Tagged Questions

For questions about approaches and techniques for discovering a proof, as opposed to writing it down clearly (which involves (proof-writing)). Should not be used unless the focus is on the technique of the proof instead of the solution.

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0
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0answers
195 views

what does “k” usually mean in mathematics

I am green in the theory of computation and was hoping that somebody could give me a better isight as to what is meant when $k$ is used in a mathematical sentences. Specifically I was trying to ...
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1answer
42 views

big o statement prove or disprove (impossible)

This question is harder than it looks folks for all a in the reals and for all b in the reals, [(a <= b) => (n^a is O(n^b))] n^a is O(n^b) if n^a <= cn^b for some n>= n, (n less than or equal ...
0
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1answer
51 views

Proof for a function $m:2^S\to R$

Let $S$ be a finite, non-empty set and $m:2^S\to R$ a function with the following properties $M1$: $\forall A\in2^S, m(A)\ge0$ $M2$: $\forall A, B\in 2^S, A\cap B=\varnothing\Longrightarrow m(A\cup ...
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1answer
43 views

Borsuk–Ulam theorem for $n=2$

How one can intuitively prove the following statement: At any moment there is always a pair of antipodal points on the Earth's surface with equal temperatures. What about a rigorous proof?
2
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3answers
60 views

Proving $\neg A\vee(A\wedge \neg B)= \neg A \vee \neg B$.

How do I prove using boolean algebra that $\neg A\vee(A\wedge \neg B)= \neg A \vee \neg B$? I can see it in the logic table and it is logical, but I can't prove it mathematically.
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2answers
34 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: ...
3
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2answers
64 views

Prove that $\lim\limits_{x\to\infty} \frac{\Gamma(x+1,x(1+\epsilon))}{x\Gamma(x)}=0$.

I conjecture that for any $\epsilon>0$, we have $$ \lim_{x\to\infty} \frac{\Gamma(x+1,x(1+\epsilon))}{x\Gamma(x)}=0 $$ where $\Gamma(x,a) = \int_a^\infty t^{x-1}e^{-t} \mathrm{d}t$ denotes the ...
1
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1answer
47 views

Condition to separability of a Banach space.

I am trying to prove the following statement: Let X be a Banach space and $X^{*}$ its topological dual space. If there exists a countable family of functions $(f_{n})_{n} \subset X^{*}$ such that ...
0
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1answer
45 views

Induction to prove that something is not true?

This is maybe a very basic question, but I have never seen it done before. Can you use induction to prove that something is not true? In particular if something does not hold in dimension n=1, can I ...
0
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1answer
23 views

Property of GCD in ring

Let D be a domain and $\emptyset \subset A \subseteq D^*$ Show that CD(A)={$d\in D$ | $(A)\subseteq (d)$} I know that I'll need to show both containments to show that the two statements are ...
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2answers
43 views

Abstract Algebra Subgroup Proof Help

Show that if N is a normal subgroup of G and |N| = 2, then N is a subgroup of Z(G). proof: Let N be a normal subgroup of G. Then N is a subgroup of G and g is in G. So gN = Ng for all g in G. Suppose ...
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1answer
46 views

Calculating Variance and Standard Deviation with probability distribution

The age [in years] $X$ of sewing machines to be reconditioned is a random variable with the following probability distribution: $f(x)=(1/972)x(18-x)$ for $0<x<18,$ and $f(x)=0,$ elsewhere. The ...
0
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0answers
52 views

Big-O and Big-Omega Proof

Trying to refresh my memory on proofs, I got a feeling this is true but can't remember how to prove it: If f(n)∈Ω(n) how do I prove f(n)∈O(n) Thanks for help (I do remember definitions of big omega, ...
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2answers
32 views

define $f :R\to R$ by $f(x)=\frac{1}{(x-1)}$ when $x<1$ and $f(x)=\sqrt{(x-1)}$ when $x\geq 1$. Show that $f$ is a bijection and determine its inverse

A bonus Q on a discrete math/proofs test, I know I must prove injectivity and surjectivity, but am not exactly sure how to do so. Please help, this will be covered on the upcoming final exam in April. ...
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1answer
52 views

Let $X \neq \emptyset$, define the relation$A\sim B$ if there exists a bijection $f : A \to B$, Show that $\sim$ is an equivalence relation on $X$.

A question on my last proofs midterm, I know I must prove injectivity and surjectivity, but there aren't really any obvious conditions or descriptions on S that helped me to manipulate it to try and ...
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2answers
103 views

Combinatorics proof of “sum of (k choose m) with k from m up to n is equal to n+1 choose m+1”

I've already proved this statement algrebraically. I'm asked to prove it with combinatorics. So far I came up with, LHS= # ways to choose m apples from a total of m,m+1,...,n RHS= # ways to choose ...
0
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1answer
51 views

GCD Domain Proof

Let $D = \mathbb{R} + X \mathbb{C}[X]$ Show that $\gcd_D(X^2,iX^2)=\emptyset $ Here is my plan so far... (and my questions) Suppose $f \in \gcd_D(X^2,iX^2) $. How do I show that because X is ...
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2answers
208 views

$\epsilon - N$ definition of a limit of sequence problem

i have a question i cannot seem to solve! i would really appreciate help if possible. please explain how to solve this question from textbook, i really want to learn but i cant $$\lim \limits_{n \to ...
2
votes
2answers
56 views

Irreducibililty of $X^6+X^3+1$ in $\mathbb{Q}[X] \ $

Could anyone advise me on how to prove $X^6+X^3+1$ is irreducible in $\mathbb{Q}[X] \ ?$ I'm thinking of substituting $X=Y+1$ into the equation, do some tedious computations to simplify and use ...
0
votes
0answers
22 views

Operations in a polynomial ring over $\mathbb{F}_5$

Let $f(x)=3x^2 + 4x + 2$ and let $g(x) = 2x + 3$. Perform the following operations in $\mathbb{Z}/5\mathbb{Z}[x]$. (a) $f(x) + g(x)$ (b) $f(x)g(x)$ (c) divide $f(x)$ by $g(x)$. What is the ...
0
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1answer
64 views

Homework excercise, completeness in Vector-spaces, is it correct?, long, but can it be simplified?

I have a very difficult excercise. I see now that it became too much text for someone to might go through it, if you can please help me, but don't want to read all, can you please then only answer my ...
0
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3answers
21 views

Modular Equivalence

Prove that if a and b are integers such that a|b and b > 0, then (x mod b) mod a = x mod a for any x. Solution: As a|b, we have b = pa for some integer p. Let x mod b = r, then we have x = bq + r = ...
1
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1answer
53 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
97 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 ...
1
vote
2answers
441 views

why area under curve or riemann sum equals to definite integral

i do get that Riemann sums is sum of infinite triangles with with infinitely small length. But definite integral is completely different you are taking anti derivative of f(x) at b and subtract anti ...
1
vote
0answers
49 views

expected value with integration

For the exponential distribution, $f(x)=(1/\theta) e^{-x/\theta}$ for $x>0,$ and $f(x)=0$ for $x \leq0$ $(i)$ Determine the exact value for the probability $P(0<X<3\theta).$ I need help ...
13
votes
5answers
1k views

Why is one proof for Cauchy-Schwarz inequality easy, but directly it is hard?

Let's say you are in $\mathbb{R}^n$ and you define the norm as $||x||=\sqrt{x_1^2+x_2^2...+x_n^2}$. This we recognize as the usual norm from the inner product: $||x|| = \sqrt{\langle x, x \rangle}$, ...
1
vote
1answer
63 views

Open Subsets of open sets

How does one go about proving/disproving that given $(X,d)$ a metric space that a subset $S$ is open. Given the following definitions: A set $X$ is open $\iff \forall x \in X, x\in int(X)$ i.e. x ...
2
votes
1answer
35 views

Continuous map of a compact set

Claim: If $f:X \to Y$ is continuous, where $X$ is compact, and $Y$ is Hausdorff, then $f$ is a closed map. Proof: Take $A \subset X$ to be closed in $X$. Now as $X$ is compact and by choice of $A$ we ...
0
votes
2answers
20 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
2answers
41 views

Algebraic proof with matrices

I need to proof the following: Given $A$ is a $n\times n$ matrix so that $A^2 - 3A + I = 0$ Prove that $A^{-1} = 3I - A$ So I laid out a matrix: $$ A =\begin{pmatrix} a & b \\ c & d ...
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0answers
32 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
239 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....
0
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1answer
35 views

Binomial coefficient proof2

Having difficulty with starting off this proof. Let n be a positive whole number. Prove that $$n\dbinom{2n}{n}=(n+1)\dbinom{2n}{n-1}$$. Any help would be greatly appreciated.
0
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2answers
86 views

Irreducible elements and Associates

Show that, in a domain, every associate of an atom is an atom. An atom is the same thing as an irreducible element. I think these two facts will be important to prove this statement: A nonunit ...
0
votes
2answers
35 views

Proof that $t^ne^{-t}\leq Ce^{-t/2}$ for all $n\geq 1$ and $t\geq 0$

How do I prove that $t^ne^{-t}\leq Ce^{-t/2}$ for all $n\geq 1$ and $t\geq 0$. I am not sure which type of proof to use, for example induction with two variables. The graphs suggest C can always be ...
-1
votes
2answers
71 views

Equivalence Classes

The scenario is a follows: I am given a set $X$ along a map $d$ defined as $d:X \times X \to \Bbb R^+$ for all $x,y,z \in X$ with the following properties: $d(x,y) =0 \Leftarrow x =y \\ d(x,y) ...
0
votes
1answer
43 views

$X^4-5X^2+X+1$ is irreducible in $\mathbb{Q}[X]$

Could anyone advise me on how to efficiently prove $X^4-5X^2+X+1$ is irreducible in $\mathbb{Q}[X] \ ?$ Hints will suffice. Thank you.
1
vote
1answer
63 views

Proof by induction: $2^n > n$

Base is $2^1 > 1$. Now we assume $2^n > n$ and try to obtain $2^{n+1} > (n+1)$. If I can use $2^n > 1$, I could just add that to $2^n > n$ and get $2^{n+1} > (n+1)$ but I don't ...
1
vote
2answers
69 views

Having Trouble Forming Mathematical Proof

I'm having trouble forming a mathematical proof for a question. I can write down thousands of examples with various values of n that shows it's correct, but I'm not ...
1
vote
1answer
35 views

Roots of polynomial in $R[X],$ where $R$ is $\text{UFD}.$

Let $R$ be a $\text{UFD},$ with field of fractions $F$ and let $f(X)=a_0+a_1X+...+a_nX^n \in R[X]$ such that $a_n \neq 0.$ Let $x\in F$ be a root of $f(X).$ Could anyone advise me on how to show ...
0
votes
1answer
102 views

A finite set of wffs has an independent equivalent subset

This seems to be a common exercise among many books (cf. Enderton, p. 28, van Dalen, p. 45, Hinman, p. 51, Chang and Keisler, p. 18), with some minor variations among them. The idea is simple. Say ...
0
votes
2answers
18 views

$A^c$ and $B^c$ are independent

I am trying to prove that, $A^c$ and $B^c$ are independent. My approach: $P(A^c \cap B^c)=P(A \cap B) - P(A \cap B)=P(A \cap B) \times (1-P(A \cap B) = P(A)P(B) \times ...
0
votes
3answers
78 views

epsilon delta to prove $\lim_{x \rightarrow a} \frac{1}{f(x)}$

i was solving problems on my textbook.... and i became stuck. The question is: Let $a\in (- \infty , \infty ).$ Suppose $\lim_{x \rightarrow a} f(x)=L \neq 0$. Use the $\epsilon - \delta$ arguement ...
0
votes
2answers
61 views

Infinimum of a set

Given the following conditions: $x \in \Bbb R$ and $y\in (0,1)$ I was asked to prove that inf $ |x-y |=0 $ My Attempt: By the elementary properties of the modulus function , we know that $ 0 ...
1
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1answer
45 views

How to show whether a statement is true or false(Example question inside)?

So I'm reading How to Read and Do Proofs by Solow and I'm on the exercises now. So far it has been good but I'm stuck on how to answer a question. There are no answers for even numbered questions in ...
0
votes
0answers
39 views

What's the best approach to self-learn from How to read and do proofs (D.Solow)

I got this book and I'm still on page 4. However, I've enjoyed every bit of what has been written so far and everything just makes perfect sense. However I'm not sure how to proceed with further ...
1
vote
0answers
88 views

Proof about lognormal distribution

I'm trying to prove a result about the lognormal distribution that seems to me to be fairly intuitive, but I can't get the proof to work. Basically, I'd like to prove that as the mean increases, the ...
0
votes
1answer
34 views

Isomorphism of direct product of semigroups

I would appreciate some help with the following problem. Consider four semigroups $A,B,C,D$. I was able to prove that $A\cong C\wedge B\cong D$ implies $A\times B\cong C\times D$. But does also ...
0
votes
2answers
25 views

negation of powersets

If given two power sets P(A) and P(B), and told that the Union of these two sets was a subset of another powerset P(C), what would be the negation of this statement? Would the Union go to an ...