2
votes
1answer
37 views

Find the permutation

This is part of an exercise I did on an assignment but I am having trouble remembering how to complete the exercise (even though I got full marks on my assignment). Let $P_1=(3\,4\,1\,2\,5), ...
0
votes
2answers
66 views

How many coefficients are in the expansion $(x + y + z)^{10}$

I need to find the number of coefficients in the expansion $(x + y + z)^{10}$. I had this exercise on a recent assignment. The answer I gave is: $3^{10} = \binom {3 + 10 - 1}{10} = \binom{12}{10} = ...
0
votes
1answer
28 views

Why is this a boolean algebra

Let $A = \{a,b\}$. The $\mathcal P(A) = \{\emptyset,\{a\},\{b\},A\}$. Let $+$ be $\cup$, $\cdot$ be $\cap$, complement be set complement, $1$ be $A$, and $0$ be $\emptyset$. I need to explain why ...
0
votes
0answers
48 views

Describe the language generated by the grammar $G = \{\Sigma, \Delta\ S, I \} $

I need to describe the language generated by the grammar $G = \{\Sigma, \Delta\ S, I \} $, where $$\Sigma=\{0,1\epsilon \}, \Delta = \{S,X,Y,Z\}$$ and $$I = \{S \to0X|1Y, x \to1Y|1Z, Y \to0X|0Z, Z ...
1
vote
0answers
46 views

Create a finite-state machine

I need to create a finite-state machine which accepts strings whose characters are in {a,b,c} and produce output strings of T's and F's. The machine outputs a T once the characters ab is encountered ...
0
votes
1answer
46 views

Determine the languages for the given alphabet

For the alphabet $\sum = \{0,1\}, let A,B,C \subseteq \sum^*$ be the languages below. $i. A = \{1, 0, 00, 11, 000, 111, 0000, 1111\}$ $ii. B = \{w \in \sum^*|||w|| \ge 2 \}$ $ii. C = \{w \in ...
1
vote
1answer
95 views

Give an example of a relation R on $A^2$ which is reflexive, symmetric, and not transitive

I am just looking for some clarification on this exercise: Let $A = \{a,b,c,d\}$. Give an example of a relation $R$ on $A^2$ which is reflexive, symmetric, and not transitive. I understand that if I ...
1
vote
1answer
26 views

Determine the set $A$ = {$m\in Z|mR52$} and give its cardinality $|A|$.

Given a relation R on $Z^+$ defined as: $mRn$ if and only if $m|n$, I need to determine the set $A$ = {$m\in Z|mR52$} and give its cardinality $|A|$. I know that $mR52$ = $m|52$ and that $52 = mk$ ...
2
votes
2answers
80 views

find the coefficient of the given term when the expression is expanded by the binomial theorem

I am just trying to understand why the term is $\binom{15}8$(3p$^2$ - 2q)$^7$. I need to find the coefficient in $p^{16}q^7$ in $(3p^2 - 2q)^{15}$ So, I know that $n = 15$ and I have $a^{n - k}b^k$ ...
0
votes
1answer
46 views

Pre requisites of linear algebra

I want to learn abstract linear algebra. Do I require the knowledge of discrete mathematics before I start? I have the impression that abstract maths and their proofs can be understood easily by the ...
2
votes
1answer
128 views

How many positive integers n can we make with the digits 3, 3, 4, 5, 5, 6, 7, if the number n > 4, 000, 000?

According to my study guide the answer to the exercise, How many positive integers, (n), can we make with the digits 3, 3, 4, 5, 5, 6, 7, if the number n > 4, 000, 000, : The total of numbers n > ...
0
votes
1answer
53 views

How many different 5 characters words are there with only one letter a?

I just need to clarify my answer to this exercise. This is a permutations exercise. If we define a word to be a string of 5 letters of the English alphabet, regardless of meaning, then mnnnw is a ...
0
votes
1answer
187 views

Book on discrete mathematics for self study

I am searching for book on discrete mathematics which is suitable for self study. This mean I want it to have exercises with answers (It would be ideal if it had solutions). I have already read ...
0
votes
2answers
39 views

Trying to understand an example of an equivlance relation that is symmetric

I am just tying to figure our this example but am having difficulty understand the math being used. The example state: Let R be a relation on the set $\mathbb{Z}$ defined as (m,n)$\in$ R if and only ...
3
votes
2answers
67 views

one-to-one and onto functions help

I am trying to understand this exercise. Define $S : \mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ by the rule: For all integers $n$, $S(n) =$ the sum of the positive divisors of $n$. a. Is $S$ one-to-one? ...
0
votes
2answers
57 views

one-to-one and onto function question

I am trying to understand this exercise: Let S be the set of all strings of 0's and 1's, and define f: S -> $Z^{nonneg}$ by f(s) = the length of s, for all string in S. a. Is f one-to-one? The ...
0
votes
2answers
133 views

Pigeonhole Principle and maximum length of the repeating section

The question I have is, when 5 / 20483 is written as a decimal, what is the maximum length of the repeating section of the representation? I believe I need to divide 5 by 20483 which is equal to ...
0
votes
1answer
134 views

Prove that the three statements are equivalent

I need to show that the following statements are equivalent. A $\subset$ B, A $\cap$ B$^c$ = $\emptyset$, and A$^c \cup$ B = U (U is the universal set) So to show that A $\subset$ B is true I said ...
2
votes
1answer
83 views

Help understanding a counting and probability exercise

I need help in trying to understand the answer to this exercise. [Question] A club is considering changing its bylaws. In an initial straw vote on the issue, 24 of the 40 members of the club favored ...
1
vote
1answer
50 views

counting and probability question - help needed

I am stuck on how to start this exercise. Any help is welcome. An instructor gives an exam with 14 questions. Students are allowed to choose any 10 to answer. Suppose the exam instructions specify ...
1
vote
3answers
219 views

Solutions to $x+y+z=31$ and $x+2y+3z=41$

For the equations $$x+y+z=31$$ $$x+2y+3z=41$$ is there a elegant way or method to find all the positive solutions in integers? Thus far, I have been using trial and error (which is time consuming). ...
1
vote
1answer
124 views

Trying to understand an exercise using factorials with induction

Exercise: Prove that (n + 1)! - n! = n(n!) for any n $\ge$ 1 Given Answer: I will skip the basic step since I understand that part. (n + 2)! - (n + 1)! = (n + 1)!(n + 2) - n!(n + 1) I understand ...
1
vote
1answer
201 views

Venn diagram question

Here is my question. A math examination has three questions. Twenty-six students took the examination, and every student answered at least one question. Six students did not answer the first ...
2
votes
2answers
943 views

MOOCs for college-level discrete math?

Specifically I am looking for short lectures (and quizzes) on specific topics. (like Khan Academy offers) Topics I am learning about include; Intro logic-theory + set-theory Notation, negation, ...
3
votes
1answer
399 views

Concrete Mathematics Prerequisite Question

I've been very interested in the book Concrete Mathematics (Graham,Knuth,Patashnik) and I've been reading it for the past few weeks. I'm at the chapter about Sums (Chapter 2), specificaly, the lesson ...
1
vote
2answers
239 views

Study regimen for discrete mathematics? - Lack high-school maths…

I have just gotten into college, and will be studying mathematics from next semester. (this course) Unfortunately I did not study mathematics for the last 2-3 years of high-school mathematics. What ...
3
votes
2answers
231 views

find recurrence relation $T(n)=2T(n/2) +\log_2(n)$

$$\begin{align*} &T(n) = 2T(n/2) + \log_2(n)\\ &T(1) = 0 \end{align*}$$ $n$ is a power of $2$ solve the recurrence relation my work so far: unrolling this, we have $$\begin{align*} ...
2
votes
1answer
139 views

Recurrence relation: $T(n) = T(n-1) + 1/n$

\begin{align} T(0) & = 0 \\ T(n) & = T(n-1) + \dfrac{1}{n} \end{align} solve the recurrence relation My work so far: \begin{align} T(1) & = 1 \\ T(2) & = 1 + \dfrac{1}{2} \\ T(3) ...
1
vote
4answers
257 views

Which books /tutorials will be good for these topics for AI computer science student

I have found from the internet that I need to know these topics for understanding Artificial Intelligence: Matrix algebra: most machine learning models are represented as matrices and vectors. ...
3
votes
2answers
90 views

Conditions for Moving Function Outside Sine Argument

Are there multiple discrete functions $f[n]$ such that \begin{align} \sin \left( f[n] x[n] \right) = f[n] \sin \left( x[n] \right) \end{align} I know that the above equation holds for $f[n] = 0$, ...
5
votes
4answers
3k views

Where can I download Discrete Mathematics lecture videos?

Good morning, I'm doing a course in Discrete Mathematics (so far: Four Colour Theorem, Intro Graph Theory, Intro Logic Theory, Intro Set Theory and Intro Proofs) at University, but unfortunately they ...
8
votes
2answers
161 views

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 $0\leq m < N$, ...