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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4
votes
2answers
206 views

Find the sum of the series.

I need to find the following sum: $$\sum_{s=0}^{n+1}{(-1)}^{n-s}4^s\binom{n+s+1}{2s}$$ First I tried to simplify this: $$\begin{split} \sum_{s=0}^{n+1}{(-1)}^{n-s}4^s\binom{n+s+1}{2s} &= ...
4
votes
2answers
130 views

Proof sought for a sum involving binomials that simplifies to 1/2

A proof of: $$\begin{align*}(1/2)^{2m+1} \sum_{k=0}^{m} \binom{m}{k} \sum_{j=0}^{k} \binom{m+1}{j} = \frac{1}{2} \end{align*} $$ Conjecture based on the following Maple code: ...
-4
votes
2answers
34 views
2
votes
2answers
160 views

Initial value of Newton Raphson Method

I am currently studying Newton-Raphson Method. I feel that I understand the concept of it. Somehow, I am facing some question in my head about how to actually apply it. The questions that I have are ...
4
votes
0answers
45 views

How do i find formula for the recurrence relation :$x_{n+1}= x^2_{n}-x^2_{n-1}$ with :$x_{-1}=0,x_{0}=\frac{3}{4}$?

How do i find formula for the recurrence relation according to the below initial conditions: $x_{n+1}= x^2_{n}-x^2_{n-1}$ with :=$x_{-1}=0,x_{0}=\frac{3}{4}$ ? Note:$n=0,1,2\cdots$ Thank you for ...
0
votes
0answers
43 views

Number of graphs with 5 vertices

Let $v_i$ where $i=1,2,3,4,5$ be vertices of a graph. Each vertex makes only one directed edge to any other vertex. For instance $v_1 \to v_2 \to v_3 \to v_4 \to v_5 \to v_1$ and $v_1 \to v_3 \to v_4 ...
0
votes
1answer
151 views

King Arthur and knights at the round table puzzle

Can you help me with this math problem : Each of the $K$ knights from the round table needs to choose a card which is marked with a number from $1$ to $N$, $N \ge K$. The cards all have a different ...
2
votes
1answer
339 views

How to find a closed form formula for the following recurrence relation?

I have to find a closed form formula for the following recurrence relation which describes Strassen's matrix multiplication algorithm - $$T(n) = 7\,T\left(n \over 2\right) + \frac{18}{16}n^2$$ with ...
0
votes
2answers
16 views

Invalid function or invalid domain

Let $ f : A \rightarrow B $ What happens if $\exists\ a\in A $ which doesn't map to any element in B ?
0
votes
2answers
40 views

Proving 9 divides a cubic by Induction

I have just started to cover induction mathematics in my Discrete Mathematics class and I'm a little confused as to where to go with this problem. Am I on the right track? Prove that 9 divides (n^3 ...
-2
votes
2answers
45 views

An injection from R × {0, 1} to R [on hold]

What would be an example of this An injection from R × {0, 1} to R i think it is all real numbers f(x) = x Can some one help me on this. Thanks in advance
9
votes
2answers
203 views
+50

How do I prove this combinatorial identity using inclusion and exclusion principle?

$$\binom{n}{m}-\binom{n}{m+1}+\binom{n}{m+2}-\cdots+(-1)^{n-m}\binom{n}{n}=\binom{n-1}{m-1}$$ Note that we can show this with out using inclusion and exclusion principle by using Pascal's Identity ...
1
vote
1answer
22 views

Proof by contradiction - Predicates and quantifiers

Consider statement, For all integers, b,c,d, if x is a rational number such that $x^2+bx+c=d$, than x is an integer. a) express above statment in the form, $Q_1 b,c,d\in U_1 ( Q_2 x\in ...
1
vote
2answers
117 views

Calculating a recursive power term binomial sum

Could someone please help me or give me a hint on how to calculate this sum: $$\sum_{k=0}^n \binom{n}{k}(-1)^{n-k}(x-2(k+1))^n.$$ I have been trying for a few hours now and I start thinking it may ...
1
vote
2answers
1k views

Solution Verification: Maximum number of edges, given 8 vertices

Suppose a simple graph G has 8 vertices. What is the maximum number of edges that the graph G can have? The formula for this I believe is n(n-1) / 2 where n = number of vertices. 8(8-1) ...
1
vote
1answer
27 views

Lines in the plane from Concrete Mathematics. How many bounded regions are there?

I'm studying Concrete Mathematics by Donald Knuth. While doing the warmup questions, from the recurrence chapter, I stumbled upon an interesting problem. I have an intuition about the solution but I ...
1
vote
0answers
19 views

Maximum number of edges in a subgraph of hypercube

Let $H_n$ is an $n$-dimensional hypercube, $|V(H_n)|=2^n, |E(H_n)|=n2^{n-1}$. Let $M\subset V(H_n), |M|=2^k, 1\le k<n$, and $G_M$ is a subgraph of $H_n$ induced by $M$, $V(G_M)=2^k$. Prove that ...
1
vote
0answers
98 views

Maths puzzle 1: smart play with sets

Let $$X=\{ a, b, c, d, e, f, {ab}, {ac}, {ad}, {ae}, {af}, {bc}, {bd}, {be}, {bf}, {cd}, {ce}, {cf}, {de}, {df}, {ef}, {abc}, {abd}, {abe}, {abf}, {acd}, {ace}, {acf}, {ade}, {adf}, {aef}, {bcd}, ...
0
votes
1answer
381 views

Principle of Inclusion and Exclusion

Annually, the 65 members of the maintenance staff sponsor a “Christmas in July” picnic for the 400 summer employees at their company. For these 65 people, 21 bring hot dogs, 35 bring ...
0
votes
2answers
34 views

Solve it by using logical proposition

Show that given logical proposition is tautology $((A \implies C) \land (B \implies C) \land \lnot C) \implies \lnot (A \lor B) $ I can apply the implication rule first and got $\lnot((A \implies ...
0
votes
5answers
68 views

How can I prove that $4^{2012} \mod 8$ is $0$

Prove that $4^{2012} \mod 8 = 0$ I'm not really sure what rule I should use to prove this.
5
votes
3answers
685 views

Number of 11-digit length number with all 10 digits and no consecutive same digits

Here is the question: In how many ways we can construct a 11-digit long string that contains all 10 digits without 2 consecutive same digits. Initially, I came up with $10!9$. I thought that there ...
-2
votes
1answer
29 views

What is the best answer from choices for 15:220 :: 100:? [on hold]

This question is from "DEO General Intelligence Exam" Held on 31 August 2008 by Staff selection commission of India. So, please help me solve this, which of the option best suits for this question. ...
-2
votes
1answer
46 views

How many elements are in the set $S^S$, where $S=\{a,b\}$? [on hold]

If set $S =\{a,b\}$, then how many elements will be in set $S^S$? Here $S^S$ is {Set S is Exponent of S}. Do we need to do cross product like $S*S$ when it says $(S^S)$. Please advise.
0
votes
2answers
39 views

Big-O Question 1

We have to find the least integer such that $f(x)$ is $O(x^n)$ for the given function. We also have to find the smallest corresponding witnesses $C$ and $K$. Here is what I have, let me know where I ...
0
votes
2answers
19 views

Double modular exponent with Euler-Fermat

$$7^{3^{18}} \pmod{9}$$ Using this formula : $a^{\phi(m)} \equiv 1 \pmod m$ I got $7^6 \equiv 1 \pmod{9}$ and I can write $3^{18}$ as $3^6 \cdot 3^3$ And what are next steps? I got stuck here.
0
votes
1answer
29 views

What are the composite functions

f : $\mathbb{R} \to \mathbb{R}$ $$g(x)=\begin{cases} \frac1n,&x\in\Bbb Q\text{ and }x=\frac{1}n\text{ in lowest terms}\\ \sqrt{2},&x=0\ \end{cases}$$ g(x) is the inverse of f(x) determine ...
-3
votes
0answers
34 views

Show that $\log_ax \in \operatorname{\Theta}(\log_bx)$ [on hold]

Suppose $a$ and $b$ are greater than $1$ and that $f(x) = \log_ax$ and $g(x) = \log_bx$. Prove $f \in \operatorname{\Theta}(g)$. Edit: I fail to see how this is off-topic. This is the entire ...
0
votes
2answers
38 views

What is the domain of the function

I think the subset D is 1/n where n is an element of natural numbers. Can someone help me with this, thanks in advance
1
vote
0answers
41 views

how to sum the floors of ratios n/k when prime factorization of n is known

According to @harald-hanche-olsen the sum of the floors of ratios of $n/k$ is approximately: $$n(\ln n-1-\ln2)<\sum_{k=2}^{n-1}\Bigl\lfloor \frac nk\Bigr\rfloor<n\ln n.$$ If the prime ...
0
votes
1answer
32 views

Is the function invertible?

$$f(x)=\begin{cases} \frac1q,&x\in\Bbb Q\text{ and }x=\frac{p}q\text{ in lowest terms}\\ 0,&x\notin\Bbb Q\;. \end{cases}$$ Is the function $f|_D$ invertible? If so, describe its inverse ...
1
vote
1answer
54 views

Bit strings of even length that start with 1

We have to give the recursive definition of the set of bit strings of even length that start with 1 We were shown an example that showed the set of all bit strings with no more than a single 1 can be ...
0
votes
1answer
24 views

Show that $[(P→Q)∧P)]→Q$ is a tautology using rules of replacement

Show that $[(P→Q)∧P)]→Q$ is a tautology using rules of replacement I've done this so far, from $[(P→Q)∧P)]→Q$ to $[(~P∧Q)∧P)]→Q$ by Mat. Imp. to $[P∧(~P∧Q))]→Q$ by Commutation. After that ...
0
votes
0answers
19 views

RSA number sequence encryption

Encrypt the following number sequence $3,9,27$ with key $m=33$ and $r=7$ It's about RSA encryption. How should I encrypt this? Should I find the key $s$ (inverse key) and what then? $r \cdot s + ...
1
vote
1answer
30 views

Graph and tree computation

A graph is given with set of nodes $[x_1,x_2,x_3,\ldots,x_6]$ and with set of edges: $$\{[x_1,x_2], [x_1,x_3], [x_1,x_4], [x_1,x_5], [x_1,x_6], [x_2,x_3], [x_2,x_6], [x_3,x_4], [x_4,x_5], ...
0
votes
1answer
14 views

Solve the relation with congruence

On $\Bbb Z$ consider the relation $xRy \Leftrightarrow x-y \not\equiv 0 \mod 3$. Prove (with explanation), whether the relation reflexive, symmetric, antisymmetric transitive is and prove if they are ...
1
vote
2answers
447 views

Find recurrence relation for ternary strings that don't have substrings 00, 01 and last symbol is not 0

I am preparing for my finals for discrete mathematics and I came across this exercise in textbook. Let $s_{n}$ denote all ternary strings of length $n$, such that any string in $s_{n}$ does not ...
-1
votes
2answers
23 views

What is the adjacency matrix and number of paths of length $4$ between vertex $2$ and vertex $5$ in the null graph on $\{1,2,3,4,5\}$? [on hold]

Given the following graph 1) Compute adjacency matrix 2) Compute the number of paths of length 4 from knot Nr.2 to knot Nr.5 Can anyone provide a solution how to do it?
0
votes
1answer
20 views

Bridge hands (13) Discrete Mathematics [on hold]

How many bridge hands contain four cards of the three suits and one card of the fourth suit?
-4
votes
1answer
17 views

Compound interest in half yearly [on hold]

In what time will $64000 amount to $68921 at 5% per annum interest being compounded half yearly.
-4
votes
0answers
13 views

Discrete mathematics: propositional calculus [on hold]

Please state and explain the duality law and De Morgan's theorem for propositional calculus
-4
votes
0answers
31 views

find a method for twin primes and with Golbach conjecture [on hold]

There are infinitely many twin primes. Two primes (p, q) are called twin primes if their difference is 2. Let be the number of primes p such that p<= x and p + 2 is also a prime. a sample ...
-1
votes
0answers
25 views

Quadratic recurrence inequality

I have the recurrence relation: $r_{k+1} \leq r_k^2+ (1/2)r_k \quad (k =1,2,\ldots)$, where each $r_k$ is non-negative and $r_1<1$. I have the following questions in this regard: A simple plot ...
1
vote
2answers
40 views

In how many ways I can write a number $n$ as sum of $4$ numbers?

The precise problem is in how many ways I can write a number $n$ as sum of $4$ numbers say $a,b,c,d$ where $a \leq b \leq c \leq d$. I know about Jacobi's $4$ square problem which is number of ways ...
0
votes
1answer
26 views

Relation $R = \{(1,2), (2,2), (2,3), (3,2)\}$ is defined on $A = \{1,2,3\}$. Draw the digraph for the relation $R^3$.

So I don't want an explicit answer, but I do need help getting it from $R^1 \rightarrow R^2 \rightarrow R^3$. Relation $R = \{(1,2), (2,2), (2,3), (3,2)\}$ is defined on $A = \{1,2,3\}$. Draw the ...
-3
votes
2answers
36 views

What is the matrix and directed graph corresponding to the relation $\{(1, 1), (2, 2), (3, 3), (4, 4), (4, 3), (4, 1), (3, 2), (3, 1)\}$?

Let $R$ be a relation on set $A = \{1, 2, 3, 4\}$ defined by $$R = \{(1, 1), (2, 2), (3, 3), (4, 4), (4, 3), (4, 1), (3, 2), (3, 1)\}.$$ Find the matrix and directed graph of relation $R$.
0
votes
1answer
30 views

Euler-Fermat with exponents

How to solve $6^{(3^{17})}$ mod 11 with Euler-Fermat? Note: If not possible with Euler-Fermat than with Chinese Remainder Theorem I know that that they are coprime and I computed $\varphi(11)$, so ...
3
votes
2answers
80 views

Probability with changing number of marbles

Given a bag containing 20 marbles of 5 different colors in this configuration: 8x Blue 6x Red 3x Green 2x White 1x Black How would you determine the probability of picking a marble of a specific ...
0
votes
2answers
26 views

Student card handing Inclusion–exclusion principle

I got the following question and would very much appreciate any help with understanding it solution. "5 Student cards are handed to 5 students so that each student gets 1 student card, what is the ...
4
votes
3answers
2k views

Recurrence relation (linear, second-order, constant coefficients)

Q1. Find the general solution to the difference equation $$ a_{n} - 4a_{n-1} + 3a_{n-2} = 6 $$ Q2. Solve the difference equation $$ a_{n} - a_{n-1} - 2a_{n-2} = 0, a_0 = 2, a_1 = 4 $$ I am ...