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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1answer
147 views

finding the minimal property of a graph

While working out on a problem, I found that cycles $C_n$ are minimally self-centered graphs, as if we remove any edge then it is paths $P_n$ and $P_n$ are not self-centered graphs. My question is ...
3
votes
3answers
214 views

Gerrymandering urns (redux)

This is a rehash of this question (and probably the intent of this, and several other similar questions), but I'd like: a more detailed answer that builds from the simplest cases to potentially ...
2
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9answers
137 views

If $n\ge2$, Prove $\binom{2n}{3}$ is even.

Any help would be appreciated. I can see it's true from pascal's triangle, and I've tried messing around with pascal's identity and the binomial theorem to prove it, but I'm just not making any ...
2
votes
2answers
73 views

Prove that if an average of a thousand numbers is less than 7, then at least one of the numbers being averaged is less than 7 [closed]

I tried proving this by contraposition, by saying, "If every number that is being averaged is greater than 7, then the average of a thousand numbers is less than 7." This seems easier to prove, but I ...
2
votes
2answers
546 views

If $x \in\mathbb{Z}$ has the property that for all $m \in\mathbb Z$, $mx = m$, then $x = 1$

I am learning proofs, and I am stuck with this proposition: Let $x \in\mathbb{Z}$. If $x$ has the property that for all $m \in\mathbb Z$, $mx = m$, then $x = 1$. I want to use the additive ...
2
votes
4answers
78 views

Clarifying on how if p,q is logically equivalent to p only if q [duplicate]

Here is what my book says about the different ways implications are worded I am struggling with how "if p, then q" is logically equivalent to "p only if q" The example I came up with With "if ...
2
votes
2answers
36 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 ...
2
votes
1answer
77 views

Asymptotic Function proof?

I am doing questions from past exams and I stumbled upon this one. I have no idea how to go about solving it.I never had any logarithmic functions in my previous bigOh proofs nor have I had to use ...
2
votes
1answer
68 views

Express $\forall n \in \mathbb N$, $\exists m \in \mathbb N$, $n^4 = m^2$ in words without using the symbol $\mathbb N$

Express $\forall n \in \mathbb N$, $\exists m \in \mathbb N$, $n^4 = m^2$ in words without using the symbol $\mathbb N$. My Solution: For all $n$ that is an element of Natural number there is ...
2
votes
1answer
252 views

Pigeonhole Principle

Let $X = {x_0, x_1, · · · , x_m}$ be a subset of ${1, 2, · · · , n}$, where $m > n/2$, and $x_0$ is the smallest number in $X$. Use the pigeonhole principle to show that $X$ contains two numbers ...
2
votes
2answers
142 views

At a party $n$ people toss their hats into a pile in a closet.$\dots$ [duplicate]

Question: At a party $n$ people toss their hats into a pile in a closet. The hats are mixed up, and each person selects one at random. What is the expected number of people who select their own hats? ...
2
votes
1answer
100 views

Fibonacci proof by induction

I have fibonacci numbers defined as such: $$ f(n) = f(n-1) + f(n-2)$$with$$ f(0) = 0 $$$$f(1) = 1$$ I have to prove that $$ F(n) \geq 1.5^{n-1}, n \geq 6$$ Base Case: $$ f(6) = 8 \geq (1.5)^5 ...
2
votes
1answer
45 views

Need help with Proof by Strong Induction question

So, here is the question: For any position integer $n$, let $T(n)$ be the number 1 if $n<4$ and the number $T(n-1) + T(n-2) + T(n-3)$ if $n \geq 4$. We have $T(1)=1, T(2)=2, T(3)=3$ ...
2
votes
2answers
201 views

Infinite Sum with Combination

I am trying to figure out what the following sum converges to: $$\sum_{n=0}^\infty {6+n\choose n}x^n(6+n),\qquad\qquad0<x<1$$ An answer would be great, but if you have an explanation, that'd ...
2
votes
4answers
184 views

For the Fibonacci sequence prove that $\sum_{i=1}^n F_i= F_{n+2} - 1$

For the Fibonacci sequence $F_1=F_2=1$, $F_{n+2}=F_n+F_{n+1}$, prove that $$\sum_{i=1}^n F_i= F_{n+2} - 1$$ for $n\ge 1$. I don't quite know how to approach this problem. Can someone help and ...
2
votes
2answers
180 views

Proving a recurrence relation for strings of characters containing an even number of $a$'s

We consider strings of $n$ characters, each character being $a$, $b$, $c$, or $d$, that contain an even number of $a$'s. (Recall that $0$ is even.) Let $E_n$ be the number of such strings. ...
2
votes
1answer
75 views

Bijective function proof in $R\times R$ and $Z\times N$

How can I verify if these functions are bijective? $ f_4:\Bbb{R^2} \rightarrow \Bbb{R^2}, \ (x,\ y)\mapsto (x+y,\ x-y)$ $ f_5:\Bbb{Z} \times \Bbb{N^*} \rightarrow \Bbb Q, \ (p,\ q)\mapsto p + ...
2
votes
4answers
126 views

Fibonacci sequence proof

Prove the following: $$f_3+f_6+...f_{3n}= \frac 12(f_{3n+2}-1) \\ $$ For $n \ge 2$ Well I got the basis out of the way, so now I need to use induction: So that $P(k) \rightarrow P(k+1)$ for some ...
2
votes
1answer
112 views

Solve $x^2$ $mod$ $23 = 7^2$

What is the procedure to solving $x^2$ $mod$ $23 = 7^2$? According to WolframAlpha, there is no integer solution but I am completely confused as to what steps was taken to determine that. Before ...
2
votes
1answer
91 views

Planar Realization of a Graph in Three-Space

We call a planar graph one that we can draw in two-space such that no two edges intersect. I was told that we're not so interested in drawing graphs in three-space, because it is "intuitively obvious" ...
2
votes
3answers
306 views

Demonstrate that if $f$ is surjective then $X = f(f^{-1}(X))$

I haven't been able to do this exercise: Let $f: A \rightarrow B$ be any function. $f^{-1}(X)$ is the inverse image of $X$. Demonstrate that if $f$ is surjective then $X = f(f^{-1}(X))$ where $X ...
2
votes
1answer
135 views

pigeonhole principle divisibility proof

Let n be some positive odd number, prove that there exists some positive integer k such that n|(2k-1), prove in terms of the pigeonhole principle
2
votes
1answer
159 views

Possible ways to walk to school

I am not sure how to approach this problem. Every day, a dance student walks from her home to dance-school, which is located $12$ blocks east and $16$ blocks north from home. She always takes the ...
2
votes
2answers
219 views

Parity function proofing for every n>=1 using only AND, OR, 0, and 1

Consider the parity function: $F_n$($x_1$, $...$ ,$x_n$) $=$ $\oplus_{i=1}^n$$x_i$ where each $x_i$ is boolean. Prove that, for every $n \ge 1$, there is no way to compute $F_n$ using only ...
2
votes
2answers
2k views

Example of Left and Right Inverse Functions

I am independently studying abstract algebra and came across left and right inverses. I was hoping for an example by anyone since I am very unconvinced that $f(g(a))=a$ and the same for right ...
2
votes
4answers
1k views

How to find a closed form solution to a recurrence of the following form?

I need to find the closed form solution to the following recurrence -: $ T(n) = 8*T(n/2) + 0.25*n^2$ with $T(1) = 1$ and $n=2^j$ and this is what I have tried so far but just can't seem to get a ...
2
votes
1answer
1k views

Prove that the maximum number of edges in a graph with no even cycles is floor(3(n-1)/2)

The question is in the title. I can see why the bound is sharp (for example, a lot of triangles sharing one common vertex if n is odd, or the same but with one spare edge hanging out if n is even). ...
2
votes
1answer
635 views

Find Total number of ways out of N Number taking K numbers every M interval

I have been stuck in a problem, that has thrown my brain out of the coding. This problem is at very high priority and I need the solution as early as possible. Problem is as : There are exactly N ...
2
votes
1answer
1k views

probability of hand with at least 2 kings

A hand H of 5 cards is chosen randomly from a standard deck of 52. Let E1 be the event that H has at least one King and let E2 be the event that H has at least 2 Kings. What is the conditional ...
2
votes
1answer
464 views

Factoring a number $p^a q^b$ knowing its totient

We are given: $n=p^aq^b$ and $\phi(n)$, where $p,q$ are prime numbers. I have to calculate the $a,b,p,q$, possibly using computer for some calculations, but the method is supposed to be symbolically ...
2
votes
3answers
461 views

Proof by Induction: Solving $1+3+5+\cdots+(2n-1)$

The question asks to verify that each equation is true for every positive integer n. The question is as follows: $$1+ 3 + 5 + \cdots + (2n - 1) = n^2$$ I have solved the base step which is where ...
2
votes
3answers
201 views

recursion need a closed form

Is there a closed formula to the problem $f(1)=1, f(2n)=f(n), f(2n+1)=f(2n)+1$. So far i have found a solution for $n$, which is the number of power of $2$'s needed to add up to the number starting ...
1
vote
1answer
17 views

One hypothesis concerning Hamming distance matrix

Suppose $a_1, a_2, \ldots, a_m$ are different strings of the same length n. And let $V = [v_1, v_2, \ldots, v_n]$ be a matrix such that $V_{i, j}$ is a Hamming distance between $a_i$ and $a_j$. ...
1
vote
1answer
20 views

Proof of cardinalities sets

Prove that the cardinality of set $A^{B+C}$ is equal to the cardinality of $A^{B}\times A^{C}$. I think I need to make functions from $B+C$ to $A$ and one from $B$ to $A$ and one from $A$ to $C$. I ...
1
vote
1answer
75 views

how to solve generating function for odd number?

im working on this question and i don't know where to start the question is: a) Find a closed form for the generating function $c(x)$ for counting compositions with $k$ parts, where each part is ...
1
vote
6answers
121 views

The function $G: x \mapsto 2^{x^2}$ maps $\mathbb{R}$ onto $\{ x \in \mathbb{R} : x \geq 1 \}$

Let $X = \mathbb{R}$ and $Y = \{x \in \mathbb{R} :x ≥ 1\}$, and define $G : X → Y$ by $$G(x) = e^{x^2}.$$ Prove that $G$ is onto. Is this going along the right path and if so how do get the ...
1
vote
1answer
48 views

Is this quantifier is true?

"Every mail message larger than one megabyte will be compressed". Let $M(x) = x$ mail message $L(x) = x$ larger than one megabyte will be compressed $ \forall x \space (M(x) \rightarrow L(x))$
1
vote
1answer
152 views

Prove this equality by using Newton's Binomial Theorem

Let $ n \ge 1 $ be an integer. Use newton's Binomial Theorem to argue that $$36^n -26^n = \sum_{k=1}^{n}\binom{n}{k}10^k\cdot26^{n-k}$$ I do not know how to make the LHS = RHS. I have tried ...
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1answer
44 views

Is Proof by Resolution really needed here?

So I'm doing a problem in the book but this problem (where they ask me to use proof by resolution) seems unnecessary: $p\iff r$ $r$ $\therefore p$ By definition of IFF, this seems true, but they ...
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4answers
73 views

Prove by induction that $n < 2^n$ for all $n \ge 1$

I'm trying to do homework problems and for the most part I've been getting the results. For this one though, I am having some trouble since its $2^n$ and I can't relate it properly: Prove using ...
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vote
2answers
91 views

Finding a formula for $1+\sum_{j=1}^n(j!)\cdot j$ using induction

I need help with finding the formula and proving it by induction. Am stuck, but the professor says we should know this by now.
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vote
2answers
92 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 ...
1
vote
3answers
185 views

Draw a finite state machine which will accept the regular expression $(a^2)^* + (b^3)^*$

Draw a finite state machine which will accept the regular expression: $(a^2)^* + (b^3)^*$ In particular, I am confused by the $+$ sign, what does it exactly mean? Most literature I could find about ...
1
vote
5answers
214 views

Show {$ ax + by | x, y \in \mathbb{Z}$} = {$n$ gcd$(a,b)|n\in \mathbb{Z}$}

I have the following problem: Let $a, b \in\mathbb{Z}$. Show that {$ ax + by | x, y \in \mathbb{Z}$} = {$n$ gcd$(a,b)|n\in \mathbb{Z}$} I understand that the Bezout's lemma says that $gcd(a,b) = ...
1
vote
1answer
44 views

Question over induction, suppose $P(n)$ is true for all positive integers $n$ that is a power of 2.

Suppose, that $P(k+1) \Rightarrow P(k)$ for all positive integers $k$. How would I prove $P(n)$ is true? I am getting confused since this is going the 'other way'. Usually $P(k)\Rightarrow P(k+1)$
1
vote
3answers
84 views

How many solution of a equations?

I have the following question: Let $n$ and $k$ be integers with $n \geq k$. How many solutions are there to the equation $$ x_1 + x_2 + \cdots + x_k = n $$ where $x_1, x_2, x_k$ are integers $\geq ...
1
vote
1answer
113 views

Discrete Math- Four different dice are rolled

Four different dice are rolled. a) In how many outcomes will at least one five appear? b) In how many outcomes will the highest die be a five? I think i figured out the answer for how many outcomes ...
1
vote
1answer
115 views

Different arrangements of the word PHILOSOPHY

I want to figure out the number of different arrangements using all the letters in PHILOSOPHY such that the letters H,I,S,Y always stick together. The way I solved this is given below ; Selecting a H ...
1
vote
1answer
104 views

Probability - Testing for diseases

I am just learning probability in my Discrete Structures class and am very lost. This is the example given in the book and I have no idea how to solve this problem. Problem: Suppose one in 1000 ...
1
vote
2answers
497 views

How to prove a total order has a unique minimal element

Let $R$ be a total order on set $S$. Prove that if $S$ has a minimal element, than the minimum element is unique. I have difficulties with proofs. I know any graph of a total order is a straight ...