7
votes
4answers
163 views

Let $a_n$ be the $nth$ term of the sequence $1, 2, 2, 3, 3, 3, 4, 4, 4, 4, \dots$

Question:Let $a_n$ be the $nth$ term of the sequence $1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6\dots$, constructed by including the integer $k$ exactly $k$ times. Show that $a_n = ...
1
vote
2answers
26 views

If $R$ is a transitive realation, then $R\circ R\subseteq R$

Here's the question I'm struggling with: Let R be a transitive relation on a set A. Prove the R composed with R is a subset of R. I'm kind of lost on how to prove this. I've started with saying: ...
2
votes
1answer
23 views

Minimum score for winner and maximum score for loser in a round-robin tournament.

I have just correctly solved this programming problem. The problem is the following: $N$ teams play a round-robin tournament, i.e. each pair of teams plays exactly one game and the winner gets 3 ...
3
votes
2answers
69 views

Prove $A = (A \setminus B) \cup (A \cap B)$

Prove $A = (A \setminus B) \cup (A \cap B)$ Logically, this is clearly true. I can explain why: start with $A$, remove all elements in $B$ and then add in any elements in both $A$ and $B$, which ...
0
votes
0answers
40 views

How do we prove that, if $\mathcal{P}(A) \sim \mathcal{P}(B)$, then $A \sim B$? [duplicate]

The converse--if $\ A \sim B$ then $ \mathcal{P}(A) \sim \mathcal{P}(B)$--is very easy to prove. I can't see an immediate, simple proof for the converse case. It seems like a potentially good strategy ...
1
vote
0answers
15 views

Existence of a particular transformation

I've a set of data points $S = \{ x | x\in [0,1]\}$ (i.e. real values from the unit interval). In some cases I've big clusters in the data and I want to spread the values in between the unit interval ...
2
votes
2answers
53 views

Prove that n!+1 contains a prime factor greater than n and use this to prove that there are infinte many primes [duplicate]

Prove that $n!+1$ contains a prime factor greater than $n$ and use this to prove that there are infinitely many primes. I said assume that $n!+1$ contains a prime $p$ which is less than or equal to ...
0
votes
1answer
35 views

Disjunctive Normal Form (DNF) of a boolean combination

Upon revisiting chapter 1 of Robert S. Wolf's "A tour though mathematical logic" I sumbled upon the following Proposition on page 13 : Suppose that $P$ is a Boolean combination of ...
1
vote
1answer
26 views

Show that if the diameter of an undirected graph is $d$ then there exists some vertex separator $S\subseteq V$ of size $|S| \leq { n\over d-1} $

Show that if the diameter of an undirected graph is $d$ then there is some set $S\subseteq V$ with $|S| \leq \frac{n}{d-1} $ such that removing the vertices in S from the graph would break it into ...
2
votes
1answer
61 views

Help Needed Showing that $\chi(\overline{G \times H}) \leq \chi(\overline{G}) \times \chi(\overline{H})$

Where $\chi(G)$ denotes the chromatic number, $\overline{G}$ the graph complement, and $\times$ the Cartesian Graph Product: I need to show that $(\forall G,H)( \chi(\overline{G \times H}) \leq ...
2
votes
1answer
17 views

Need help proving that $ fRg \Leftrightarrow fg = f $ on $ B^{n} $ to $ B $ if and only if $ f(b_1,…,b_n) \leq g(b_1,…,b_n) $

I'm trying to gather my thoughts for proving the following claim: For $ fRg \Leftrightarrow fg = f$ on $B^{n}$ to $B$, show that $ fRg $ if and only if for any input values $ b_1,...,b_n $, we ...
1
vote
1answer
42 views

Proving breath first traversal on graphs [duplicate]

I am trying to proof the following algorithm to see if a there exists a path from u to v in a graph G = (V,E). ...
0
votes
1answer
54 views

Proofing a Reachable Node Algorithm for Graphs

I am trying to proof the following algorithm to see if a there exists a path from u to v in a graph G = (V,E). ...
1
vote
1answer
18 views

proving tautology for $¬((∀x.(Q(x)∧P(x)))∧∃y.¬P(y))$

I need to prove that the following is a tautology. Not sure even where to start. Can I prove it by deduction? $$¬\Big(\big(∀x.Q(x)∧P(x)\big)∧∃y.¬P(y)\Big)$$ Thank you!
0
votes
2answers
24 views

equivalence properties of $\equiv \pmod n$ proofs

Prove the identity: $$a \equiv b \pmod n \wedge a\equiv c \pmod n\implies b\equiv c \pmod n$$ I need to prove this property of $\equiv \pmod{n}$ along with a few others can someone link me to a ...
0
votes
2answers
95 views

Prove $x$ and $y$ in $y = x^2 + 2$ are prime only for $x = 3$ and $y = 11$?

Let $x$ be a positive integer and $y = x^2 + 2$. Can $x$ and $y$ be both prime? The answer is yes, since for $x = 3$ we get $y = 11$, and both numbers are prime. Prove that this is the only value of x ...
0
votes
1answer
19 views

Proving claims about sequences by induction?

I am learning how to prove claims about finite sequences right now. Can you help me prove or disprove the following claim? ...
2
votes
1answer
89 views

Can you draw the e-NFA from the following definition?

I am trying to understand the solution, because I think I got it completely wrong. I wrote we could take the initial DFA and replace the normal transitions with epsilon transition except for all ...
1
vote
2answers
42 views

one to one positive integers and positive rationals

How would you go about proving that there is a 1 : 1 correspondence between the set of positive integers and the set of positive rationals. I know there are a lot of ways to do this but I am looking ...
0
votes
2answers
34 views

Proving that $S_k = \{A \subset \mathbb{N} : |A| = k\}$ for $k\in\mathbb{N}$ is denumerable. [duplicate]

I am having trouble with this problem for quite some time. I posted this question before but I still can not figure out this problem. So far,from the suggestion of user134824, I have tried to define ...
5
votes
0answers
55 views

How can I better solve proofs requiring the introduction of algebraic assumptions?

Today I decided to binge on discrete mathematics after a three year hiatus. I tackled three proofs, and all of them required the introduction of assumptions that seemed to not be found in the givens ...
1
vote
1answer
43 views

commutative ring and unity elements proof

So this is a review problem in our book I came across and i really want to understand it but I am just not having any luck, I did some research and found a guide on solving it but that's not really ...
2
votes
1answer
85 views

Prove $\gcd(ka,kb) = k*\gcd(a,b)$

For all $k > 0,\ k\in \Bbb Z$ . Prove $$\gcd(k*a,\ k*b) = k *\gcd(a,\ b)$$ I think I understand what this wants but I can't figure out how to set up a formal proof. These are the guidelines we ...
0
votes
1answer
33 views

proof with divisibility

this is the original question prove: $\forall c \in Z, a\neq 0 $and b both $ \in Z$ $a|b \iff c\cdot a|c\cdot b$ Then he corrected himself by saying for problem 1: to show that ca | cb implies a | b ...
1
vote
2answers
136 views

Every planar graph has a vertex of degree at most 5.

I am trying to prove the following statement, any help!? Prove that every planar graph has a vertex of degree at most 5.
-4
votes
3answers
73 views

Graph Theory - Proof - Isomorphism [closed]

If anyone can help me prove the following: Suppose that $G$ is a plane graph which is isomorphic to its dual. Prove that $G$ has $2n-2$ edges. I thank you for your time!
1
vote
2answers
74 views

Graph Theory - Proof

I am need help to Prove the following statement: Let G be a $k$-regular graph with $n$ vertices and $k \geq 1$. Prove that $G$ does not have an independent set of size greater than $\dfrac{n}{2}$. ...
1
vote
2answers
86 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 ...
0
votes
2answers
37 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 ...
0
votes
0answers
42 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 ...
1
vote
1answer
38 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 ...
1
vote
1answer
28 views

Help with composite identity functions in discrete mathematics

I am having trouble with the following problem: For nonempty sets $A$ and $B$ and functions $f : A \rightarrow B$ and $g: B \rightarrow A$ suppose that $g \circ f = i_A$, the identity function on ...
1
vote
1answer
43 views

Help with identity functions in discrete mathematics

I have trouble with trying to solve the following problem: For nonempty sets $A$ and $B$ and functions $f:A\rightarrow B$ and $g:B\rightarrow A$ suppose that $g\circ f=i_A$, the identity function on ...
-1
votes
3answers
59 views

Proving functions are injective and surjective

I am having trouble with the following problem: For nonempty sets $A$ and $B$ and functions $f:A \rightarrow B$ and $g:B \rightarrow A$ suppose that $g\circ f=i_A$, the identity function of $A$. ...
0
votes
1answer
20 views

Proving Integer Modulo is Well-Defined

I have trouble figuring out this problem: $h: Z_4 \rightarrow Z_6$ by $h([a])=[3a]$ for each $a\in Z$. Prove that h is well-defined thus it is a function and that h is neither injective nor ...
1
vote
2answers
36 views

$K$ events that are $(K-1)$-wise Independent but not Mutually/Fully Independent

I had the following question: Construct a probability space $(\Omega,P)$ and $k$ events, each with probability $\frac12$, that are $(k-1)$-wise, but not fully independent. Make the sample space as ...
1
vote
1answer
55 views

Proving a Bound for Oddtown-Eventown or Clubtown

Suppose we have a town with a set of residents $V$, where $|V| = n$. The residents like forming clubs, and we have clubs $C_1,C_2,\ldots,C_m \subseteq V$. We are interested in the maximum number of ...
0
votes
3answers
22 views

Prove that a function is a bijection?

I am having trouble with this problem: Prove that the function $f(x)=x^2-2x+3$ with domain $x\in(-\infty, 0)$, is a bijection from $(-\infty, 0)$ to its range. Work: Basically, I try to use the ...
1
vote
2answers
25 views

Help with Discrete Math Functions and Bijections

I have trouble with the following problem: Prove that the function $f(x)=x^2-2x+3$, with domain $x\in (-\infty, 0)$, is a bijection from $(-\infty, 0)$ to its range. Work: I tried to first prove ...
0
votes
0answers
14 views

Help with Integer Modulo Proof

I am stuck on this problem for a while and need some help: Prove that for any prime $p$, if $[a]*[b]=[0]$, does it follow that $[a]=[0]$ or $[b]=[0]$? Work: I do not know where to start. I was ...
1
vote
4answers
103 views

proof by induction: sum of binomial coefficients

Question: Prove that the sum of the binomial coefficients for the nth power of $(x + y)$ is $2^n$. i.e. the sum of the numbers in the $(n + 1)^{st}$ row of Pascal’s Triangle is $2^n$ i.e. prove ...
1
vote
1answer
37 views

Proving modulo equation with x-power

I'm trying to prove following equation: $$ (g^{y} \mod n)^x \mod = g^{xy} \mod n $$ I tried many multiple approaches, all of them failed, and there is waaay too much of them to write them here, so I ...
0
votes
2answers
31 views

Proof by Induction $4^n \geq 16n^2$

Prove that for an integer $n \geq 4$, $4^n \geq 16n^2$ Base Case: For $n = 4$, $4^4 \geq 16(4)^2$ $256 = 256$ Induction Hypothesis: Suppose this statement hold up to $4^k > 16k^2$ Then: ...
1
vote
1answer
42 views

How would I solve this mathematical induction proof? I am stuck after the first part of the induction.

$$1 + 5 + 5^2 + \ldots + 5^n = \frac{5^{n+1}-1}{4}$$ Basis case $n= 0$: $1^0 = 1 \;\;\;\;\;\;\;\;\;\;\;\; \frac{5^{1+1}-1}{4}=1$ Assume true for $n=k$: $$1 + 5 + 5^2 + \ldots + 5^k = ...
2
votes
4answers
49 views

Help with discrete math proof?

I am having trouble proving the following: If $x\in R$ and $x > 0$, then $x^4+1 \geq x^3+x$. Work: I tried to rearrange the equation as $x^4-x^3-x+1 \geq 1$, but that does not really help. I ...
0
votes
2answers
18 views

Show that for any $s \in S$ where $S = $ {$s \in \mathbb{N} | \exists m,n \in \mathbb{Z}$, we have $d|s$ where $d$ is the smallest element

Explain why the set $S = $ {$s \in \mathbb{N} | \exists m,n \in \mathbb{Z}, s = ma+ nb $} has a smallest element, call it d, so we know there exists $x,y \in \mathbb{Z}$ such that $d = ax + by$, and ...
0
votes
0answers
31 views

Prove that $s \not= \emptyset $ by showing that at least one of $|a|$ or $|b|$ is an element of $S$.

. I'm trying to show an alternative proof to Bezout's Lemma (let $a, b \in \mathbb{Z}$, then there exists $x,y \in \mathbb{Z}$ such that $gcd(a, b) = ax + by$). Heres one of the steps in proving it: ...
-1
votes
2answers
101 views

Quotient-Remainder Theorem Proving [closed]

This theorem is obviously correct. Now I try to prove it by well-ordering principle. But I don't know where to start the proving....
3
votes
1answer
68 views

Proving a statement about prime numbers

Let $p_1,p_2,p_3,\cdots$ be all the primes sorted in an increasing order. Is $p_1p_2p_3\cdots p_i + 1$ is always prime? Why? How can I prove that?
1
vote
0answers
212 views

Can I use Strong Induction to prove graph theory? - Hamilton Path/Tournament Graphs

I am working on the below proof. I am still new to proves so I am wondering if you guys can answer the following questions: 1) Did I use the inductive hypothesis correctly here? (By the inductive ...