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

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1answer
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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 “...
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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 ...
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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) ...
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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 $i=0,\ldots,l$...
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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^...
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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 $(...
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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 ...
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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) + ...
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0answers
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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 ...
1
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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(...
1
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2answers
23 views

Let $S = { r \in \mathbb{Q} : r \lt 2}$. Prove that $S$ does not have a largest element.

Let $S$ = $[{ r \in \mathbb{Q} : r \lt 2}]$. Prove that $S$ does not have a largest element. My method: Assume to the contrary that $S$ does have a largest element, where $S$ = $[{r \in \mathbb{Q} :...
0
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2answers
15 views

Show that $\frac{\alpha+y}{\alpha+n+\beta}\in (\frac{\alpha}{\alpha+\beta};\frac{y}{n})$

Suppose you assign a $Beta(\alpha,\beta)$ prior distribution for $\theta$, and the you observed $y$ heads out of $n$ spins. Show algebraically that your posterior mean of $\theta$ always lies ...
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1answer
27 views

Proof for statement: It's impossible to find a pair of consecutive natural numbers whom digit sums would divide without reminder by 3

I am searching for a mathematical proof of this statement: It's impossible to find a pair of consecutive natural numbers whom digit sums would divide without reminder by 3. I have tried: To make a ...
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1answer
29 views

Prove $O(f(n)+g(n)) = O(f(n))$ when $g(n)=O(f(n))$

Given $g(n) = O(f (n))$, how can I prove that the following expression is true: $O(f (n) + g(n)) = O(f (n)) \tag1$ So I just write down what it says: $g(n) = O(f (n)) <=> f(n) \le c_1 g(n)...
2
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2answers
34 views

Can$A \cap (B' \cap C')$ be $(A \cap B') \cap (A \cap C')$?

If I use the above statement, provided that it is right, in a question, would I have to prove it as well?
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1answer
36 views

Proof formalization help: Given a vector $u$ in $\mathbb{R}^3$ and a compact 2 dimensional manifold $m$, $u$ is normal to the $m$ at 2 points.

Proof formalization help: Given a vector $u$ of Euclidean length $1$ in $\mathbb{R}^3$ and a compact 2 dimensional manifold $m$, $u$ is normal to the $m$ at at least 2 points. I've thought about the ...
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2answers
29 views

How to show that countable union of $F_\sigma$ is $F_\sigma$

On https://www.physicsforums.com/threads/countable-intersection-of-f-sigma-sets.666055/ Is it claimed that it is obvious that countable union of $F_\sigma$ is $F_\sigma$ Can someone elaborate why ...
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7answers
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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 ...
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1answer
73 views

What kind of proof is this?

Let's say that we want to prove that object A is blue. Is the following reasoning true? First assume that $A$ is indeed blue. Then, use other axioms to show that depending on a control parameter $p$,...
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0answers
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$\max\{c^Tx:Ax\le b,x\ge 0\}=+\infty$ iif it exists $j\in\{1,…,n\}$ such that $\max \{x_j:Ax\le b, x\ge 0\}=+\infty$

Show a vector $\vec c$ exists such that $\max\{c^Tx:Ax\le b,x\ge 0\}=+\infty$ if and only if it exists $j\in\{1,...,n\}$ such that $\max \{x_j:Ax\le b, x\ge 0\}=+\infty$ I'm only asking for a hint ...
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0answers
28 views

Proving a basis and dimension

Hi I'm currently looking at proofs in linear algebra and came across this one and I'm compketely baffled Suppose $ C_{ij} $ is the $2\times3$ matrix with $1$ in the $ i,j^{th} $ entry and zero ...
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3answers
27 views

Proving a basis in linear algebra

So at the moment I'm trying to go through proofs and I came across this one: Suppose $ P_n $ is the vector space of all polynomials with degree less than or equal to n. Prove that $ \{1, x − 1, ...
2
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1answer
41 views

Show that for propositional logic $\vdash_i \neg \varphi \Leftrightarrow \vdash_c \neg \varphi$.

As the title says, where $\vdash_i$ is derivations in Intuitionistic logic and $\vdash_c$ is derivations in Classical logic. I am allowed to use a corollary that states that $\vdash_i \varphi \...
0
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1answer
26 views

How to use monotone convergence theorem to show that $\int \sum |f_n| = \sum \int |f_n|$

In https://math.la.asu.edu/~quigg/teach/courses/473/2009/lectures/11integral.pdf, pg 4 The monotone convergence theorem was used to state $$\int \sum |f_n| = \sum \int |f_n|$$ without proof. Can ...
2
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1answer
25 views

How to show that all sequentially compact spaces are bounded?

I want to show given a sequentially compact subset $A \subseteq M \implies A$ is bounded. I read this Every sequentially compact set is closed and bounded. but the proof is poorly written and jumpy ...
0
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1answer
29 views

Clean proof for showing $f^{-1}([a,\infty))$ measurable implies $f^{-1}([a,b))$ measurable

I wish to show that for $f:\mathbb{R} \to \mathbb{R}$, $f^{-1}([a,\infty))$ measurable implies $f^{-1}([a,b))$ measurable Looks fairly easy if $f^{-1}([a,b))$ is one piece. Suppose $f^{-1}([a,b)...
0
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2answers
44 views

How to prove that the square matrix $A_{n}$ matrix is nilpotent such that $A^{(n-1)}=0$

The matrix A looks like this: $$A=\begin{bmatrix} 0 & 1 & 0 & 0 & .&.&. &0\\ 0 & 0 & 2 & 0 & .&.&. &0\\ 0 & 0 & 0 & 3 &...
0
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1answer
29 views

Verifying multiplicative inverses of modulo n are the elements that are relatively prime to n

A proposition in my book states: $(\mathbb{Z}/n\mathbb{Z})^{\times} = \{a \in \mathbb{Z}/n\mathbb{Z}~|(a,n) = 1\}$ which I want to prove. I start by defining $a$ in terms of prime factors $$a = p_1^{\...
2
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3answers
45 views

Proof help: Prove that $x^2+y^2+z^2 \geq xy+xz+yz$ [duplicate]

$x^2+y^2+z^2 \geq xy+xz+yz $ for all real numbers, x, y, and z. I'm not very good with working inequality proofs. Can someone help me prove this? The technique doesn't really matter.
0
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0answers
18 views

Existence of convergent sequence in a closed set?

Is it true that if $C$ is a closed set, and $x \in {C}$ then there exist a sequence $\{x_i\}$ which tends to $x$? I'm not sure about the correctness of this statement, and whether it is true for ...
0
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3answers
51 views

Prove that graph with odd number of odd degree vertices does not exist

I need to prove that it is impossible to have a graph in which there are an odd number of odd degree vertices. What is the easiest way to formally prove this? I feel that I can prove it just by ...
0
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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 ...
2
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2answers
29 views

Proof outer measure satisfies monotonicity: $A \subseteq B \implies m^*(A) \leq m^*(B)$

Theorem: $$A \subseteq B \implies m^*(A) \leq m^*(B)$$ Proof Attempt: By definition, $m^*(B) = \inf\{\sum\limits_{k=1}^\infty |J_k||\{J_k\} \text{ is a cover of B }\}$, $m^*(A) = \inf\{\sum\limits_{...
0
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2answers
30 views

Moments of a continuous random variable (Exercise 4.3.3 from Grimmett and Stirzaker)

Let $X$ be a non-negative random variable with density function $f$. Show that $$ E(X^r) = \int_0^\infty r x^{r-1} P(X > r)\,dx. $$ I tried using integration by parts to obtain \begin{align} \...
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0answers
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Suppose G and G' are isomorphic and G has m vertices of degree k. Prove that G' has m vertices of degree k

Suppose G and G' are isomorphic and G has m vertices of degree k. Prove that G' has m vertices of degree k. I don't know how to start this one
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1answer
27 views

Need help understanding algebra steps taken in proof of why an even minus an odd is odd

I don't understand the algebra used in the below example proof from my textbook. Where does the + 1 come from? Is it okay to just add 1 anywhere you want? Or is there some rule here or reason you ...
5
votes
1answer
90 views

Is my proof correct? If $f$ has a finite number of discontinuities on $[a, b]$, then it is integrable on $[a, b]$

Question: Suppose a function $f(x)$ over the interval $[a, b]$ is bounded and has only a finite number of discontinuous points on $[a, b]$. I intend to prove that it must be integrable on $[a, b]$. ...
0
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1answer
27 views

Proof of Interceting Lines

I have this practice problem from a final exam study guide. Let $f,g$ be continuous on $[a,b]$ and $f(a)>g(a)$ but $g(b)>f(b)$. Prove that $\exists c \in [a,b]$ such that $f(c)=g(c)$. My ...
2
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0answers
35 views

Simple Vacuous Proof, Correct Approach?

I am doing some practice exercises as I am starting out on proofs but I noticed that though I am getting the correct approach between vacuous and trivial proofs, I am not doing it in the same format ...
1
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1answer
16 views

How does one consider what a graph looks like in a mathematical proof

Mostly I am wondering for example what it would be like to prove that a linear graph (negative slope) shifted right would look the same as one shifted up. Can you consider how a graph looks when ...
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0answers
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When is it appropriate to write “Then it follows”

I am reading a proof, and before the proof fully finishes, the author writes "Then it follows [the statement we are trying to prove] is true" I have been spending the last three hours justifying the ...
1
vote
1answer
29 views

How to verify this relationship between area under the graph and the preimage?

Define $h : \mathbb{R} \to [0, \infty)$, Let $H = \{(x,y)| 0 \leq y \leq h(x)\}$ be the area under the graph (including the boundary) I wish to show the following is true: $$H = \bigcup_{c&...
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0answers
40 views

Why is this proof that a circular cone is not a surface not rigorous?

In example $4.1.5$, page $73$ of Pressley's Elementary Differential Geometry, a "heuristic" argument is given to prove that the circular cone with vertex the origin and angle $\pi/4$, is not a surface....
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0answers
63 views

I did not understand one thing in the proof of substitution lemma?

The substitution lemma in lambda-calculus is proved by the following way, but I just did not understand the application of induction hypothesis in it. The lemma as shown below, where x and y are ...
0
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1answer
15 views

Prove that multiplication by an integer $a$ that is relatively prime to $n$ defines a bijection from $\mathbb{Z}_n-\{0\}$ to itself

If gcd$(a,n)=1$, then multiplication by $a$ defines a bijection from $\mathbb{Z}_n-\{0\}$ to itself. My working: If $n=p$ a prime, then we can use the Fermat's Little Theorem. If $n$ is not prime in ...
0
votes
1answer
12 views

Show that monotonicity implies positive definiteness of the Jacobian

Given $f: \mathbb{R}^n \to \mathbb{R}^n$, $f$ differentiable, $x,y, p \in \mathbb{R}^n$, show that $(x-y)^T(f(x) - f(y)) \geq 0 \Leftrightarrow p^TDf(x)p \geq 0, \forall p \in \mathbb{R}^n$ This ...
0
votes
3answers
43 views

How to prove that $\sum_{k=0}^{n-1}\cos\left(\frac{2\pi k}{n}+\phi\right)=0$ [duplicate]

Prove that $\displaystyle\sum_{k=0}^{n-1}\cos\left(\frac{2\pi k}{n}+\phi\right)=0$ for $n\in\mathbb{N},n>1$ I'm thinking at a demonstration by induction, as base case $n=2$ $$\sum_{k=0}^{n-1}\cos\...
0
votes
1answer
25 views

Proof on showing the integral of f(x)=0

I am having difficulties showing that if $f$ is continuous such that $f(x)<0$ for every $x \in (a,b)$ then $\int_a^bf(x)\;dx <0$ I am given the theorem that if $f$ is continuous such that $f(x)&...
2
votes
2answers
22 views

Why Is This Squared Modulo Prime Number Equation True?

I recently was trying to figure out if there was an simple way to tell how many unique outcomes can be produced from the following equation: $k^2 \mod m$ where $m$ is some odd prime number and $k$ ...