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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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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4answers
3k views

There are 31 houses on north street numbered from 1 to 57. Show at least two of them have consecutive numbers.

I thought to use the pigeon hole principle but besides that not sure how to solve.
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9answers
2k views

What do we actually prove using induction theorem?

Here is the picture of the page of the book, I am reading: $$P_k: \qquad 1+3+5+\dots+(2k-1)=k^2$$ Now we want to show that this assumption implies that $P_{k+1}$ is also a true statement: ...
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6answers
2k views

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 ...
7
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9answers
6k views

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.
7
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9answers
717 views

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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3answers
7k views

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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4answers
7k views

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 ...
7
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4answers
18k views

How many distinct functions can be defined from set A to B?

In my discrete mathematics class our notes say that between set A (having 6 elements) and set b (having 8 elements), there are $8^6$ distinct functions that can be formed, in other words: $|b|^{|a|}$ ...
7
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6answers
131 views

How am I supposed to know that $\frac{1}{(1-x)^2} = \sum_{n=0}^{\infty } \binom{n+1}{n}x^n$?

I'm currently reading through the solution to a problem that involves finding generating functions. In some of the intermediary steps, it is written that $$\frac{1}{(1-x)^2} = \sum_{n=0}^{\infty } ...
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3answers
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Counting zero-digits between 1 and 1 million

I just remembered a problem I read years ago but never found an answer: Find how many 0-digits exist in natural numbers between 1 and 1 million. I am a programmer, so a quick brute-force would ...
7
votes
5answers
141 views

Show that if $g \circ f$ is injective, then so is $f$.

The Problem: Let $X, Y, Z$ be sets and $f: X \to Y, g:Y \to Z$ be functions. (a) Show that if $g \circ f$ is injective, then so is $f$. (b) If $g \circ f$ is surjective, must $g$ be surjective? ...
7
votes
5answers
610 views

How does one find a formula for the recurrence relation $a_{1}=1,a_{2}=3, a_{n+2}=a_{n+1}+a_{n}?$

I am self-studying Discrete Mathematics, and I have two exercises to solve. Find a formula for the following recurrence relation: (translated from Portuguese) a) $a_{1}=3,a_{2}=5, ...
7
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2answers
216 views

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
7
votes
4answers
281 views

Not understanding the concept of equivalence class

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 ...
7
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3answers
958 views

Getting exactly one pair in a poker hand

I am not understanding this problem: In a deck of 52 cards, of 13 ranks, and 4 suits, how many different 5 card hand can we get such that, there is always exactly one pair. There is a similar ...
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5answers
1k views

discrete math book suitable for younger person?

When I took discrete math as an adult I realized that this was a subject I would have enjoyed and done well at much earlier in life, even in my early teens. Does anyone know if there are good books, ...
7
votes
3answers
748 views

Fibonacci numbers divisible by $9$

The $n$th Fibonacci number $F_n$ is defined as follows,$$F_1=F_2=1\mbox{ and } F_{n+2}=F_{n+1}+F_{n}\mbox{ for } n\geq 1$$ I want to know how many of the first $1000$ Fibonacci numbers are divisible ...
7
votes
4answers
724 views

Closed-form expressions for sums

I've got a problem with deducing closed-form expressions for sums: $1) \ \sum_{k=0}^{m}(-1)^k {n \choose k} {n \choose m-k}$ $2) \ \sum_{A,B\subseteq X} |A\cup B|$ where $|X|=n$ Can anyone help me? ...
7
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2answers
386 views

Given three integers in $\{0,\ldots,100\}$ which sum up to $100$. What is the probabilty that two of them are the same?

We pick $3$ numbers (one by one) from set $\{0,1,...,100\}$. What is probabilty that two numbers are the same if sum of those $3$ numbers is $100$? My solution: Which two are the same we can pick in ...
7
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1answer
330 views

Spectrum of a real number

Whilst reading Concrete Mathematics, the authors mention something which they refer to as the "spectrum" of a real number (pg. 77): We define the spectrum of a real number $\alpha$ to be an ...
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2answers
375 views

Watchdog Problem

I just came up with this problem yesterday. Problem: Assume there is an important segment of straight line AB that needs to be watched at all time. A watchdog can ...
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5answers
3k views

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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2answers
1k views

What exactly is the difference between weak and strong induction?

I am having trouble seeing the difference between weak and strong induction. There are a few examples in which we can see the difference, such as reaching the $k^{th}$ rung of a ladder and ...
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2answers
5k views

Prove $\sum \binom nk 2^k = 3^n$ using the binomial theorem

I'm studying for a midterm and need some help with proving summation using the binomial theorem. $\sum\limits_{k=0}^n {n \choose k} 2^k = 3^n$ This is what I'm thinking so far: In the binomial ...
7
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2answers
150 views

Every 33-length subsequence of $1,2,\dotsc,122$ contains a three term arithmetic progression

Is it possible to prove that every 33-length subsequence of the sequence $1,2,3,\dotsc,122$ contains a three term arithmetic progression? Maybe I should post it on mathoverflow
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votes
2answers
310 views

Existence of infinitely many integers $n$ such that $2^n$ ends with $n$

Can anyone please help me on the following proof: Prove that there exist infinitely many positive integers $n$ such that $2^n$ ends with $n$ in decimal notation, i.e. $2n = \ldots n$.
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5answers
1k views

Inherently discrete concepts

Are there any concepts which are naturally defined only for the integers and so far has resisted any attempts at extension to other fields such as rationals or reals? Does not meet criteria: ...
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2answers
224 views

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 ...
7
votes
3answers
181 views

20 balloons are distributed amongst 6 children: Probability that one child gets no balloon?

20 balloons are randomly distributed amongst 6 children. What is the probability, that at least one child gets no balloon? What's the mistake in the following reasoning (I know there has to be a ...
7
votes
6answers
132 views

Solve $\lfloor \sqrt x \rfloor = \lfloor x/2 \rfloor$ for real $x$

I'm trying to solve $$\lfloor \sqrt x \rfloor = \left\lfloor \frac{x}{2} \right\rfloor$$ for real $x$. Obviously this can't be true for any negative reals, since the root isn't defined for such. My ...
7
votes
2answers
245 views

Number or regions formed when $n$ points on a circle are joined

The maximum number $R_{n}$ of regions formed when $n$ points on a circle are joined in pairs is $\frac{1}{24}\left(n^{4}-6n^{3}+23n^{2}-18n+24\right)$. This is a fact that I have read in several ...
7
votes
4answers
417 views

Let $a_n$ be the $nth$ term of the sequence $1, 2, 2, 3, 3, 3, 4, 4, 4, 4, \dots$

Question:Let $a_n$ be the $nth$ term of the sequence $1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6\dots$, constructed by including the integer $k$ exactly $k$ times. Show that $a_n = ...
7
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2answers
657 views

If we have $m$ indistinguishable objects how many ways is it possible to put them in $n$ indistinguihable positions?

if we have $m$ indistinguishable objects, how many ways is it possible to put them in $n$ indistinguishable positions? (for 2 cases 1: without empty position allowed 2: empty positions are allowed.) ...
7
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4answers
3k views

Big-O notation Basics, is it related to derivatives?

I am having the hardest time with Big-O notation (I am using this Rosen book for the class I am in). On the surface, Big-O reminds me of derivatives, rate of change and what not; is this proper ...
7
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3answers
855 views

Decomposing a discrete signal into a sum of rectangle functions

Hello math@stackexchange community ! I have a simple question that seems to have a non trivial answer. Given a discrete one dimensional signal $w(x)$ defined in a finite range, and the boxcar ...
7
votes
2answers
482 views

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 ...
7
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3answers
300 views

Prime numbers divide an element from a set

Show that if $p$ is a prime number different from 2 and 5, then it divides at least one of the elements of the set $\left \{ 1,11,111,1111,...\right \}$.
7
votes
2answers
653 views

For all $n>2$: there exists $p$ prime: $n<p<n!$

The question is: For all $n>2$, where $n \in \mathbb Z$: there exists $p$ prime such that $n<p<n!$ Here is my Proof: $\forall$ $p<n: p|n!$, or $p$ divides $n!$ Since $n!$ and $n!-1$ ...
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1answer
1k views

How many possible arrangements for a round robin tournament?

How many arrangements are possible for a round robin tournament over an even number of players $n$? A round robin tournament is a competition where $n = 2k$ players play each other once in a heads-up ...
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1answer
1k views

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 ...
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1answer
57 views

Partitioning $n$ naturals summing $2N$ into two sets summing $N$

I'm trying to solve this problem: Let $a_1, \ldots , a_n$ be natural numbers such that $a_k \le k$ for every $k = 1,\ldots,n$, and $\sum_{k=1}^{n} a_k=2N$. Show that there exists a partition of ...
7
votes
1answer
181 views

Occupying seats in a classroom

Here's a nice probability puzzle I have thought about for a class I'm TAing, I'm curious to see different solutions :) It goes like this: We have a classroom with $n$ seats available and $m \leq n$ ...
7
votes
1answer
149 views

Number of permutations $\langle a_1,\ldots,a_n\rangle$ of $\{ 1,\ldots ,n \}$ with $a_{i+1} - a_i \ne 1$

Prove that for $n>0$, the number of permutations $\langle a_1,\ldots,a_n\rangle$ of the set $\{ 1,\ldots ,n \}$, where $a_{i+1} - a_i \ne 1$ for $ i = 1, \ldots, n-1$ is equal to: $$D_n + ...
7
votes
1answer
177 views

Graph, two colors, no path length 3

I've just begun studying graph theory and I have some difficulty with this problem. Could you tell me how to go about solving it? In a graph $G$ all vertices have degrees $\le 3$. Show that we can ...
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2answers
316 views

Sum(Partition(Binary String)) = $2^k$

So given any binary string B: $$b_1 b_2 \dots b_n$$ $$b_i \in \{0,1\}$$ It would seem it is always possible to make a partitioning of B: $$ b_1 b_2 \dots b_{p_1}|b_{p_1 + 1}b_{p_1 + 2}\dots ...
7
votes
1answer
221 views

Complicated Multivariate Recurrence Relations For Generating Polynomials

I have the following multivariate recurrence relations all from the same system: First, suppose that $0\le k\le j\le m$, and let $N$ be an independent integer. Then we have for expressions $a(k,~ m,~ ...
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votes
3answers
838 views

Proving statements by its contrapositive

Prove the following statement by proving its contrapositive: “If $n^3 + 2n + 1$ is odd then n is even” Therefore: $\lnot q \rightarrow \lnot p =$ "if $n^3 + 2n + 1$ is even then $n$ is odd. So ...
7
votes
3answers
424 views

Changing Summation Index Question

I'm sorry if this seems like a very novice question, but I am still relatively new to the world of discrete math ( still in 9th grade). I've been reviewing some of the concepts I learned in a ...
7
votes
2answers
846 views

Looking for a bijective, discrete function that behaves as chaotically as possible

I need to write a coupon code system but I do not want to save each coupon code in the database. (For performance and design reasons.) Rather I would like to generate codes subsequent that are ...