equals to sign for granted in mathematics and all of science. So I started to study about it in detail, and came to the understanding that -...
1 vote
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### Set Theory: with an introduction to Real point sets

In the book Set Theory: with an introduction to real point sets (problem 15 on page 9) The problem states: If $\mathcal{R}$ is a relation, then $\mathcal{R}$ is a relation on some set $A$. How can I ...
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### Can a relation be transitive without a direct relation?

Say there is a set {x,y,z} and there is a relation R {(x,y), (y,z)}. Does that make the relation transitive, as x=y and y=z, making x=z? Or is it not transitive because there is no (x,z) relation?
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### Is 1-place relation just a single element set of N ？N is the set of natural numbers

I am confused about the 1-place relation. In the textbook, the definition of the 1-place relation is a subset of natural numbers. In the book, {1} is a an example oof 1-place relation. So according to ...
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### Why function is separated from relation in model definition

A model is defined as a universe together with symbols of relations, functions, and constants. I find this definition odd because a function can generally be considered a special case of a relation (...
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The Wikipedia definiton of equality gives it as a 'relationship between two expressions' This confuses me as when we define mathematical expressions like $2+2=4$ it makes no sense to say that '=' or '...