Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous.

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Which non-Abelian finite groups contain the two specific centralizers? - part II

This is a question requiring the good knowledge of group theory: (Q1) Which finite groups $G$ contains some specific centralizers isomorphic to both of these two groups (but may contain other ...
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1answer
22 views

Tanh function representation for conditional function

I have a condtion as $$T(x)= \begin{cases} -1 & \text{if }x <a \\ 0 & \text{if }a\le x \le b \\ 1 & \text{if }x >b \end{cases} $$ I want to approximate the above condition as ...
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3answers
61 views

How to prove a function from $\mathbb N\times \mathbb N$ to $\mathbb N$ is bijective. [duplicate]

I am having trouble with this problem: $f\colon \mathbb N\times \mathbb N \rightarrow \mathbb N$ is defined by $f(i,j)=\dfrac{(i+j-1)(i+j-2)}{2}+i$. How do you prove that $f$ is a bijection from ...
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1answer
31 views

How to represent the condition by mathematical

I have a condtion as $$T(x)= \begin{cases} -1 & \text{if }x <a \\ 0 & \text{if }a\le x \le b \\ 1 & \text{if }x >b \end{cases} $$ I want to represent the above condition as one ...
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1answer
17 views

How do I use Graph theory to determine the minimum amount of moves needed to swap chess pieces?

On 3x4 chessboard (see below) there are 3 Black knights (B B B) and 3 white knights (WWW), exchange knights in the min # of turns (hint: use graph representation) B B B -> WWW 0 0 0 -> 0 0 0 0 0 0 ...
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some past paper questions in Discrete Time Systems i couldnt solve.

I am working on past papers of my exam which is in two days, there was one particular year , 2009, which I could not solve quite a lot of its questions... i only could solve 5 out of 10, can anyone ...
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20 views

how to solve the current problem of evaluating limits approching zero [on hold]

$\sum{(\dfrac{1}{pi}\cdot \tau\cdot \alpha)X}, $$ \ \ \sum(\sqrt{(1-(\mho/2\alpha\tau) {k_a}^2)})$ first summation limit is $\tau=0$ second summation limit is $\mho=-2\alpha\tau$ to $+2 \alpha \tau$ ...
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35 views

Boole's functions' domain is D = {1, 2, 3, 4}. Find ∃xF(x, 2), when F(x, y) = 1100 1111 0011 0101. [on hold]

The problem is, I actually do not understand this problem very well. When the logical function is given, making truth table is not a problem for me at all. I wonder, if this exercise requires to make ...
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2answers
32 views

Counting problem involving sets

Let $S$ be a set of size $37$, and let $x$,$y$, and $z$ be three distinct elements of $S$. How many subsets of $S$ are there that contain x and $y$, but do not contain $z$? How many subsets of $S$ ...
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1answer
23 views

When proving a partial order relation is a total order do we have assume both elements are distinct?

Consider the "divides" relation on the set $A=\lbrace 1,2,2^2,.\;.\;.,2^n\rbrace$, where $n$ is a non-negative integer. Prove that this relation is a total order on $A$. First we prove $A$ is a ...
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2answers
24 views

Different ways of arranging a group of 10 people

In how many ways can a photographer arrange $8$ people in a row from a family of $10$ people, if (a) the bride and groom are in the photo. This would be $9*8*7*6*5*4*3*2*1=362880$, correct? (b) the ...
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0answers
11 views

How do I draw a Hasse Diagram for the given PO-set?

Copied from my homework: Draw a Hasse Diagram for the PO-set: ({$p, r, p \lor r, p \land r, p \to r$}, $\Rightarrow$) where {$p, r, p \lor r, p \land r, p \to r$} is a set of propositions and ...
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2answers
22 views

Proving that a set is denumerable without using a particular theorem

this question may seem like a duplicate of another one that I asked, but it is not. In this question, I am not allowed to use the Theorem which states: Every infinite subset of a denumerable set is ...
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1answer
23 views

Equivalence Relations and distinct equivalence classes

$A=\lbrace(1,3),(2,4),(-4,-8),(3,9),(1,5),(3,6)\rbrace$. $R$ is defined on $A$ as follows: For all $(a, b)\;(c, d) \in A$, $(a, b) R (c, d) \iff ad=bc$ I know what they are asking but I cannot see ...
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2answers
38 views

Proving that a function from $N\times N$ to $N$ is bijective.

I am stuck on this problem: Define $f: N\times N \rightarrow N$ by $f(i,j)=\frac{(i+j-1)(i+j-2)}{2}+i$. How do you prove that $f$ is a bijection thus $N\times N$ and $N$ are numerically ...
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1answer
13 views

How many way can 7 friends line up if there are certain conditions?

How many ways can 7 friends line up if Ann, Beth, and Chris have to stand next to each other where Ann is ahead of Beth and Beth is ahead of Chris? Would it simply be $5*4*3*2*1=120$ ways? Expanding ...
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0answers
36 views

Proving that the set of irrational numbers is uncountable [duplicate]

Work: Assume that the set of irrational numbers is countable. Since $Q$ is infinite, it is therefore denumerable. Therefore, there exists a bijective function $f: N \rightarrow Q$. From here I am ...
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2answers
23 views

How many strings of 8 digits end with an even digit?

So there are $10$ combinations for each digit except the last which has 5 possibilities ($0,2,4,6,8$). Thus $10*10*10*10*10*10*10*5=50000000$ combinations right? As a follow up, how many strings of 8 ...
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1answer
45 views

$(p \implies q) \wedge (q \implies r) \implies (p \implies r)$

Show that $(p \implies q) \wedge (q \implies r) \implies (p \implies r)$ is a tautology. I have the truth tables but cannot algebraically manipulate the language itself to prove it. What I ...
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25 views

Find the formula for the given sum of series

Find the sum of the series: $$\sum_{i=2}^{n}\binom{i}{2}= \,^{2}C_{2}+\cdots+\,^{n}C_{2}$$ I did try expanding it and see if I could simplify it further.I am unable to find a formula for it? Can ...
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13 views

Simplify the following summation involving the Floor function

Let $x,y,n \in \mathbb{Z}$ and $a\in [0,1/3).$ Further assume that $x<0,$ and $y>-2x.$ Is there any significant way to simplify the following: $\left(\sum\limits_{i=\lceil 1/3-(x+a) ...
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1answer
34 views

How to come up with a probability distribution knowing the mean value? [on hold]

I would like to know about some algorithms or techniques to find a discrete probability distribution knowing the mean value. Let's say given the mean=2.5. The probability distribution can be $x_1=2, ...
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1answer
47 views

help in approaching this problem

This is a new scheme started by AirTel in Karnataka which allowed institutions to host websites on their servers by paying for only 4 of the 7 days per week. However, the service cannot be ...
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18 views

In how many ways can i build this String : abbcccdddd? [on hold]

In how many ways can i build this String : abbcccdddd ?
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1answer
49 views

Confusion regarding steps in bipartite matching proof

Can someone please explain how it follows that $|N(S)|x \geq |S|x$? What I'm asking is why is it necessary to use the value of x to derive the inequality? Theorem 5.2.7. Let G be a bipartite graph ...
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2answers
37 views

How to find the Equivalence class for a given set?

I'm really having trouble understanding these equivalence classes. Could someone please guide me through the following problem step by step, and help explain this. I have a final exam next week, and ...
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2answers
39 views

Solving a Linear Non-Homogeneous Recurrence

How can I solve the following recurrence? $$a_n = 121a_{n-2} + 14400 n$$ I derived this: $$\frac{1228}{11} (-11)^n + \frac{-4044}{11}11^n + 4800n$$
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Solving Linear Recurrences

I have the following recurrence: $$a_n = 49 a_{n-2}, n \geq 2, a_0 = -8, a_1 = 14$$ I was able to derive the following: When $n$ is even then, $a_n = -8(7^n)$, and for the odd values of $n$, $a_n = ...
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1answer
35 views

Proving that $f: N\times N \rightarrow N$ is surjective [duplicate]

I am having trouble proving that the function $$f: N\times N \rightarrow N, \ \ f(i,j)=2^{i-1}(2j-1)$$ is surjective. Work: I know that using the theorem in which $n$ is the product of prime numbers ...
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2answers
43 views

Proving that $f$ is a bijection from $N$x$N$ to $N$.

I am having trouble with the following problem: $f: N\times N\rightarrow N$ and $f(i,j)=2^{i-1}(2j-1)$. Prove that $f$ is a bijection thus $N\times N$ and $N$ are numerically equivalent. Work: I ...
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1answer
30 views

Determining the total degree of a tree

At the start of the solution, I understand that any tree with four vertices has three edges. I don't understand the next statement: "Thus the total degree of a tree with four vertices must be 6." ...
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1answer
30 views

Expected number of rolls when repeatedly rolling an $n$-sided die

Suppose I roll an $n$-sided die once. Now you repeatedly roll the die until you roll a number at least as large as I rolled. What is the expected number of rolls you have to make? I know the answer ...
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1answer
26 views

Chromatic Equivalence Requirements

I have searched and searched and am unable to find the answer that I am looking for. I am trying to determine the conditions required for two graphs to have the same chromatic polynomial. On both ...
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2answers
23 views

How many bit strings of length $7$ either begin with two $1's$ or end with three $1's$?

So for the first case (beginning with 2 $1's$) there are: $2*2*2*2*2=32$ ways Second case (end with three $1's$): $2*2*2*2=16$ And then we can just sum it $32+16=48$ different bit string of length 7 ...
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2answers
11 views

Combinations questions

a. How many different 4 letter codes can there be? b. What if letters cannot be repeated? c. What if, in addition, 2 of the letters are x and y? For a, it would simply be $26*26*26*26=456976$ For ...
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1answer
15 views

Question over combinations

A t-shirt is being sold in 8 colors, 4 sizes, collared or tee, and long sleeve or short sleeve. a. How many different shirts are being sold? b. What if collared shirts only come in 5 colors and 2 ...
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1answer
36 views

feedback on my answer regarding set intersections.

Prove or find a counter-example to the claim that for all sets $A,B,C$ if $A\cap B = B \cap C = A \cap C = \emptyset$, then $A \cap B \cap C=\emptyset $. the above statement is not true so i need a ...
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2answers
29 views

proving discrete mathematics or giving counter example

Prove or find a counterexample: For all real numbers x and y it holds that x + y is irrational if, and only if, both x and y are irrational. can anyone explain to me or give a hint on how to start ...
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3answers
43 views

proving or providing counter example in disrete mathematics

Prove or find a counterexample: The product of any three consecutive natural numbers is divisible by 6. if we take a few consecutive natural numbers such as 1 ,2 ,3. and multiply i get 6 which is ...
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Finding mathematical relation of matrices with reverse indices

I am designing a simple game, I have faced this problem to get the mathematical relation between two kind of tables: MATRIX A MATRIX B As you can see the table A (or Matrix A) is the normal ...
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0answers
36 views

Write a non-recursive algorithm to compute n! [on hold]

I am having problems writing a code in java to compute n! without recursion. I know how to do it in loops, but I am not sure how to do it non-recursively. Thanks. ...
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Write a recursive algorithm to calculate the number of ways the robot can walk n meters. [on hold]

I am having problems trying to solve these two questions. I am writing the code in java to produce this and have not gotten very far. Can someone please help? Thank you. a) A robot can take steps of ...
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5answers
624 views

Explanation of recursive function

Given is a function $f(n)$ with: $f(0) = 0$ $f(1) = 1$ $f(n) = 3f(n-1) + 2f(n-2)$ $\forall n≥2$ I was wondering if there's also a non-recursive way to describe the same function. WolframAlpha tells ...
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3answers
41 views

Proof by Direct Method

If $(3n+2)$ is odd then, prove $n$ is odd. $$3n+2 = (2n+1)+(n+1)$$ We already have a fact that $2n+1$ is always odd. So, for $3n+2$ to be odd, $n+1$ should be even (For $x+y$ to be odd then ...
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27 views

Coefficient question in generating functions [on hold]

In each of the following, find the coefficient of $x^{2005}$ in the generating function $A(x)$. (a) $A(x) = (1 – 2x)^{5000}$ (b) $A(x) = \frac{1}{1 + 3x}$ (c) $A(x) = \frac{1}{(1 + 5x)^2}$ (d) ...
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1answer
30 views

terms/names for these *things* [on hold]

So I am having difficulty finding the correct terms to describe the following things. multiple planes make up a space multiple spaces make up a universe (maybe? if not, what is it?) multiple ...
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1answer
32 views

Closed form questions [on hold]

Please could you help me to find the generating functions of the following sequences in closed form: (a) 1, 0, 1, 0, 1, 0, … (b) 2, –4, 6, –8, 10, –12, …
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1answer
39 views

Number of Solutions in counting problem [on hold]

Find the number of solutions to the equation a + b + c + d = 50 if each variable is: (a) a non-negative integer (b) a positive integer (c) an odd positive integer (d) an integer between 4 and ...
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3answers
561 views

Proof about prime numbers

Can we prove that every prime larger than 3 gives a remainder of 1 or 5(edited) if divided by 6 and if so, which formulas can be used while proving?