Questions tagged [discrete-mathematics]
The study of discrete mathematical structures. Consider using a more specific tag instead, such as: (combinatorics), (graph-theory), (computer-science), (probability), (elementary-set-theory), (induction), (recurrence-relations), etc.
33,605 questions
7
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Non-recursive, explicit rational sequence that converges to $\sqrt{2}$
Is there a non-recursive, explicit sequence of rational numbers that has $\sqrt{2}$ as a limit?
I know of rational sequences such as $x_{n+1}=(x_n+2/x_{n})/2$ and $q_n=[10^n\sqrt{2}]/10^n$ that have $\...
1
vote
0
answers
29
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VC exponent of a set system
I would like to prove that the VC dimension of a set system $(X,\mathcal{R})$ never takes values in $(0,1).$
For the sake of completeness, I'll define some basic ideas in this context.
Definition: A ...
- 354
-1
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1
answer
33
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Justification for Definition of Directed and Undirected Graph
It is commonly known that directed graphs are defined as a double $G_d:=(V,E)$ such that $E \subseteq V^2$, and that undirected graphs $G_u:=(V,E)$ such that $E \subseteq \left\{ \{a,b\}\Big\vert a \...
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0
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18
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Confusion Matrix word Problem [closed]
Problem #4: Autonomous Vehicle Pedestrian Detection An autonomous vehicle's pedestrian detection system is being tested. Out of 150 pedestrian detections, 20 were missed (actual pedestrians not ...
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0
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1
answer
93
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Defining ordinal multiplication for infinite families of ordinals without transfinite recursion
In Naive Set Theory p.82-83, Halmos defines ordinal addition by defining the ordinal sum of an infinite family ${A_i}$ of well-ordered sets, indexed by some well ordered set $I$. He then proceeds to ...
- 23
6
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1
answer
81
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Finding $\lim_{n \to \infty} \frac{l(n)}{n^2}$ for minimal vertex coloring satisfying a property for all $k \times k$ sub-squares
I am working on the following grid coloring problem and am stuck on finding the general form of $l(n)$.
The Problem
Some of the vertices of the unit squares of an $n \times n$ chessboard are colored ...
- 296
1
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2
answers
73
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Is there a better lowerbound and upperbound for the minimum value and maximum value of a magic square given the target sum?
I have been playing a bit with magic squares, we will consider this definition:
Take any $9$ distinct positive integers and put them into a $3\times3$ square. If the sum of the columns, rows, and ...
- 3,662
4
votes
4
answers
288
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Minimizing the sum of cubes minus squares under a fixed sum constraint
Let $x_1, \dots, x_n \ge 0 $ be non-negative real numbers satisfying
$\sum_{i=1}^n x_i = S,$
where $S \ge 0 $ is fixed.
Consider the function
$$
F(x_1, \dots, x_n) = \sum_{i=1}^n (x_i^3 - x_i^2).
$$
...
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6
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2
answers
192
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A conjecture about a difference expression of $n$ positive numbers
The starting point for this question is a set of $n$ positive ordered numbers:
$$ x_{1} \, \ge \, x_{2} \, \ge \, \ldots \, \ge x_{n} \, > \, 0 \; .$$
From these numbers difference expressions ...
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1
answer
321
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What is the origin and interpretation of this piecewise formula for Rule 54?
I'm researching the properties of the single-cell evolution of ECA Rule 54 and its connection to the Collatz conjecture.
The MathWorld page for Rule 54 1 and OEIS A118108 2 both present (or are ...
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1
answer
65
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If a graph has at least 3 vertices (A,B,C) and there are 2 paths between A,B as well as 2 paths between B,C prove there is/n’t 2 paths between A,C
An undirected graph G contains at least 3 vertices (A,B,C). A and B have two edge-disjoint paths; B and C also have two edge-disjoint paths. Can I conclude that A and C also have two edge disjoint-...
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0
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36
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Maximum visible cells in a 3D grid
Given:
A cubical 3D grid made of N×N×N cubical cells.
A point P outside of the bounds of the grid.
Question:
What is the maximum number of cells I can fill such that a line can be traced from any ...
- 101
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0
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46
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Closed form of Homogeneous Linear Recurrence Relations
I know that given a linear homogeneous recurrence relation of order k:
$$a_n = c_1 a_{n-1} + c_2 a_{n-2} + \cdots + c_k a_{n-k}$$
We can get the characteristic equation:
$$r^n = c_1 r^{n-1} + c_2 r^{n-...
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0
answers
140
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A new proof for the common identity involving the sum of the first $n$ squares
Sum of Squares Proof
A New Proof for the Sum of Squares Formula
Observation: Every perfect square can be expressed as a sum of consecutive squares:
\[
1^2 + 2^2 + 3^2...k^2 = 1 + 1 + 3 + 1 + ...
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1
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1
answer
111
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What is the number of points that can be placed in the unit square such that there exists a pair of points is at most λ apart?
Can somebody help me to check whether the idea is correct or not?
Let $S$ be an $n\times n$ square containing $k$ points. We will show that there
exist two points whose distance is at most $\lambda$.
...
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0
answers
57
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Determining the Element Position in a Repetitively Structured Integer Sequence
Problem:-
Packets of numbers are being sent in a special sequence controlled by a number n: first comes 1, 2, 3,…, n; then 1, 1, 2, 2, 3, 3,…, n, n; then 1, 1, 1, 2, 2, 2, 3, 3, 3,…, n, n, n, and so ...
0
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1
answer
48
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How to account for mirror images when arranging people around a table
Let's suppose we have 4 people. Those 4 people have 2 pairs of couples that must sit in front of each other. Because of this, there should only be one way to combine them, right? But if I mirrored one ...
1
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0
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97
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Looking for guidance: academic career prospects [closed]
I’m writing here because I could really use some advice. I got my bachelor’s and master’s degrees in math from a small university in the EU. Following the advice of some good professors, I came to UC ...
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0
answers
51
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About Fuzzy Logic [closed]
I am reading fuzzy logic and completed basics like fuzzy sets , fuzzy arithmetic , operations and other things. I want to study advanced topics like Interval type 2 Fuzzy sets, ordered fuzzy numbers ...
1
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0
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41
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What is the maximum size of a basis for the class T₁ (1-preserving Boolean functions)?
I was able to figure out the maximum basis size for the classes L (Linear) and M (Monotonic), but I'm having trouble with the class T₁ (1-preserving).
The problem can't be solved using the five basic ...
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0
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32
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Characterizing Binomial Coefficients [duplicate]
Is there a way of characterizing Binomial coefficients into even or odd, i.e. when is $\binom{n}{k}$ even or odd?
Observation: When $n$ is even and $k$ is odd, then $\binom{n}{k}$ is even. Is there a ...
- 141
0
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0
answers
29
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coefficient for series expansion [duplicate]
I have to proof that $a_n$ are coefficients for the series expansion of $\sqrt{\frac{1+z}{1-z}}$ are
$$a_n = \begin{cases}{\binom{n}{\frac{n}{2}}\frac{1}{2^n}} & \textrm{for even n} \\
\binom{n-1}...
- 9
3
votes
1
answer
87
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Expected number of unique items in sample without replacement
I am stuck at this problem:
The set S contains N unique items $\{x_1, x_2,...,x_N\}$, and each item appears exactly 5 times in the set.
I randomly select k items from S (without replacement).
What is ...
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6
votes
2
answers
224
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Distribution of number of ties in sum of two random permutations?
I have two random vectors $x$ and $y$ which are permutations of $\{1,...,n\}$. How is the number of ties in the component-wise sum $s$ with $s_i = x_i + y_i$ distributed? I am interested in the ...
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1
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0
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48
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General book graph
In graph theory, usually a book graph $B_p$ implies $p$ 4-cycles $(C_4)$ sharing an edge. I saw in wikipedia that this could also be called a quadrilateral book and there's a variant called triangle ...
2
votes
1
answer
184
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Why are the values of $P_n(2m+1)$ integers?
From the following formula for the Legendre polynomials $P_n(x)$:
$$P_n(x) = \frac{1}{2^n}\sum_{k=0}^{\lfloor n/2 \rfloor} (-1)^k \binom{n}{k}\binom{2n-2k}{n} x^{n-2k}\tag{*}$$
we can see that when $x\...
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votes
5
answers
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Professor insists $-0$ does not exist, am I crazy?
I'm currently in a discrete structures class. On our first midterm, we had the following true or false question:
$$\exists x(\sqrt{x^2} = \pm x), x \in \mathbb{R}$$
I answered true, and was marked ...
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10
votes
1
answer
441
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Total numbers of paths in a grid without any restrictions
I'm trying to solve the following problem:
Given an $n \times m$ rectangular grid, find the number of directed paths that go through every square exactly once and only go up, down, left or right.
I ...
- 351
2
votes
1
answer
38
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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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1
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The Treasure Hunter Problem: Prove that the treasure hunter can reach a treasure located any distance away from the oasis. [closed]
A treasure hunter is attempting to reach a treasure believed to be located in the middle of a desert, which requires h hours of walking from the nearest oasis. Because the desert is extremely hot, the ...
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1
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306
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Does there exist a triple (a, b, c) such that, together with any fourth number from 1 to 9, the Math 24 Game is unsolvable?
Does there exist a triple $((a,b,c) \in \{1,\dots,9\}^3)$ such that ({a,b,c,d}) cannot make 24 for any $d \in \{1,\dots,9\}$?
Body:
In the classic 24 Game, one is given four integers between 1 and 9 ...
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5
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1
answer
203
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(Hidden) symmetries examples in probability
I am looking for some probability problems in which there is a symmetry that you don't notice in the first place, but by noticing it, you can easily get the answer.
I have two examples in mind, and in ...
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0
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240
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Removing Vertices from Graphs by Reductions - a Continuation of an Unsolved Problem
Let $G$ be a simple graph. A reduction consists of choosing an even-degree vertex $v$, say with degree $2m$, and a number of its neighbouring vertices $u_1, u_2, ..., u_k$, where $k$ is either $1, 3, ...
- 947
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1
answer
157
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$u_{2n}=u_{n+1}+u_{n+k}$ general form
Let $k$ be a positive integer. Is there a general form for the sequence $(u_{n})_{n \geq 0}$that satisfies this recurrence:
$$u_{2n}=u_{n+1}+u_{n+k}$$
My end-goal is to prove that any such sequence ...
- 493
1
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3
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Combinatorics problem: How many different ways can you choose the pizzas?
A famous pizza restaurant is running a monthly promotion, advertised on social networks as follows:
“We have 9 toppings to choose from. Buy 3 large pizzas at the regular price and add as many toppings ...
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0
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123
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Relationship between the the total number of functions $A \to B$ and the number of surjections $A \twoheadrightarrow B$
Let $A$ and $B$ be two totally ordered finite sets having, respectively, $|A|=p$ and $|B|=q$ elements with $p > q$. The set $B^{A}$ of all functions $f\colon A \to B$ has $|B^{A}|=q^{p}$ elements, ...
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votes
1
answer
400
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Why does this discrete product built from floor and ceiling of squares converge to pi or 1?
I have been experimenting with a structure I call the Discrete Square Residual Structure (DSRS).
For a fixed integer $\mu > 0$, define
$U(n) = \lceil \tfrac{n^2}{\mu} \rceil, \quad L(n) = \lfloor \...
- 91
1
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1
answer
134
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Translating "Someone has visited every country in the world except Libya"
Let $V(p, c)$ mean that person $p$ has visited country $c$ in the world.
Is the following deconstruction correct?
Someone has visited every country in the world except Libya.
There is a person $p$ ...
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6
votes
1
answer
249
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A version of Van der Waerden's theorem
Recall that Van der Waerden's theorem states: Whenever one partitions $\mathbb{N}=A_0\sqcup A_1$, there is $i\in\{0,1\}$ such that $A_i$ contains arithmetic progressions of arbitrary length.
Recently, ...
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0
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44
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Should I also include a Jacobian when rescaling sums?
Suppose $S \subset \mathbb{Z}^{d}$ is a finite set and we have an expression of the form:
$$\sum_{x \in S}\alpha(x)f(a^{-1}x)$$
where $a > 0$ is fixed. If I want to rescale $S$ by $a$, does the ...
- 1,053
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votes
2
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Are all monotone Boolean functions weighted threshold functions?
$\DeclareMathOperator{\th}{th}$
A Boolean function $f:\{0,1\}^n \to \{0,1\}$ is monotone if for all $a_1,\dots,a_n,a'_1,\dots,a'_n$, if $a_i \leq a'_i$ for all $1 \leq i \leq n$ then $f(a_1,\dots,a_n) ...
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1
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125
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Is Negation Laws same as Law of Excluded Middle?
Negation Laws shown in Table-6 here are stated as:
$$p ∨ ¬p ≡ \mathbf{T} \text{ and } p ∧ ¬p ≡ \mathbf{F}$$
Is it same as the Law of Excluded Middle? This gets confusing for a novice like me when same ...
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votes
1
answer
95
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Prob. 17 (c), Sec. 1.5, in Rosen's DISCRETE MATHS, 8th ed: How to translate this statement into a logical expression [duplicate]
From Prob. 17 (c), Sec. 1.5, in the book Discrete Mathematics and Its Applications by Kenneth H. Rosen, 8th edition:
Express the following system specification using predicates, quantifiers, and ...
- 27.8k
4
votes
2
answers
282
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Statistical test whether puzzle pieces drawn from an urn belong to one puzzle or are random
I was inspired to ask this by my answer to this question: https://math.stackexchange.com/a/5094818/870285
The pieces of a torroidal $m\times n$ puzzle with square pieces which can have a $\cup$ or $\...
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8
votes
3
answers
632
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How to formalise “There are two students who between them have chatted with everyone else”?
From Discrete Mathematics and Its Applications by Kenneth H. Rosen, 8th edition :
Let $C(x, y)$ be the predicate “$x$ has chatted with $y$”, where the domain for $x$ and $y$ consists of all the ...
- 27.8k
4
votes
3
answers
559
views
Can the words: set, collection, list and family be synonyms, representing the same concept in mathematics?
Looking at my favorite programming languages, they have all their definitions of what they are calling being a set, a collection ...
- 269
5
votes
1
answer
738
views
How to translate the idiosyncratic set notation in Neville Dean’s textbook into standard set-builder notation?
I am working on the book Essence of Discrete Mathematics by Neville Dean and I keep coming across this notation for what he calls set replacement. He uses it in set builder notation in the following ...
- 73
0
votes
1
answer
180
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An Alternative Binary Representation
Recently I have been thinking about writing numbers using the digits $1,-1(=\bar{1}),0$, and combining them similarly as one would for binary representations. For example,
$$(1\bar{1}0)_2 = 1\cdot 2^2+...
- 687
3
votes
1
answer
79
views
"Without loss of generality" in reference to division of inheritance
I am in the process of solving a warm-up problem (not graded) for a course I'm hoping to self-study this semester. I believe I have a solution sketch, and several older posts like here and here were ...
- 83
1
vote
0
answers
38
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Minimal size of S∪T given functional closure constraints on S and T with prescribed intersection size
Question
Let $S, T$ be finite sets of real numbers satisfying:
For every $x \in S$, we also have
$\frac{x-1}{x} \in S.$
For every $y \in T$, we also have
$\frac{y-1}{y+1} \in T.$
Let $n = |S \cap T|$...
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