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.
33,604 questions
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How many odd numbers are there in one row in Pascal's triangle?
I was looking at the pattern of odd entries in Pascal’s triangle and noticed that every row contains an even count of odd numbers. This is easy to justify, but it led me to wonder how the exact count ...
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As a high school student, what resources would help me learn the foundation of maths? (axioms and definitions) [closed]
I am a high school sophomore, and I am currently trying to prove everything I have learned about math until now starting from basic axioms and definitions.
However, I found it extremely hard to find ...
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What are the building blocks of the Integers set? [closed]
Wen looking at the integers set $\mathbb{Z}$, what are its fundamental building blocks?
Is it the positive subset $\mathbb{Z}^+$ or is it the negatives $\mathbb{Z}^-$ that form the foundation of the ...
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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 $\...
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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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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 ...
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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 ...
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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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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 ...
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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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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 ...
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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 ...
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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 ...
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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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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 ...
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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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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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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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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, ...
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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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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 \...
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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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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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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 ...
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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 ...
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