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.
4,712 questions
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Necessary and sufficient condition for a subset of the powerset of $S$ to be the set of relation classes of some binary relation $R$ on $S$?
Let $S$ be a non-empty set, and let $R$ be a binary relation on $S$. Generalizing equivalence classes, let a "relation class" of $S$ under $R$ be some subset of $S$ which is the forward ...
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A non transitive game league 2
Cf. introducing question, but now make the more realistic assumption that strengths of a team $T=(a,b,c)$ are bound from both sides, WLOG $[0\le a\le b\le c \le 6]$. You can show a team as a point in ...
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A non-transitive game league
This is probably known:
A team $T$ is a tuple $(a,b,c)$ with $a\le b \le c\in\mathbb{R}$ where the mean $m=(a+b+c)/3$ is fixed.
$T_1=(a,b,c)$ "beats" $T_2=(d,e,f)$ (short $T_1\rightarrow T_2)...
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Number of surjective mappings from set $A$ to set $B$
I was solving a question which said :
Set $A$ consists of $6$ different elements, set $B$ consists of $4$ different elements, then the number of surjective mappings from set $A$ to set $B$ is ??
My ...
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Suppes's concept of a relation: Why are domain and range sets?
This question is a succession question to Problems with Munkres's definitions of "rule of assignment" and "function".
Mauro Allegranza directed my attention to his answer to ...
- 5,780
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How can this theorem be proven geometrically?
Here's a problem I just came up with:
A chord, passing through the point of intersection of the two diagonals of a cyclic quadrilateral $ABCD$, intersects the circumcircle at $N$ and $P$, as shown in ...
- 3,518
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Operators and how they work on functions [duplicate]
I have recently learned about operators or functions that take a function as input and output a function.
My question is how do they work with function, especially with the definition of a function ...
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Can this metric relation be proven synthetically?
In the attached figure, circle $O$ passes through vertex $A$ of $\triangle ABC$, intersecting sides $AB, AC,$ and $BC$ at $\{A, M\}, \{A, N\},$ and $\{P, Q\}$.
Using complex numbers, I found:
$|AB| \...
- 3,518
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Finding an orientation on some edges in a mix digraph to avoid an alternating cycle
I encountered the following graph theoretical problem while I was working on my proof for a completely different topic. I found some configurations as necessary conditions on the initial graph. ...
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Is every serial relation a generalized divisor relation?
First, some preliminary definitions. A serial relation is a binary relation $R$ on a set $S$ where this property holds: $(\forall x)(\exists y)xRy$, where the quantifiers range over $S$. Now, let $*$ ...
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Is the class of cycle-free binary relations the same as the class of binary relations whose transitive closure is a strict partial order?
A binary relation $R$ is said to be cycle-free if there are no cycles in the relation, meaning, for every positive integer $n$, there are no $x_1,...,x_n$ such that $x_1Rx_2...Rx_1$. Also, a strict ...
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Reflexive and Antisymmetric relation question
I'm reading Invitation to Discrete Mathematics (2nd edition) by Matousek and Nesetril. Page 41, problem #2 asks:
Prove that a relation $R$ on a set $X$ satisfies $R ◦ R^{-1} = ∆X$ if and only
if $R$ ...
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Are $(c_0^+, \prec)$ and $(c_0^+, \succ)$ isomorphic?
This is a segue from this question I posted the other day.
For $a, b:\mathbb N\to\mathbb R$, we write $a\prec b$ iff $\{n\in\mathbb N:a_n<b_n\}$ is cofinite. It is easy to see $\prec$ is a strict ...
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Are $\ell^1_+$ and $\ell^\infty_+$ order-isomorphic?
For $a, b:\mathbb N\to\mathbb R$, we write $a\prec b$ iff $\{n\in\mathbb N:a_n<b_n\}$ is cofinite. It is easy to see $\prec$ is a pre-order relation over $\mathbb R^\mathbb N$.
Let
$$\begin{aligned}...
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Geometry - Quadrilateral diagonals such that one divides the other into equal parts
Given is a quadrilateral ABCD.
Its diagonals intersect at point O.
It is given that: AO = CO
and: angle(ADC) > angle(ABC)
From the given data, what can we deduce about the relationship between DO ...
