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

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0
votes
2answers
647 views

Using the Pumping Lemma to Prove $L = \{a^ib^jc^k \mid i < j < k\}$ is not Context-Free

I want to use the Pumping Lemma to prove that $$L = \{a^ib^jc^k \mid i < j < k\}$$ is not context-free. I think I have the intuition, but I don't know how to prove it. Help?
1
vote
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
votes
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
votes
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
votes
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 $a$. ...
18
votes
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
votes
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
vote
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 $=1$...
0
votes
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 ...
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 ...
2
votes
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?
1
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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 ...
1
vote
2answers
45 views

Probability Proof about A and B

I have to formally prove that: $$P(A) = P(A\wedge \neg B) + P(A\wedge B)$$ so I did like this: $$P(A\wedge \neg B) + P(A\wedge B)$$ $$=P(A\wedge \neg B) + P(A)\cdot P(B)$$ $$=P(A)\cdot P(\neg B) + ...
1
vote
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 ...
0
votes
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
votes
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\...
3
votes
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}\...
3
votes
4answers
332 views

Uniqueness proof for $\forall A\in\mathcal{P}(U)\exists!B\in\mathcal{P}(U)(C\setminus A=C\cap B)$

I managed to prove existence for the following theorem: $$\forall A\in\mathcal{P}(U)\ \exists!B\in\mathcal{P}(U)\ \forall C\in\mathcal{P}(U)\ (C\setminus A=C\cap B)$$ where U is any set. My ...
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 ...
0
votes
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 $E$ ...
0
votes
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
votes
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 $n$ ...
1
vote
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 ...
1
vote
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} + ...
-1
votes
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 $\...
1
vote
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
votes
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 \...
3
votes
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\...
1
vote
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 ...
-2
votes
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 ...
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 ...
1
vote
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) = ...
0
votes
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
4answers
137 views

Proving $\lfloor 2x \rfloor = \lfloor x \rfloor + \lfloor x+0.5\rfloor$

I can intuitively see that this is true, but I'm having a very hard time proving it. I'm actually not even quite sure where to begin. I tried using the inequalities that define the floor function, and ...
2
votes
1answer
331 views

Group Order and Least Common Multiple

Let $G_1,G_2,...,G_n$ be groups. Show that the order of an element $(a_1,a_2,...,a_n)$ $\in$ $G_1 \times G_2 \times\cdots\times G_n$ is lcm($o(a_1),...,o(a_n))$. I know I need to use the fact that ...
0
votes
3answers
39 views

Showing proof using contrapositive

Tell if the statement is true or false. If true provide a proof. $\forall x$ $\in R$ $\left(\forall M > 1 \left(x \ge 1-\frac{1}{M}\right) \to x \ge 1 \right)$ I believe this statement is True. ...
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 “...
2
votes
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 ...
23
votes
7answers
2k views

Can you use both sides of an equation to prove equality?

For example: $\color{red}{\text{Show that}}$$$\color{red}{\frac{4\cos(2x)}{1+\cos(2x)}=4-2\sec^2(x)}$$ In high school my maths teacher told me To prove ...
2
votes
2answers
56 views

Prove that L is a sub-language of the CFG G by using induction. (CFG,Induction,School)

i am asking for help with a question from a course in Logic im reading at university. I am aware that this type of question is frequently asked here(i have looked at alot of other questions/answers) ...
2
votes
3answers
103 views

Prove that there is no largest irrational number

I have to prove that there is no largest irrational number from the result of the a previous proof: Prove that if $x$ is rational and $y$ is irrational then, $x+y$ is irrational. I was able to prove ...
0
votes
1answer
21 views

How to show if two polynomials are equal for all substituted real numbers, then all the coefficients are equal

Let $p(x)=c_0+c_1x+\ldots+c_lx^l$ and $q(x)=d_0+d_1x+\ldots+d_mx^m$ be polynomials with real coefficients. Suppose $\forall x\in\mathbb{R}$, $p(x)=q(x)$. Show that $l=m$ and that for all $i=0,\ldots,l$...
0
votes
1answer
24 views

Proof for $\forall x \in R^+, x^2 + y^2 + z^2 \geq xy + xz + yz$

I am doing a question to practice doing proofs with real numbers as I am still not so good at it. I ran into some problems for the following question where it is as given: $\forall x \in R^+, x^2 + y^...
4
votes
1answer
48 views

Would this be a valid proof for $\left| x + y \right| \geq \left| x \right| - \left| y \right|$

I wanted to check if this was a valid proof for considering whether $\left|x + y \right| \geq \left| x \right| - \left| y \right|$. My proof is as follows: Case 1: Assume $x > 0, y>0$,then $(...
0
votes
0answers
18 views

Would this be considered a valid proof for $\forall r \in R$ if $0 < r < 1 $, then $\frac{1}{r(1-r)}\geq 4$

I did a proof of the following $\forall r \in R$ if $0 < r < 1 $, then $\frac{1}{r(1-r)}\geq 4$ using a proof by contra-positive, which was different from the direct proof that the solutions ...
2
votes
0answers
235 views

Theoretical proof of convergence of sequential weight update procedure (Neural Networks and Machine Learning)

My question is at the bottom. (Most of the descriptive words come from Chris. Bishop's Neural Networks for Pattern Recognition) Let $w$ be the weight vector of the neural network and $E$ the error ...
1
vote
0answers
27 views

Uniqueness of sum and multiplication of numbers

so I was writing a program that took two strings and said if they were anagrams or not, and I had this idea of making each character into a number, adding them all and checking if the result was the ...
0
votes
2answers
52 views

NEUTRAL GEOMETRY PROOF. prove that a figure can have at most one center of symmetry

A center of symmetry for a figure F is a point O such that every line through it cuts F in two points, P and P', such that O is the midpoint of PP'. Prove that a figure can have at most one center of ...
1
vote
1answer
15 views

Prove that if $f : A \rightarrow B$ is a function, $D \subseteq A$, and $E \subseteq A$ then $f(D) - f(E) \subseteq f(D - E)$.

Prove that if $f : A \rightarrow B$ is a function, $D \subseteq A$, and $E \subseteq A$ then $f(D) - f(E) \subseteq f(D - E)$. My method: Let $y \in f(D) - f(E)$. Hence $y \in f(D)$ and $y \notin f(...
0
votes
2answers
37 views

Prove that if A and B are sets such that $A \cup B \neq \emptyset$, then $A \neq \emptyset$ or $B \neq \emptyset$

Prove that if A and B are sets such that $A \cup B \neq \emptyset$, then $A \neq \emptyset$ or $B \neq \emptyset$. It was suggested to me that the easiest way to approach this was with a proof by ...