# Tagged Questions

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.

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### Mathematical explanation for the Repertoire Method

There are a few questions already about this method, which has stumped me for a long while. The process is explained, for instance, here: Repertoire Method Clarification Required ( Concrete ...
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### what are the applications of the isomorphic graphs?

While studying data structures i was told my instructor that even i am given 3 hour/30 days/3 years to find out whether two graphs are isomorphic or not, it is very very complex and even after ...
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### How many rectangles are there on an $8 \times 8$ checkerboard?

How many rectangles are there on an $8 \times 8$ checkerboard? \begin{array}{|r|r|r|r|r|r|r|r|} \hline & & & & & & & \\ \hline & & & & &...
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### Correct way to calculate numeric derivative in discrete time?

Given a set of discrete measurements in time $x_t, t \in \{0,\Delta t, 2\Delta t,\ldots,T-\Delta t,T\}$, what is the correct way to compute the discrete derivative $\dot x_t$. Is it more correct to ...
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### Prove that if $2^n - 1$ is prime, then $n$ is prime for $n$ being a natural number

Prove that if $2^n - 1$ is prime, then $n$ is prime for $n$ being a natural number I've looked at http://math.stackexchange.com/a/19998 It is known that $2^n-1$ can only be prime if $n$ is prime....
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### If a plane is divided by $n$ lines, then it is possible to color the regions formed with only two colors.

I am self-studying Discrete Mathematics, and there is the following exercise. (in Portuguese) A plane is divided by many lines. Show that it is possible to color the regions formed with only two ...
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### Cardinality of a discrete subset

If I am correct, a discrete subset of a topological space is defined to be a subset consisting of isolated points only. This is actually equivalent to that the subspace topology on the subset is ...
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### What is the prerequisite knowledge for learning discrete math?

To become a better computer programmer I would like to take the time to learn discrete mathematics, but I am positive that I do not have the required existing knowledge to do so. So I would like to ...
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### What's the difference between a contrapositive statement and a contradiction? [duplicate]

I keep mixing them up, because they are very similar. Some contrapositives resemble some contradictions.
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### Symbolic coordinates for a hyperbolic grid?

Rephrasing     (one year later)    (original question is below) Apparently the original question wasn't clear, or nobody knows an answer (or both). So I will try to rephrase it. Look at your ...
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### Find Nth formula of recursive formula $a_n=a_{n-1}+n(n-1)a_{n-2}$

$$a_n=a_{n-1}+n(n-1)a_{n-2}$$ $$a_0=1, a_1=-\frac{1}{2}$$ Is it possible to find explicit formula for $a_n$ just by using $a_0$ and $a_1$? I know how to solve this problem if $a_n=Aa_{n-1}+Ba_{n-2}$ ...
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### maximum number of edges to be removed to possess a property

I am working on a problem. We know that on squaring a cycle, degree of every vertex is 4. For squares of cycles, we know if we delete any arbitrary edge then still eccentricity is same for all ...
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### Converting to Chomsky Normal Form

I am trying to learn how to convert any context free grammar to Chomsky Normal Form. In the example below, I tried to apply Chomsky Normal Form logic, to result in a grammar, where every symbol either ...
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### I derived a new formula related to arithmetic sequences, I think!

First of all, I am a 12th grader so I don't know how to write research notes. So please forgive me if my writing is not so impressive! I don't know what to do to tell the world about whatever I found....
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### Number of $n^2\times n^2$ permutation matrices with a 1 in each $n\times n$ subgrid

I found the following question in a paper I was trying to solve: The following figure shows a $3^2 \times 3^2$ grid divided into $3^2$ subgrids of size $3 \times 3$. This grid has $81$ cells, $9$ in ...
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### recurrence relation $f(n)=5f(n/2)-6f(n/4) + n$

I've been trying to solve this recurrence relation for a week, but I haven't come up with a solution. $f(n)=5f(n/2)-6f(n/4) + n$ Solve this recurrence relation for $f(1)=2$ and $f(2)=1$ At first ...
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### convert ceil to floor

Mathematically, why is this true? $$\left\lceil\frac{a}{b}\right\rceil= \left\lfloor\frac{a+b-1}{b}\right\rfloor$$ Assume $a$ and $b$ are positive integers. Is this also true if $a$ and $b$ are ...
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### Probability that a clumsy boy eats $k$ out of 20 candies

A week or two (or maybe more) ago, the following question was posted and then deleted just as I was getting to the end of my solution. Unfortunately I have now forgotten what my solution was going to ...
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### Counting number of distinct systems

This is an enumeration problem in conjunction with some lottery problems. Given an integer $N \ge 5$. Let a ticket be a set of 5 distinct integers between $1$ and $N$. Given an integer $T$ between $1$...
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### $n \times n$ lattice graph with partial derivatives bounded by $1$ has $n$ equal values.

I once proved this question many years ago but now I have completely forgotten how I did it. I remember it being a fun problem and wouldn't mind seeing a proof again, with the likelihood of it being ...
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### For how many $n \in \mathbb{N}$ is $\sqrt{n^2+2379}$ natural?

Here's my attempt at a solution: the expression $\sqrt{n^2+2379}$ is natural iff $$n^2 + 2379 = x^2, \quad \mbox{ for some } x \in \mathbb{N}.$$ Therefore $$(x+n)(x-n)=2379=3 \cdot 13 \cdot 61.$$ I ...
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### Function mapping challange

For a given set $A=\{1, 2, 3, 4, \ldots, n\}$, find the number of non-constant mappings (associations ) $f$ from $A$ to $A$ such that $f(k) \leq f(k + 1)$ and $f(k) = f(f(k + 1))$. This is the ...
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### Throwing balls into $b$ buckets: when does some bucket overflow size $s$?

Suppose you throw balls one-by-one into $b$ buckets, uniformly at random. At what time does the size of some (any) bucket exceed size $s$? That is, consider the following random process. At each of ...
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### Into how many parts do $n$ ellipsoids divide $\mathbb{R}^{3}$?

What is the maximum number of regions into which $\mathbb{R}^{3}$ can be divided by $n$ ellipsoids? (Each ellipsoid has the same size). Let´s denote this number by $r_{n}$. Clearly $r_{1}=2$. But ...
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### Given $N$, count $\{(m,n) \mid 0\leq m<N, 0\leq n<N, m\text{ and } n \text{ relatively prime}\}$

I'm confused at exercise 4.49 on page 149 from the book "Concrete Mathematics: A Foundation for Computer Science": Let $R(N)$ be the number of pairs of integers $(m,n)$ such that $0\leq m < N$, ...
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### What is an example of function $f: \Bbb{N} \to \Bbb{Z}$ that is a bijection?

Could you give me an example of function $f \colon \mathbb N \to \mathbb Z$ that is both one-to-one and onto? Does this work: $f(n) := n \times (-1)^n$? N starts with zero.
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### Number of 6-digit passwords, starting with even or ending with odd digit

My problem is A password consists of six digits, each in $\{0,\ldots,9\}$ How many passwords start with an even digit or end with an odd digit? the answer is $750,000.$ I would like to know ...
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### Prove by induction that $5^n - 1$ is divisible by $4$.

Prove by induction that $5^n - 1$ is divisible by $4$. How should I use induction in this problem. Do you have any hints for solving this problem? Thank you so much.
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### How to prove $\left \lceil \frac{n}{m} \right \rceil = \left \lfloor \frac{n+m-1}{m} \right \rfloor$?

everybody, how can I prove that, for natural $m$ and $n$, $$\left \lceil \frac{n}{m} \right \rceil = \left \lfloor \frac{n+m-1}{m} \right \rfloor \; ?$$ Thanks a lot.
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### What is a null set?

I am very confused with null sets. I get that a set which has no elements will be called a null set but I am not getting the examples given below. Please help me by explaining how $P,Q,R$ are all ...
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### Prove the cuberoot of 2 is irrational

I need to prove the cube root is irrational. I followed the proof for the square root of $2$ but I ran into a problem I wasn't sure of. Here are my steps: By contradiction, say $\sqrt[3]{2}$ is ...
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### Combinatorial proof that $\frac{({10!})!}{{10!}^{9!}}$ is an integer

I need help to prove that the quantity of this division : $\dfrac{({10!})!}{{10!}^{9!}}$ is an integer number, using combinatorial proof
Let $U$ be a set defined: $U=\{(x,y)\in \Bbb R^2\mid x^2+y^2=1; xy\neq 0\}$, and let $R$ be relation defined: $(x_1,y_1)R(x_2,y_2) \iff (x_1 \cdot x_2>0∧y_1\cdot y_2>0)$. I was to prove it's ...