For questions about the formulation of a proof, not about the mathematics behind it.

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1answer
33 views

What is a useful starting idea to think about this simple graph problem?

I would like to prove the statement that there are $2^{\binom{n-1}{2}}$ simple graphs are there with vertex set $\{1,\ldots,n\}$ such that every vertex has even degree. The thing that confuses me is ...
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1answer
20 views

Writing a proof for $f(W) \setminus f(X) \subseteq f(W\setminus X)$

I am trying to write a proof to prove/disprove the following question: Will it always be true that $f(W\setminus X) = f(W)\setminus f(X)$? I know to prove this you need to show both ways since ...
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1answer
18 views

$\mathcal B_{\mathbb Q}$ = = { [p, q) ⊆ R : p, q ∈ Q, p < q } is not a bases for the Lower Limit Topology

I'm having a bit of trouble proving this: The definition of Lower Limit Topology I am working with: $ \{[a, b) \subseteq \mathbb R \ \text s.t \ a < b\}$. The only thing I can think of is that ...
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1answer
15 views

Is this proof about the null space and column space correct?

My question asks me to show that if $A$ and $B$ are $n\times n$ matrices, and $AB=0$, then the column space of $B$ must be a subspace of the nullspace of $A$. My attempt at a proof is like this: we ...
0
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1answer
39 views

Show that $\bigcap_{r>0}B(a,r)=\bigcap_{n=1}^\infty B(a,\frac{1}{n})=\{a\}$

Let $(M,d)$ be a metric space. Show that $\bigcap_{r>0}B(a,r)=\bigcap_{n=1}^\infty B(a,\frac{1}{n})=\{a\}$ where $B(a,r)$ is a ball with center in $a$ and radius $r$. My attempt: Set $0<r\leq ...
3
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3answers
78 views

There is no uncountable collection of pairwise disjoint open sets in $\mathbb R$

Working in $R_{\text usual}$ Topology: Show that there is no uncountable collection of pairwise disjoint open subsets of $\mathbb R$. Definition of $R_{\text usual}$ I'm working with: $\{U \subseteq \...
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1answer
48 views

Not sure how to prove this statement by contradiction?

There is this a simple looking and intuitive statement but I am not sure how to start approaching this problem. Let $S=\{s_1,s_2,\ldots,s_n\}$, where $s_1,s_2,\ldots,s_n>0$ such that $s_1+s_2+\...
0
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1answer
41 views

How to prove the group of roots of unity in $\mathbb{C}$ is a group

I mostly need help with proving $G$ is closed but a verification of the other parts is appreciated. Let $G = \{z \in \mathbb{C} \mid z^n=1$ for some $n\in \mathbb{Z^+}\}$ I want to start by proving $...
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2answers
64 views

How to prove $f^{-1}(f(X)) = X$

Suppose $X \subseteq A$. Will it always be true that $f^{-1}(f(X)) = X$? I am try to prove this problem with either proofs or counterexamples. I have found a counterexample for $f^{-1}(f(X)) \...
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1answer
35 views

Prove (or check) the expression is positive given constraints on variables?

The following proof problem have taken me a few days. Perhaps it is too hard for me to overcome it. Can you help me? The expression is by the following: \begin{equation} \begin{split} &2\,x{c}^{x-...
1
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1answer
59 views

When to use $\in$ and $\subseteq$ when talking about bases and topologies

Can someone demonstrate a concrete example of when to use $\in$ and $\subseteq$ when talking about topologies and bases? When is something $\subset$ of a basis or a topology and when is something $\...
0
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1answer
16 views

Under what assumptions on φ is Tco-φ a topology

Fix a set X, and let φ be a property that subsets A of X can have. Define Tco-φ = {U ⊆ X : A = ∅, or X \ U has φ } . Under what assumptions on φ is Tco-φ a topology on X? What I think: 1. X\X has φ ...
3
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1answer
53 views

Find the mistake of the following inductive proof: all algorithms have the same time complexity

I came across this problem: Find the mistake of the following inductive proof: Theorem: all algorithms have the same time complexity. Proof: (By induction on the number of algorithms.) The ...
1
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2answers
25 views

Prove $A\setminus(B \cup C) = (A \setminus B) \cap (A \setminus C)$ using element chasing?

How can I prove $A \setminus(B \cup C) = (A \setminus B) \cap (A \setminus C)$ using element chasing? I need to verify that it is correct and show the steps of element chasing.
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0answers
77 views

Prove that there exists at least one root of $g$ between any two roots of $f$ [duplicate]

Given that $$f(x)= 1 - e^x\sin(x)$$ $$g(x)= 1 + e^x\cos(x)$$ Using Rolle's theorem, prove that there exists at least one root of $g$ between any two roots of $f$. Attempt so far: $f'(x) = -e^x(\...
5
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4answers
86 views

An antonym for “converse”

Suppose you are proving $p \leftrightarrow q$. In your first paragraph you prove $p \rightarrow q$. Your second paragraph begins, “For the converse, assume $q$ holds.” In this situation, we have a ...
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1answer
42 views

The image of an injective function whose domain is a topological space also a topology

Let $(X, T )$ be a topological space, and let $f : X → Y$ be an injective (but not necessarily surjective) function. QUESTIONS. (1) Is $T_f := \{ f(U) : U ∈ T \}$ necessarily a topology on $Y$ ? ...
2
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3answers
92 views

Using $\epsilon-\delta$ proof to prove continuity

Use an $\epsilon-\delta$ proof to show that $f : R \setminus \left \{ \frac{-3}{2} \right \} \rightarrow R$ , $$f(x) = \frac{3x^2-2x-5}{2x+3}$$ is continuous at $x = -1$ Hello there. Can anyone ...
0
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1answer
18 views

Proof using mean value theorem

Prove using the mean value theorem that $e^{x+1}\geq 2e^x$ by considering the interval $[x,x+1]$. Using the definition, there exists a $c$ in $(x,x+1)$ such that $e^{x+1} - e^x = e^c$ (this is of ...
2
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1answer
59 views

how to prove $G$ is an abelian group under $*$ (called the real numbers mod 1)

Let $G = \{x \in \mathbb{R}~|~0\leq x < 1\}$ and for $x,y \in G$ let $x*y$ be the fractional part of $x+y$ i.e $x*y = x + y - [x + y]$ where $[a]$ is the greatest integer less than or equal to $a$. ...
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7answers
71 views

I'm having trouble understanding why inductive proofs are logical [duplicate]

I am new to Mathematics, reading books in my free time. I have recently learned about proving Mathematical propositions by induction. I am having a bit of trouble understanding the process and why it ...
1
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3answers
122 views

Is there a purely algebraic proof to show that $-1\leq\sin x\leq1?$

I have to prove the boundedness of $\sin x$ (strict inequality) ie. $-1\leq\sin x\leq1$. I know a geometric proof using trianglesbut I am not too satisfied with it as it does not prove that $=1$...
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4answers
2k views

Are professional mathematicians concerned with formalizing infinitely many dependent choices?

I've noticed certain arguments in analysis textbooks which rely on the principle of being able to pick elements infinitely many times. For example, an argument might go "Pick $x_1\in S$ such that $P(...
0
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1answer
28 views

How to prove associative law for groups

I'm having trouble figuring out the proof to the proposition: for any $a_1,a_2,\ldots,a_n \in \mathbb{G}$ the value of $a_1~R~a_2~R~a_3~R\cdots R~a_n$ is independent of how the expression is bracketed ...
0
votes
2answers
158 views

Can some inequalities help to pin down an unique solution in a linear system of equations with infinite solutions?

I need to discuss the number of solutions of the following system of equations. Any help would be very appreciated. Consider the known parameters $a_1,...,a_4;d_1,d_2,d_3$ such that $0< a_i< ...
2
votes
1answer
544 views

Show $\nabla^2g=-f$ almost everywhere

let a continuous function $f(x,y,z)$ be absolutely continuous over every bounded region and let it be in $L^1$ that is $\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} |{f(x',...
2
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1answer
41 views

Are there other methods of proof other than contrapositive, induction, contradiction, construction, and counter example?

I have only heard of a few methods of proof, namely, contrapositive, induction, contradiction, construction, and counter example. Are there other types of proofs?
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0answers
39 views

Proving function is convex argmin

How can I show that the following function is convex in which Z is a random variable? $$\rho(Z)=\frac{2}{3}argmin_{t}\{t+10\mathbf{E}[Z-t]_{+}\}+\frac{1}{3}argmin_{t}\{t+5\mathbf{E}[Z-t]_{+}\}$$ I ...
5
votes
3answers
81 views

Proof that the Period of $\sin(x)$ is $2\pi$.

As I was walking through campus today, I had an interesting question pop into my head: How can we prove that the period of $\tan(x)$ is $\pi$ rather than $2\pi$? The answer to this was extremely ...
0
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1answer
28 views

Vector subspace and linear application proof

I would like to know if my proof is correct and, moreover, if it is well written. Let $E$ and $F$ be vector subspaces and $f: E \to F$ an application. Proof that, if $U$ is a vector subspace of $E$...
3
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2answers
56 views

Prove that $f(x)=0$ for all $x\in\mathbb{R}$.

The question is: Prove that if $f:\mathbb{R}\to\mathbb{R}$ is continuous and $f(x) = 0$ whenever $x$ is rational, then $f(x) = 0$ for all $x\in\mathbb{R}$. My proof: Let $ x\in\mathbb{R} $. If $ x\...
0
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0answers
25 views

Vector spaces and bases proof verification

I would like to know if my proof is correct and, moreover, if it is well formulated. Let $B = \{b_1 , \dots,b_n \}$ be a base of a vector space $E$. Prove that every vector of $E$ ...
3
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1answer
33 views

Show $P(\frac{1}{n}\sum_{i=1}^{n}Y_i\geq c)\leq e^{-nd}$ for constants $c$ and $d$

Let $Y_1, Y_2\ldots$ be a sequence of i.id. random variables uniformly distributed on $[0,1]$. Let $c>\frac{1}{2}$. Show that there exists $d>0$ (depends on $c$) such that $$P\left(\frac{1}{n}\...
1
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1answer
22 views

Need help proving the standard topology is a topology

Define $$\tau_s = \{ U \subseteq X| \forall x \in U, \exists \delta > 0 \text{ s.t. } B_\delta(x) \subseteq U\}$$ Show $\tau_s$ is a topology on $\mathbb{R}^n$ Can someone check if my proof ...
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2answers
55 views

regarding odd perfect numbers [closed]

$$3-1=2$$ Let $n$ be a perfect number. Subtract each proper divisor from greatest to least. Example: $n=28$ 28-14=14. 14-7=7. 7-4=3. 3-2=1. 1-1=0 With an even perfect number, we can go from $n$ ...
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0answers
26 views

Feedback of proof that $\lim \inf A_n \subseteq \lim \sup A_n$

I just wrote down the proof of the following easy proposition, and I was wondering about both the content (I would like to know if it is error-free), and the form of it. Proposition: $\lim \inf ...
0
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0answers
15 views

Acceptable notation? $\lesssim_{n} n^{-\beta}$ for constants NOT depending on $n$

I am preparing a paper and found it convenient to write things like $$ |\text{Expression of a lot of variables}|\lesssim_{n} n^{-\beta} $$ when an inequality is true up to a multiplicative factor that ...
1
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1answer
38 views

Equivalence between $(a_n)$ being Cauchy and the hypothesis that $|a_{n+2} - 2a_{n-1} + a_n|\to0$

(a) Prove that $\forall \epsilon >0, \exists N \in \mathbb{N}$ so that $|a_{n+2} - 2a_{n-1} + a_n|<\epsilon$ (b) Is the converse true? Why or why not? So for (a) I have $|a_{n+2} - 2a_{n-1} + ...
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2answers
39 views

Show that $(a,b)$ is open in the usual topology of $\mathbb{R}$ and $[a,b)$ is not

I am little bit stumped by this proof. Show that $(a,b), a < b$ and $ a, b \in \mathbb{R}$ is open in the usual topology of $\mathbb{R}$ and $[a,b)$ is not Recall that the usual topology on $\...
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0answers
22 views

Show that an event has strictly positive probability

Consider the random variables $W_i,W_j, X_i, X_j$ with $X_i\sim X_j$, $X_i\perp X_j$ and $W_i\sim W_j, W_i\perp W_j$, where $\sim$ denotes equal probability distribution and $\perp$ denotes ...
4
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3answers
61 views

In proving A = B, A, B are sets, do you always have to show $\subseteq$ and $\supseteq$?

I am trying to show the DeMorgan's Law $X \backslash \bigcup_{\alpha \in I} A_\alpha = \bigcap_{\alpha \in I} (X \backslash A_\alpha)$ It seems I could directly approach this as follows: $X \...
1
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1answer
29 views

Proof for: Let A and B be sets s.t $ A \cap B = A $ iff $ A \subseteq B $

I am practicing some proofs involving sets and I would like to see if what I did was a valid proof because it seemed to be different from the one provided in the textbook I am using given that it did ...
3
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3answers
51 views

Can three vectors have dot product less than $0$?

Can three vectors in the $xy$ plane have $uv<0$ and $vw<0$ and $uw<0$? If we take $u=(1,0)$ and $v=(-1,2)$ and $w=(-1,-2)$ $$uv=1\times(-1)=-1$$ $$uw=1\times(-1)+0\times(-2)=-1$$ $$vw=-1\...
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1answer
59 views

Proof assistance for abstract algebra [duplicate]

Suppose that $F$ is a subfield of a field $E$ and let $a \in E$. Define a map $\theta :F[x] \rightarrow E$ by $\theta(f(x)) = f(a)$ for $f(x) \in F[x]$. (1) Prove that $\theta$ is a ring homomorphism ...
1
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1answer
23 views

Let $A$ be a finite set and $\mathcal R$ an equivalence relation in A. Prove that exists a function …

Let $A$ be a finite set and $\mathcal R$ an equivalence relation in A. Prove that exists a function $f: A \to \Bbb N$ such that $\forall a, b \in A$ the following is true: $$a \mathcal R b \iff f(a) = ...
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1answer
18 views

Proof Writing: Given 2 two dimensional manifolds $\emptyset \neq N \subseteq M \subseteq \mathbb{R}^3$, $N$ is compact, $M$ is pconnected: $N = M$

Statement: Given 2 two-dimensional manifolds $\emptyset \neq N \subseteq M \subseteq \mathbb{R}^3 $, if $N$ is compact and $M$ is path-connected, then $N = M$. Proof: We know that there is at least ...
3
votes
2answers
63 views

Let $G$ be a group of order 1210 with a subgroup $H$ of order 121. Show that every element of order 11 is in $H$

Let $G$ be a group of order 1210 with a subgroup $H$ of order 121. Show that if $a\in G$ has order 11, then $a\in H$ This was a question on a test I just took and even though I spent almost all of ...
0
votes
1answer
25 views

Square root of a matrix as it relates to the identity

Prove that for any $2×2$ matrix $M$ which is “sufficiently close” to the identity matrix, there exists a matrix A such that $A^2 = M$, and that this matrix A is unique if $A$ isrequired to be “...
20
votes
3answers
2k views

How to prove that a very large number is not prime

I'm solving few math problems for an upcoming math contest . I am stuck with a short problem, where I have to prove that $A$ is not prime . $$A = 100\ 000\ 000\ 000\ 000\ 000\ 001$$ $A$ is not a ...
2
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1answer
46 views

Is there a book that teaches proofs from simple to intermediate level?

I am looking for a book that teaches proofs and the book has many exercises from very simple to more difficult? I have noticed with most math books, they seem to leave out pieces too soon before the ...