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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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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3answers
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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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10answers
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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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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 ...
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5answers
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How many bananas can a camel deliver without eating them all?

This is a fun puzzle I was assigned on the first day of highschool (over a decade ago). I just dug it up randomly from under my bed and thought I'd share it with the SE community. At the time, I ...
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5answers
537 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, ...
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4answers
275 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 ...
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5answers
72 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? ...
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2answers
500 views

Is the given relation transitive?

Consider the relation: $$R = \{(a,b), (a,c), (c,c), (b,b), (c,b), (b,c)\}$$ on the set $A = \{a,b,c\}$. Is $R$ transitive? I said no because $$[(c, c) \wedge (a, c)] \Longrightarrow (c,a)$$ is a ...
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3answers
696 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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3answers
558 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
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4answers
662 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
375 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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2answers
611 views

Visualizing Concepts in Mathematical Logic

If you were forced to speculate or offer anecdotal evidence, how would you say excellent practicioners of mathematical logic coneptually grasp statements like: $$ \vdash ((P \rightarrow Q) ...
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3answers
21k views

Is my understanding of antisymmetric and symmetric relations correct?

So I'm having a hard time grasping how a relation can be both antisymmetric and symmetric, or neither. Are my examples correct? symmetric & antisymmetric ...
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1answer
303 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 ...
7
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2answers
353 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 ...
7
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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
144 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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2answers
303 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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2answers
221 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 ...
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3answers
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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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6answers
109 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 ...
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4answers
207 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 = ...
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2answers
571 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.) ...
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4answers
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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 ...
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3answers
741 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 ...
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2answers
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Proof of clockwise towers of Hanoi variant recursive solution

This is from one of the exercises in "Concrete Mathematics", and is something I'm doing privately, not homework. This is a variant on the classic towers of Hanoi, where all moves must be made ...
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2answers
413 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 ...
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470 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 $n!-1$ ...
7
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1answer
165 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 $m \leq n$ ...
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1answer
224 views

A finite sum involving the binomial coefficients and the harmonic numbers

Wikipedia has a proof of the identity $$ H_{n} =\sum_{k=1}^{n} (-1)^{k-1} \binom{n}{k} \frac{1}{k}$$ http://en.wikipedia.org/wiki/Harmonic_number#Calculation Curiously, there is also the identity ...
7
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1answer
98 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
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1answer
162 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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3answers
638 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
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3answers
330 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
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2answers
743 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 ...
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2answers
442 views

A magic trick with synchronizing words

See the following magic trick. http://www.speedyadverts.com/SAEntertainment/html/realmagic4.html Spoiler Alert Believe it or not, the lady didn't really read your mind; she is not even a real lady ...
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2answers
361 views

Combinatorial proof for identity $\left(\!\!\binom{n\vphantom{1}}{k}\!\!\right)=\left(\!\!\binom{k+1}{n-1}\!\!\right)$ (multiset coefficients)

In class we have recently started using combinatorial proofs. I have tried this problem that our teacher has assigned as a "challenge". I understand how to receive the left hand side, but am ...
7
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1answer
143 views

Equation $\displaystyle\sum_{k=0}^{n-1}(-1)^{n-k-1}\dfrac{(n+k)!}{(k!)^2(n-k-1)!}=n^2$

I think this equality is very inters prove that: $\displaystyle\sum_{k=0}^{n-1}(-1)^{n-k-1}\dfrac{(n+k)!}{(k!)^2(n-k-1)!}=n^2$
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1answer
160 views

Connecting cells by line and column permutations in a finite grid

I'd like to know whether the following simple problem has been studied before and if any solution is known. Let G be a finite (MxN) grid, S a subset of G's cells (the "crumbs"). Two crumbs are said ...
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0answers
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Showing that poset of set of supports of a vector space is semimodular

Let $W$ be a subspace of the vector space $\mathbb{K}^n$, where $\mathbb{K}$ is a field of characteristic $0$. The support of a vector $v = (v_1,\ldots, v_n) \in \mathbb{K}^n$ is given by ...
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0answers
70 views

For what numbers is $a_{b}= b_{a}$? (Reference?)

A student recently asked me about solutions to the equation $$a_{b} = b_{a},$$ where the subscript notation $a_{b}$ denotes interpreting the digits of $a$ in base $b$. It turns out there are tons of ...
7
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1answer
90 views

Prove this is an equivalence relation.

Define a relation of $\mathbb{Q} -$ {$0$} as follows: $x$ ~ $y$ $\Leftrightarrow$ $\dfrac {x} {y} = 2^k $ for some $k \in \mathbb{Z}$ Prove this is an equivalence relation. ATTEMPT: Reflexive: ...
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0answers
194 views

Citation for subset complement result

Let $S = \{s_1, \ldots, s_n\} \subset \{1, \ldots, 2n\}$. Consider two operations on $S$, unfortunately both called complement in different setting: let $A(S) = \{1, \ldots, 2n\} \setminus S$ (set ...
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0answers
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A variation on the Look and Say Sequence and some questions about it.

For information on the sequence mentioned in the title, see http://en.wikipedia.org/wiki/Look-and-say_sequence. This is an original problem. Suppose instead of "describing" the numbers in a string in ...
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8answers
199 views

Why is $n^2+4$ never divisible by $3$?

Can somebody please explain why $n^2+4$ is never divisible by $3$? I know there is an example with $n^2+1$, however a $4$ can be broken down to $3+1$, and factor out a three, which would be divisible ...
6
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3answers
1k views

Is it possible to have a rule which generates: 2, 4, 6, 8, 10, 12, 14, 16, -23?

This is on Lagrange Interpolations . . . Is it possible to have a rule which generates the sequence: 2, 4, 6, 8, 10, 12, 14, 16, -23? The hint that he gave us is to use Summation Products, the only ...
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7answers
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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.
6
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9answers
655 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.