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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114
votes
17answers
9k views

Zero to the zero power - Is $0^0=1$?

Could someone provide me with good explanation of why $0^0 = 1$? My train of thought: $x > 0$ $0^x = 0^{x-0} = 0^x/0^0$, so $0^0 = 0^x/0^x = ?$ Possible answers: $0^0 * 0^x = 1 * 0^x$, so ...
37
votes
20answers
8k views

Proof that $\sum\limits_{k=1}^nk^2 = \frac{n(n+1)(2n+1)}{6}$?

I am just starting into calculus and I have a question about the following statement I encountered while learning about definite integrals: $$\sum_{k=1}^n k^2 = \frac{n(n+1)(2n+1)}{6}$$ I really ...
11
votes
8answers
1k views

Showing that an equation holds true with a Fibonacci sequence: $F_{n+m} = F_{n-1}F_m + F_n F_{m+1}$

Question: Let $F_n$ the sequence of Fibonacci numbers, given by $F_0 = 0, F_1 = 1$ and $F_n = F_{n-1} + F_{n-2}$ for $n \geq 2$. Show for $n, m \in \mathbb{N}$: $$F_{n+m} = F_{n-1}F_m + F_n F_{m+1}$$ ...
4
votes
8answers
617 views

How to compute the formula $\sum \limits_{r=1}^d r \cdot 2^r$?

Given $$1\cdot 2^1 + 2\cdot 2^2 + 3\cdot 2^3 + 4\cdot 2^4 + \cdots + d \cdot 2^d = \sum_{r=1}^d r \cdot 2^r,$$ how can we infer to the following solution? $$2 (d-1) \cdot 2^d + 2. $$ Thank you
3
votes
3answers
469 views

Prove $f(S \cup T) = f(S) \cup f(T)$

$f(S \cup T) = f(S) \cup f(T)$ f(S) encompasses all x that is in S f(T) encompasses all x that is in T thus the domain being the same, both the LHS and RHS map to the same ys, since the function ...
4
votes
2answers
2k views

Counting number of moves on a grid

Imagine a two-dimensional grid consisting of 20 points along the x-axis and 10 points along the y-axis. Suppose the origin (0,0) is in the bottom-left corner and the point (20,10) is the top-right ...
8
votes
2answers
1k views

Prove that $\mathcal{P}(A)⊆ \mathcal{P}(B)$ if and only if $A⊆B$. [duplicate]

Here is my proof, I would appreciate it if someone could critique it for me: To prove this statement true, we must proof that the two conditional statements ("If $\mathcal{P}(A)⊆ \mathcal{P}(B)$, ...
34
votes
7answers
30k views

What is the best book for studying discrete mathematics?

As a programmer, mathematics is important basic knowledge to study some topics, especially Algorithms. Many websites, and my fellows suggest me to study Discrete Mathematics before going to ...
26
votes
4answers
2k views

A stronger version of discrete “Liouville's theorem”

If a function $f : \mathbb Z\times \mathbb Z \rightarrow \mathbb{R}^{+} $ satisfies the following condition $$\forall x, y \in \mathbb{Z}, f(x,y) = \dfrac{f(x + 1, y)+f(x, y + 1) + f(x - 1, y) +f(x, ...
5
votes
5answers
6k views

Number of relations that are both symmetric and reflexive

Consider a non-empty set A containing n objects. How many relations on A are both symmetric and reflexive? The answer to this is $2^p$ where $p=$ $n \choose 2$. However, I dont understand why this is ...
3
votes
2answers
398 views

What is the probability that $x_1+x_2+…+x_n \le n$?

Given that $X_1, X_2...$ are mutually independent random variables. For each $i$ with $1\le i \le n$ the variable $X_i$ is equal to either $0$ or $n+1$ $E(X_i)$ = $1$ also.. if $X_i$ is equal to ...
13
votes
1answer
476 views

Can $\sqrt{n} + \sqrt{m}$ be rational if neither $n,m$ are perfect squares?

Can the expression $\sqrt{n} + \sqrt{m}$ be rational if neither $n,m \in \mathbb{N}$ are perfect squares? I doesn't seem likely, the only way that could happen is if for example $\sqrt{m} = ...
1
vote
2answers
166 views

Let $(a,b)$ and $(c,d)$ be intervals in $\Bbb R$, and find an injective and surjective function from $(a,b)$ to $(c,d)$

So here is this question I got stuck on: Let $(a,b)$, $(c,d)$ be intervals (not sure if that's the correct term) on $\Bbb R$, so that $a<b$, $c<d$. Find an injective and surjective function ...
2
votes
3answers
100 views

How to prove this statement: $\binom{r}{r}+\binom{r+1}{r}+\cdots+\binom{n}{r}=\binom{n+1}{r+1}$

Let $n$ and $r$ be positive integers with $n \ge r$. Prove that Still a beginner here. Need to learn formatting. I am guessing by induction? Not sure what or how to go forward with this. Need help ...
49
votes
3answers
10k views

Proof by contradiction vs Prove the contrapositive

What is the difference between a "proof by contradiction" and "proving the contrapositive"? Intuitive, it feels like doing the exact same thing. And when I compare an exercise, one person proves by ...
3
votes
4answers
910 views

Not understanding Simple Modulus Congruency

Hi this is my first time posting on here... so please bear with me :P I was just wondering how I can solve something like this: $$25x ≡ 3 \pmod{109}.$$ If someone can give a break down on how to do ...
5
votes
2answers
860 views

Number of combinations with repetitions (Constrained)

I would like to calculate the number of integral solutions to the equation $$x_1 + x_2 + \cdots + x_n = k$$ where $$a_1 \le x_1 \le b_1, a_2 \le x_2 \le b_2, a_3 \le x_3 \le b_3$$ and so on. ...
5
votes
4answers
664 views

Sum of cubes proof [duplicate]

Prove that for any natural number n the following equality holds: $$ (1+2+ \ldots + n)^2 = 1^3 + 2^3 + \ldots + n^3 $$ I think it has something to do with induction?
4
votes
2answers
172 views

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 ...
4
votes
3answers
1k views

Questions on “All Horse are the Same Color” Proof by Complete Induction

The following has been bugging me, and I can't go to sleep until I resolve it. Here is a summary of the document on page 109 of http://courses.csail.mit.edu/6.042/spring12/mcs.pdf. False ...
5
votes
2answers
6k views

Strong Mathematical Induction: Why More than One Base Case?

I am trying to understand this example of strong induction. I know normal induction. In normal induction, if base case is true then we assume some number $n$ to be true. Afterwards, we prove $n+1$ is ...
4
votes
4answers
372 views

Determine whether $F(x)= 5x+10$ is $O(x^2)$

Please, can someone here help me to understand the Big-O notation in discrete mathematics? Determine whether $F(x)= 5x+10$ is $O(x^2)$
18
votes
6answers
540 views

How to prove $\sum\limits_{k=0}^n{n \choose k}(k-1)^k(n-k+1)^{n-k-1}= n^n$?

How do I prove the following identity directly? $$\sum_{k=0}^n{n \choose k}(k-1)^k(n-k+1)^{n-k-1}= n^n$$ I thought about using the binomial theorem for $(x+a)^n$, but got stuck, because I realized ...
15
votes
3answers
2k views

Gay Speed Dating Problem

Here's an interesting problem that I came up with the other night. With straight speed dating, (assuming the number of men and women are equal) the number of iterations that need to be made before ...
2
votes
3answers
5k views

Using Euler's Totient Function, how do I find all values n such that $\phi(n)=12$?

How do I generalize the equation to be able to plug in any result for $\phi(n)=12$ and find any possible integer that works?
9
votes
4answers
33k views

What five odd integers have a sum of $30$?

I've been asked the following question: What five odd integers from the set $\{1, 3, 5, 7, 9, 11, 13, 15\}$ that when summed together equals to $30$? Note that any integer can be used more than ...
5
votes
4answers
255 views

Prove $\binom{n}{a}\binom{n-a}{b-a} = \binom{n}{b}\binom{b}{a}$

I want to prove this equation, $$ \binom{n}{a}\binom{n-a}{b-a} = \binom{n}{b}\binom{b}{a} $$ I thought of proving this equation by prove that you are using different ways to count the same set of ...
5
votes
1answer
1k views

Asymptotics for a partial sum of binomial coefficients

Good afternoon, I would like to ask, if anyone knows how to evaluate a sum $$\sum_{k=0}^{\lambda n}{n \choose k}$$ for fixed $\lambda < 1/2$ with absolute error $O(n^{-1})$, or better. In ...
2
votes
2answers
90 views

(Counting problem) very interesting Modular N algebraic eqs - for combinatorics-permutation experts

Experts in algebra please help: we would like to know the number of solutions for this set of six of modular N algebraic equations: $$ (1) \quad x_1 y_2 \equiv x_2 y_1 \pmod{N}\\ (2) \quad x_1 y_3 ...
4
votes
4answers
686 views

Prove by Mathematical Induction: $1(1!) + 2(2!) + \cdot \cdot \cdot +n(n!) = (n+1)!-1$

Prove by Mathematical Induction . . . $1(1!) + 2(2!) + \cdot \cdot \cdot +n(n!) = (n+1)!-1$ I tried solving it, but I got stuck near the end . . . a. Basis Step: $(1)(1!) = (1+1)!-1$ $1 = ...
1
vote
3answers
4k views

Prove $(n^5-n)$ is divisible by 5 by induction.

So I started with a base case $n = 1$. This yields $5|0$, which is true since zero is divisible by any non zero number. I let $n = k >= 1$ and let $5|A = (k^5-k)$. Now I want to show $5|B = ...
5
votes
1answer
995 views

Show that $n$ lines separate the plane into $\frac{(n^2+n+2)}{2}$ regions…Induction!

Show that $n$ lines separate the plane into $\frac{(n^2+n+2)}{2}$ regions if no two of these lines are parallel and no three pass through a common point. I know we start with the base case, where, ...
3
votes
4answers
2k views

Proving the sum of the first $n$ natural numbers by induction

I am currently studying proving by induction but I am faced with a problem. I need to solve by induction the following question. $$1+2+3+\ldots+n=\frac{1}{2}n(n+1)$$ for all $n > 1$. Any ...
1
vote
2answers
919 views

Discrete Math - Hasse Diagrams

This is one of many questions of similar type I have to do for an assignment and im troubled with what to do. The question is as follows: Consider a relation R defined on the set A = {−7, −6, −5, ...
30
votes
9answers
64k views

Maximum board position in 2048 game

A game called 2048 is making rounds on social media. I am trying to determine the maximum score attainable for this game. Let's assume WLOG that only 2s are returned (if 4s are possible the max score ...
15
votes
2answers
1k views

Combinatorial interpretation of Binomial Inversion

It is known that if $f_n = \sum\limits_{i=0}^{n} g_i \binom{n}{i}$ for all $0 \le n \le m$, then $g_n = \sum_{i=0}^{n} (-1)^{i+n} f_i \binom{n}{i}$ for $0 \le n \le m$. This sort of inversion is ...
12
votes
2answers
792 views

9 pirates have to divide 1000 coins…

A band of 9 pirates have just finished their latest conquest - looting, killing and sinking a ship. The loot amounts to 1000 gold coins. Arriving on a deserted island, they now have to split up the ...
7
votes
5answers
482 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, ...
8
votes
1answer
203 views

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 ...
4
votes
3answers
426 views

Proof by induction or contradiction?

I have to prove that $(4k + 3) ^2 - (4k + 3)$ is not divisible by $4$. What would be the best approach for this, proof by induction or contradiction? I've tried both and haven't got very far. Any ...
4
votes
3answers
693 views

Cardinality of a set A is strictly less than the cardinality of the power set of A

I am trying to prove the following statement but have trouble comprehending/going forward with some parts! Here is the statement: If $A$ is any set, then $|A|$ $<$ $|P(A)|$ Here is what I ...
2
votes
2answers
10k views

Proof by induction: $2^n > n^2$ for all integer $n$ greater than $4$ [duplicate]

I am a CS undergrad and I'm studying for the finals in college and I saw this question in an exercise list: Prove, using mathematical induction, that $2^n > n^2$ for all integer n greater than ...
2
votes
3answers
7k 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| ...
1
vote
3answers
850 views

$k$-element subsets of $[n]$ that do not contain $2$ consecutive integers

Let $k,n\in \mathbb{N}$ with $k\leq n$. Find the total number of $k$-element selections from $[n] = \{1, 2, \ldots, n\}$ that do not contain any 2 consecutive integers
0
votes
2answers
216 views

Is it possible to play the Tower of Hanoi with fewer than $2^n-1$ moves?

The Tower of Hanoi game consists of three identical upright pegs and n rings all of different diameters that can be stacked over any of the pegs. Initially, all of the rings are stacked around one of ...
4
votes
1answer
1k views

Combinatorics - pigeonhole principle question

This is for self-study. This question is from Rosen's "Discrete Mathematics And Its Applications", 6th edition. An arm wrestler is the champion for a period of 75 hours. (Here, by an hour, we mean a ...
2
votes
1answer
121 views

Equivalence relation question with cardinality and countability $A=\mathbb R,\ aSb \iff a-b\in \mathbb Q $

Let $A=\mathbb R,\ aSb \iff a-b\in \mathbb Q $ What is the cardinality of $[\pi]_S$ ? Prove that the quotient group $\mathbb R/S$ is uncountable. Well I think that cardinality is ...
2
votes
3answers
461 views

Equivalence Relation problem [duplicate]

Let $S$ be the relation on $\mathbb{N} \times \mathbb{N}$ defined by $(a,b)S(c,d)$ if and only if $ad=bc$. Prove this is an equivalence relation on $\mathbb{N} \times \mathbb{N}$. I think I've found ...
2
votes
1answer
363 views

Card probability problem [duplicate]

Possible Duplicate: Probability problem I found the following problem in Rosen's Discrete Mathematics and Its Applications 6th ed.: There are three cards in a box. Both sides of one card ...
2
votes
3answers
207 views

What is the expression for this summation?

Known that $\sum_{n=0}^{\infty}{x^n}{z^n}=\frac{1}{1-xz}$. If we have $\sum_{n=0}^{\infty}\frac{{x^n}{z^n}}{n\beta + \alpha}$ where $\beta, \alpha $ are element of real numbers but not equal $0$. ...