All questions with tag [math: discrete-mathematics]


Card probability problem

Possible Duplicate: ¢ I located the adhering to trouble in Rosen is Discrete Mathematics and also Its Applications 6th ed. : There are 3 cards in a box. Both sides of one card are black, both sides of one card are red, and also the 3rd card has one black side and also one red side. We select a card randomly and also observe just one side. ...
2022-07-25 20:47:14

Towers of Hanoi - are there configurations of $n$ disks that are more than $2^n - 1$ moves apart?

This is a workout from Chapter 1 of "Concrete Mathematics". It worries the Towers of Hanoi. Exist any kind of beginning and also finishing arrangements of $n$ disks on 3 fixes that are greater than $2^n - 1$ actions apart, under Lucas is initial regulations? My first hunch is no, there are no beginning and also finishing arrangemen...
2022-07-25 20:21:28

Solve the recurrence: $Q_0=\alpha$; $Q_1=\beta$; $Q_n=(1+Q_{n-1})/Q_{n-2}$, for $n>1$.

This one is from "Concrete Mathematics": Solve the recurrence: $Q_0=\alpha$; $Q_1=\beta$; $Q_n=(1+Q_{n-1})/Q_{n-2}$, for $n>1$. Assume that $Q_n \neq 0$ for all $n \geq 0$. I have solved it as follows: $Q_0=\alpha$ $Q_1=\beta$ $Q_2=(1+\beta)/\alpha$ $Q_3=[1+(1+\beta)/\alpha]/\beta=(1+\alpha+\beta)/\alpha\beta$ $Q_4=\frac{[1+(1...
2022-07-25 19:56:20

Factoring a number $p^a q^b$ knowing its totient

We are offered: $n=p^aq^b$ and also $\phi(n)$, where $p,q$ are prime numbers. I need to compute the $a,b,p,q$, perhaps making use of computer system for some estimations, yet the method is intended to be symbolically legitimate and also correct. I recognize that $\phi(n)=p^{a-1}q^{b-1}(p-1)(q-1)$, yet I do not recognize what can I subtract from ...
2022-07-25 19:37:21

Sum to closed form?

Is there a basic method for getting rid of an amount from an expression to generate a shut kind? As an example I required to "unroll" the adhering to expression in a current shows competitors (as $k_1$ and also $k_2$ are huge) and also could not do it $$ \sum_{w=2}^{k_1}\sum_{h=2}^{k_2}(w-1)(h-1) $$ Do I miss out on some distinct m...
2022-07-25 16:31:41

Proving instability of a discrete system

For my intro to DSP class, the TA gave us this discrete system and asked to prove whether or not it's BIBO-stable: $$T \left\{ x[n] \right\} = \sum \limits_{k=\text{min}(n, n_0)}^{\text{max}(n, n_0)} {x[k]}$$ for some integer constant $n_0$. Intuitively, it's unstable, and it can be easily proven by a counterexample if $x[n]$ is the unit step an...
2022-07-25 15:58:24

Meaning of Nc in Paul Erdős and Rényi's paper On Random Graphs

In Paul Erdős and Rényi's 1959 paper On Random Graphs I, they describe the number of edges in a random graph by the function (1) Nc = [1/2 * nlogn + cn] where n is the number of nodes in the graph, c is "an arbitrary fixed real number" and [x] denotes the integer part of x. They go on to use a number of graphs of the form G(n, Nc), w...
2022-07-25 15:55:02

Prove that the Stirling numbers of the first kind satisfy $\displaystyle \sum_{k}\left[n\atop k\right]a_k=n!2^{n-1}$

Let $a_n$ is the variety of organized departments of set $\left\{ 1,2,...,n \right\}$ (which suggests that the series of blocks is necessary, yet not the order of components in blocks). Confirm that: $\displaystyle \sum_{k}\left[n\atop k\right]a_k=n!2^{n-1}$ for $n\ge 1$. Is it feasible to confirm this by induction on $n$? I assume combinator...
2022-07-25 15:47:45

Adapted Towers of Hanoi from Concrete Mathematics - number of arrangements

I have a doubt concerning an exercise from Chapter 1 of "Concrete Mathematics". Actually, my doubt is in one exercise (exercise 3), but, since it depends on the previous exercise (2), I'm including it here too, with the respective solution that I found. Yes, I know that I could simply check the answer in Appendix A, since "Concret...
2022-07-25 15:32:00

Sets and element of.

I am doing the adhering to inquiry, which is fairly standard, but also for some factor I can not function it out. \[\{x \in \mathbb N : x + x= x ^2\} \cup \{x \in \mathbb N: 3 x = x^2\}\] Am I right in claiming that for $\{x ∈ \mathbb N : x + x= x ^2\}$ the solution is $2$ due to the fact that $2+2 = 4$ and also $2^2 = 4$?
2022-07-25 10:58:54

Formal Definition/counter part in mathematics for "Objects" of Object Oriented Models

I'm a rookie in both official maths and also academic computer technology, so please bear with me if you locate my inquiry is not effectively mounted. Object Oriented Modeling appears really valuable in specifying intricate communications when imitating real life. Yet it is primarily made use of in shows. I was asking yourself if we have a compa...
2022-07-25 10:57:06

Convex hull of balls

The convex hull is defined as the smallest convex set containing a set of points. Now we want to generalize it to a set of balls. If these balls have the same radius, then it can be shown that a ball lies on the boundary of the convex hull of balls if and only if its center lies on the boundary of the convex hull of center. My question is: what ...
2022-07-25 10:46:32

Integers $1, 2, \ldots, 10$ are circularly arranged in an arbitrary order.

Possible Duplicate: ¢ Some three consecutive numbers sum to at least $32$ The integers $1, 2, \ldots, 10$ are circularly prepared in an approximate order. Show that there are constantly 3 succeeding integers in this setup, whose amount goes to the very least $17.$
2022-07-25 10:40:01

To find the number of points on a 2D grid?

Given N points on a 2D grid of the form (X,Y) we need to find to find all the points (R,S) such that the sum of the distances between the point (R,S) and each of the N points given is as small as possible. Distance measurement b/w 2 points is done as follows distance between the points (A,B) and (C,D) = |A-C| + |B-D|. Also : 1) In the N given po...
2022-07-25 10:31:22

How many ways there are to open a room with ten doors?

An area has $10$ doors. I intend to locate the distinctive opportunities to open this area. My strategy. For each and every door we have 2 distinctive opportunities. Hence we have $2^{10}-1$ opportunities, bacause all doors might be opened up. Is this strategy deal with?
2022-07-25 10:24:53

Boys dancing with girls

In a party there are $r$ boys and $m$ girls. The first boy dances with $5$ girls, the second boy dances with $6$ girls, and the last boy dances with all girls. What is the relation between $r$ and $m$? I managed to find that $m=r+4,$ but it was not using combinatorics. Let me explain how I found that answer: first boy dances with $a_{1} $ girls...
2022-07-25 10:24:46

Questions about maximal element and minimal element

I have question about set theory again. I have non-empty set $A$, and set $K$ of all equivalence relations on $A$. $K$ is partially ordered set regarding subsets ($\subseteq$). now I need to find the greatest element and the least element in $K$. so I think that the greatest element is the relation $A \times A$ and the least element is $\varnoth...
2022-07-25 10:23:45

Arithmetic progression in a set or its complement

Prove or disprove: For all features $f:\Bbb Z^+ \to \{0,1\}$, the adhering to holds true: There exists favorable integers n and also d such that: $$f(n)=f(n+d)=f(n+2d)=f(n+3d)$$
2022-07-25 10:19:41

How many ways there are to put eight distinct-looking towers on a chessboard?

I am self - researching Discrete Mathematics and also I intend to address the adhering to inquiry. The amount of means there are to place 8 equivalent towers on a chessboard such that there are no 2 equivalent towers in the very same row or in the very same column? And also if the towers stand out - looking? I addressed the first component, ...
2022-07-24 09:36:33

Finite sum of reciprocal odd integers

Mathematica informs me that $\sum\limits_{i=1}^n \frac1{2i-1}$ amounts to $\frac12 H_{n-1/2}+\log\,2$, where $H_n$ is a harmonic number. Why is this real? Exists a basic approach for reviewing amounts of the kind $\sum\limits_{i=1}^n \frac1{ai+b}$?
2022-07-24 09:17:05