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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Discrete Math Elements within a set

$\{x \mid x \in\mathbb N, x \text{ is even, and } 2 < x < 11\}$ Would the elements in this set be $x$ and all positive even integers between $2$ and $10$?
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1answer
35 views

Understanding transformation as algebraic structure

I am confused about the following structure, and would be very thankful if somebody could give me a hint. Let $\mathbb{S}$ be a set with n elements $\mathbb{S}=\{a_1, a_2, ..., a_n\}$, and $(x,y) \in ...
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1answer
42 views

Equivalence relation on $\mathbb{N}\times\mathbb{N}$

Define a relation on $\mathbb N\times \mathbb N$ by $$ (a,b)\sim (c,d) \iff a+d=b+c. $$ Prove that if $(a, b)\sim(a', b')$ and $(c, d)\sim(c', d')$, then $(ac+bd,bc+ad)\sim(a'c'+b'd',b'c'+a'd')$ ...
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3answers
254 views

Proof by contradiction in Discrete Mathematics

Ok, so my college book is the worst book ever and I can only survive from this site and youtube. Could someone please explain the answer below? I really do not understand the answer and to me there is ...
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1answer
32 views

sum of a binomial coefficient [duplicate]

Trying without success to solve the following: what is the sum of $\binom{80}{0}-\binom{80}{1}+\binom{80}{2}-\binom{80}{3}...-\binom{80}{79}+\binom{80}{80}$ any help will be greatly appreciated
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2answers
26 views

Finding the error in two approximations of $e$

The number $e$ can be defined by $e=\sum_{n=0}^\infty (1/n!)$, where $n! = n(n-1)\cdots 2 \cdot 1$ for $n \neq 0$ and $0! = 1$. Compute the absolute error and relative error in the following ...
2
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2answers
42 views

Next step to take to reach the contradiction?

This problem is from Discrete Math and its Applications I am trying to use proof by contradiction to do this problem, proof by contradiction as described by the book Here is my work so far for ...
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4answers
55 views

Using proof by contraposition to show that if $n\in\mathbb Z$ and $3n+2$ is even, then $n$ is even

I have my answer below but there is one step that I am not understanding...and maybe my brain is just not trained to understand it. Prove that if $n$ is an integer and $3n+2$ is even, then $n$ is ...
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1answer
33 views

Labeling constraints in a MILP

A manufacturing company consisting of two plants intends to introduce up to three new products. The production quantity of each product can be any number, integer or non-integer, but there is an upper ...
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4answers
86 views

What is the coefficient of $x^{17}$ in the formula $(x^2+x)^{15} $?

What is the coefficient of $x^{17}$ in the formula $(x^2+x)^{15} $? Any idea how to solve this using the binomial coefficient formula?
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3answers
48 views

counting and permutation problem

i am having hard time figuring this out: a) john,tom,jessie,sam,michael,and amanda want to split among themselves 100 apples how many ways can they split the apples among themselves, if jessie cannot ...
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1answer
63 views

Puzzle Involving Infinite Grid

This is a riddle that a coworker of mine posed to me, I have a solution but I'm curious to see what you all arrive at (I'm more interested in the approach than the answer). The question (potentially ...
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0answers
105 views

Discrete Mathematics, set theory power set question.

Find the power set℘(S) for S={Ø,{Ø},{Ø{Ø}}} Since there are 4 elements would the power be 2^4 which is 16?
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1answer
57 views

counting a subset from a finite set

At a congressional hearing, there are 2n members present. Exactly n of them are Democrats and n of them are Republicans. The members want to select a smaller subcommittee of size n from within those ...
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1answer
68 views

Discrete Math - p imp q Truth Table

I'm trying to understand what imp1 is or how it is used. I'm unable to find further information in my textbook and this is an exercise question. Question: Provide further motivation for defining p → ...
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2answers
56 views

prime division methodology

Trying to solve: Find $a,b,c$ for $31|(5a+7b+11c)$ I found $a=6,b=3,c=1$ as one solution. Is there a systematic way to find all solutions? I was thinking take $5a+7b+11c=31n$ and solve by method ...
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2answers
61 views

primes equal if and only if one divides other

$p,q$ primes. prove $p=q$ if and only if $p$ divides $q$. $p|q$ stands for '$p$ divides $q$' $p|q\Leftrightarrow p=q$ $\Leftarrow$: $p(1)=q$ and therefore $p|q$ $\Rightarrow$: if $p=\pm 1$, ...
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1answer
46 views

Distinct vs Identical

In a bag containing 20 balls(6 red), (6 green), (8 purple) We draw 5 balls, put them back in the bag, then draw 5 more. In how many ways can this be done if the balls are considered distinct? My ...
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0answers
43 views

Proving the Divisibility Rule for $3$ [duplicate]

Theorem: If 3 divides the sum of the digits of a number, then 3 divides that number
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0answers
19 views

Will this predicate be true over all of the domain, as specified by quantifier?

This is a problem from Discrete Mathematics and its Applications My question is about 27a. I know what the nested quantifier is saying - for every integer n, there exists an integer m that is ...
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1answer
36 views

How to introduce bi-conditional in this proof?

This is from Discrete Mathematics and its Applications Just for context, I know that the universal set is everything and that the complement of A is just difference of the universal set and A. A ...
2
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1answer
64 views

Negation of Compound-Statements

If I have the following statements p: It is cold outside q: It is snowing p∧q = It is cold outside, and it is snowing. p∨q = It is either cold outside or it is snowing. If I were to negate p∧q ...
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1answer
21 views

Binomial theorem counting number of solutions .

How to find the number of solutions to the equation $x_1 +x_2 +x_3 +x_4 +x_5 = 37$, where $x_1, x_2, x_3, x_4, x_5$ are non-negative integers, $x_2 ≥ 8, x_3 ≥ 7, x_4 ≥ 2$ and $x_5 < 4$. Is this ...
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1answer
87 views

Recurrence relation-there is no initial condition

I want to find the exact solution of the recurrence relation: $T(n)=2T(\sqrt{n})+1$. $$m=\lg n \Rightarrow 2^m=n \\ \ \ \ \ \ \ \ \ 2^{\frac{m}{2}}=\sqrt{n}$$ So we have: ...
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2answers
101 views

How tot start proving $A \times B \times C \ne (A \times B) \times C$?

This is a problem from Discrete Mathematics and its Applications: Explain why $A \times B \times C$ and $(A \times B) \times C$ are not the same. I understand the process behind the ...
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2answers
162 views

What does this symbol mean?

This is from Discrete Mathematics and its Applications What is the symbol used in 9c, 9d, 9f, 10c, 10f, 10g? I looked through the chapter section and the closest symbol I saw to this is the subset, ...
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1answer
219 views

binary strings and counting sequence problem

Hello i'm working on these questions and I have few questions 1) A binary string is a finite sequence of 0 and 1. Ex. 001101 is a string of length 6 a) List all binary strings of length 4 (so I ...
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4answers
1k views

Is empty set element of every set if it is subset of every set?

This problem is from Discrete Mathematics and its Applications My question is on 9b. I know that the sign represents an element is a member of. (from book) I know that the O with a slash across ...
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1answer
43 views

“with n a positive integer, evaluate the sum” - bionomial quesiton

The question is, with n a positive integer, evaluate the sum $c(n,0) + 2*c(n,1) + 2^2*c(n,2) + ... + 2^k*(n,k) + ... + 2^n(n,n)$ I think since $(a+b)^n = \sum_{k=0}^n c(n,k)b^ka^{n-k}$, first thing ...
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2answers
59 views

How to iterate through all the possibilities in with this quantifier?

This is a problem from Discrete Mathematics and its Applications My question is on 9g. Here is my work so far I am struggling with the exactly one person part. The one person whom everybody loves ...
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2answers
30 views

Confused on Conditional Statements

Write these propositions using $p$ and $q$ and logical connectives (inclduing negations) $p$: You drive over $65$ miles per hour. $q$: You get a speeding ticket You will get a speeding ticket if ...
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2answers
119 views

making necklaces with beads problem

Hello im working on this problem and im completely stuck (a) there are 4 biz. How many necklase can I make? (b)consider the set of all necklase with distinct biz, where the size of a necklace is the ...
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1answer
86 views

different path permutation problem

Im having hard time with this question How many different paths in the $xy$-plane are there from $(0,0)$ to $(7,7)$ if a path proceeds either one space to the right $(r)$ or on space upward $(u)$? ...
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0answers
113 views

Inputs x1, x2, and x3 are all true then output y is true

I'm in online school there's no guidance and I'm just completely unprepared for this. I am to write this as a logical statement using logic symbols for conjunction and implication. Can someone help ...
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1answer
75 views

What is this symbol above the X?

Not sure what it is, $X$ is a set that contains ${2, 5, 6}$. This is all the text says. "Let $U =\{1, 2, 3,\dots, 10\}$ be the universal set and consider the following subsets of $U$;$X = \{2, 5, ...
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1answer
53 views

Chromatic number of a map

Since in the map each state is connect to another state we are dealing with a complete graph ($K_{12}$). Since, it is a complete graph (every state is connected to every other state), every vertex ...
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2answers
47 views

Question about combinatoris of four dice

We have $4$ dice with $4$ different dyes : Green, Yellow, Red and Blue. I need to know how many options do i get so that I'll get the number $3$ at least once. These are 6 sided dice, color does ...
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1answer
80 views

dividing objects in combinatorics

so I have this question with a lot of sub questions inside, i'm sure about my answer for most of them, still i have 2 that i dont really know how to do : There are 6 girls that robbed a bank, stole ...
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8answers
171 views

Prove that if $x+5$ is odd, then $x^2$ is even.

I've been working at this for about an hour and cannot figure it out. Problem: Prove that if $x+5$ is odd, then $x^2$ is even. I've tried induction, contrapositive and contradiction methods. I ...
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2answers
75 views

Show that $C(n,k)$ equals to $ C(n-1,k-1)+C(n-1,k)$

The question asks me to show that $C(n,k) = C(n-1,k-1)+C(n-1,k)$ by expanding out the formulas and using elementary algebra. It seems like the question requires me to use binomial expansion but Im not ...
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0answers
25 views

Binary Overflow Detection

I am trying to solve several problems, which are binary and encoded using the 2's complement system. One problem has stuck out to me: 0111 + 0001 Both are positive, with 0111 being 1+2+4 or 7, and ...
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1answer
50 views

Four dice showing 3 values

I have 4 different color dice: blue, red, yellow and green. I need to check how many possibilities we get if the set of the numbers that are on the dices consists of 3 distinct numbers. In other ...
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1answer
105 views

Time & Distance : Pokemon Hunter and the Rogue Brook

I was working my way through some Puzzles in Discrete Maths by Rosen, when I came across the following question: A Pokemon Hunter is rowing upstream a brook As he passes under the ...
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0answers
160 views

What is the counting sequence of a Binary String?

I have a two part question dealing with the binary strings for Discrete Math and I am stuck on the meaning of the very last part. (a) List all binary strings of length 3. Since a binary string is ...
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3answers
692 views

Find the highest power of two in the expression.

What would be the highest power of two in the given expression? $32!+33!+34!+35!+...+87!+88!+89!+90!\ ?$ I know there are 59 terms involved. I also know the powers of two in each term. I found that ...
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1answer
53 views

Doubt : Invariance in Geometry

I was working my way through some Proof Problems in Discrete Maths by Rosen, when I came across the following question: What Geometric proposition ( having an invariant property ) does this ...
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1answer
52 views

Combinatorics: Prove the number of matches in a Singles Tournament

I was working my way through some problems in Discrete Maths by Rosen, when I came across the following question: There are x players in a singles badminton tournament Show that there are ...
2
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2answers
55 views

How to find the numbers of Bezout identity for two numbers

I'm having troubles finding two numbers a,b such that $ 288a+177b=3=gcd(177,288) (1) $ I've been writing the equations of the Euclids algorithm one over another many times to get any pair that verify ...
2
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1answer
50 views

Next step to reach the contradiction?

This is a problem from Discrete Mathematics and its Applications Here are my notes and my current work so far for this problem. I started with an assumption that what i am trying to prove is ...
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2answers
96 views

Converting Decimal to Hexadecimal

MathExchange, I am trying to learn more about computers, and one thing I have opted to teach myself is decimal to binary, and decimal to hex conversion. From the web, I have found tutorials on ...