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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-2
votes
1answer
35 views

Let $R$ be a field. What are the $R$-submodules of $R \times R \ ?$

We know that $R$-submodule of $R$ are left ideals of $R.$ Is it also true that $R$-submodule of $R \times R$ are left ideals of $R \times R\ ?$ Please advise on the correct approach to this ...
1
vote
1answer
23 views

Question regarding Monoalphabetic Phi Test

I've been asked to prove the following system of inequalities; $$1 \ge \phi(T) \ge \frac{n-k}{k(n-1)}$$ Where $\phi(T) = \sum_{i=1}^{k} \frac{n_i (n_i -1)}{n(n-1)}$, $T =$ some text, $n = $ length ...
1
vote
1answer
102 views

No integer solutions for $x^5 - 3y^5 = 2008$

I have a tutorial question(not homework), that asks to prove that there exists no $x,y \in \mathbb{Z}$, solution for: $$x^5 - 3y^5 = 2008$$ I originally thought I would solve it by taking all cases ...
2
votes
4answers
93 views

A question about metrizability

In a lecture in Topology I had earlier this week, I was told (without proof) that not every topological space $(X,O)$ is metrizable, i.e, it is impossible to find some metric $d$ such that $O$ and ...
0
votes
2answers
53 views

Properties of $R/I$

Let $R$ be an integral domain, and let $a$ be an irreducible element of $R$. Let $I$ be the ideal of $R$ generated by $a$. 1.If $R$ is a principal ideal domain, $R/I$ is a field ? True. Since $a$ ...
2
votes
2answers
72 views

How to prove this trig identity?

If $A+B+C=\pi$ then prove:$$\sin^2A+\sin^2B+\sin^2C=2-2\cos A\cos B\cos C$$ I am completely lost on this, please help.
0
votes
1answer
57 views

Is the property of Euclidean domain inherited via surjective ring homomorphism? [duplicate]

Let $f:R \to S$ be surjective ring homomorphism and $R,S$ be integral domains. Could anyone advise me on how to prove/disprove this statement: If $R$ is Euclidean domain, then $S$ is Euclidean domain. ...
1
vote
1answer
72 views

Ascending chain condition and ring homomorphism

Let $f : R \to S$ be a surjective ring homomorphism between two integral domains. Could anyone advise me on how to prove/disprove the following statements: If $R$ satisfies the ascending chain ...
3
votes
0answers
79 views

Proof for the form of characteristic polynomial

I'd like to proof: The caracteristic polynomial of $A \in M(n\times n, K)$ has the form: $P_A(\lambda) = (-1)^n \lambda^n + (-1)^{n-1} \operatorname{tr}(A)\lambda^{n-1} +\dots +\det(A)$ My proof ...
2
votes
2answers
36 views

Let $R$ be integral domain and $r \not | a.$ If $r$ is prime and $r^k|ab,$ then $r^k|b ?$

Let $R$ be an integral domain and $a,b,r \in R.$ Let $r$ be prime. Suppose there exists positive integer $k$ such that $r^k$ divides $ab$ and $r$ does not divide $a.$ Could anyone advise me on ...
1
vote
5answers
69 views

If one of the hypotheses holds, then one of the conclusions holds. (looking for a proof)

Using a huge truth table, I proved the theorem below. I cannot find a more elegant proof. I tried to rewrite expressions; e.g. using the distributive laws and the laws of absorption - to no avail. Is ...
0
votes
1answer
20 views

Are these Cartesian Product equalities true?

Let $A, B, C, D$ be sets. (i): (A x B) $\cup$ (C x D) = (A $\cup$ C) x (B $\cup$ D) (ii): (A x B) $\cap$ (C x D) = (A $\cap$ C) x (B $\cap$ D) I have to prove these later for homework if true, and ...
0
votes
4answers
142 views

Elementary proofs of inequalities

I was just introduced into elementary proofs of inequalities, my text's explanation however feels incomplete. I did further research on the subject, my question is thus: Prove: If $0 < a < b$, ...
0
votes
1answer
34 views

If f(n)∈Ω(n) how do I prove or disprove f(n)∈O(n)

If f(n)∈Ω(n) how do I prove f(n)∈O(n) I feel it is true, but not sure how to show it the way I see it c1*n =< f(n) =< c2*n holds, but so confused on how to show it
0
votes
0answers
36 views

GCD and LCM Property

Let D = $\mathbb R + X\mathbb C[X]$ Show that $GCD(X, iX) = \mathbb R^\times$ and $LCM(X, iX) = \emptyset$ I have an outline of what to do but don't exactly know who to show all of it... First, ...
0
votes
0answers
13 views

Domain GCD Property

Let D be a domain and $\emptyset \subset A \subseteq D^*$ $d \in GCD(A)$ if and only if (d) is a minimum among the principal ideals containing (A) If $d \in GCD(A)$ then d|a for all $a \in A$ and ...
2
votes
1answer
124 views

Some burning questions on First-order logic from an amateur

I'm currently taking an introductory course in Mathematical logic(prerequisites is only advanced calculus) and my lecture notes are based on Enderton's book 'Mathematical Introduction to Logic' ...
0
votes
0answers
165 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 ...
1
vote
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
votes
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 ...
0
votes
1answer
42 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
votes
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.
0
votes
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
votes
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
vote
1answer
44 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
votes
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
votes
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 ...
0
votes
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 ...
0
votes
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
votes
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, ...
0
votes
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. ...
0
votes
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 ...
0
votes
2answers
98 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
votes
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 ...
1
vote
2answers
194 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
votes
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
votes
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
vote
1answer
52 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
436 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
62 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 ...
0
votes
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
221 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....