# Tagged Questions

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

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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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### 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 ...
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### 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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### 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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### 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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### 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 ...
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) + ... 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 ... 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(... 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} :... 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 ... 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 ... 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)... 2answers 34 views ### CanA \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? 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 ... 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 ... 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)}$$In high school my maths teacher told me To prove ... 1answer 73 views ### What kind of proof is this? Let's say that we want to prove that object A is blue. Is the following reasoning true? First assume that A is indeed blue. Then, use other axioms to show that depending on a control parameter p,... 0answers 19 views ### \max\{c^Tx:Ax\le b,x\ge 0\}=+\infty iif it exists j\in\{1,…,n\} such that \max \{x_j:Ax\le b, x\ge 0\}=+\infty Show a vector \vec c exists such that \max\{c^Tx:Ax\le b,x\ge 0\}=+\infty if and only if it exists j\in\{1,...,n\} such that \max \{x_j:Ax\le b, x\ge 0\}=+\infty I'm only asking for a hint ... 0answers 28 views ### Proving a basis and dimension Hi I'm currently looking at proofs in linear algebra and came across this one and I'm compketely baffled Suppose C_{ij} is the 2\times3 matrix with 1 in the i,j^{th} entry and zero ... 3answers 27 views ### Proving a basis in linear algebra So at the moment I'm trying to go through proofs and I came across this one: Suppose P_n is the vector space of all polynomials with degree less than or equal to n. Prove that \{1, x − 1, ... 1answer 41 views ### Show that for propositional logic \vdash_i \neg \varphi \Leftrightarrow \vdash_c \neg \varphi. As the title says, where \vdash_i is derivations in Intuitionistic logic and \vdash_c is derivations in Classical logic. I am allowed to use a corollary that states that \vdash_i \varphi \... 1answer 26 views ### How to use monotone convergence theorem to show that \int \sum |f_n| = \sum \int |f_n| In https://math.la.asu.edu/~quigg/teach/courses/473/2009/lectures/11integral.pdf, pg 4 The monotone convergence theorem was used to state$$\int \sum |f_n| = \sum \int |f_n|$$without proof. Can ... 1answer 25 views ### How to show that all sequentially compact spaces are bounded? I want to show given a sequentially compact subset A \subseteq M \implies A is bounded. I read this Every sequentially compact set is closed and bounded. but the proof is poorly written and jumpy ... 1answer 29 views ### Clean proof for showing f^{-1}([a,\infty)) measurable implies f^{-1}([a,b)) measurable I wish to show that for f:\mathbb{R} \to \mathbb{R}, f^{-1}([a,\infty)) measurable implies f^{-1}([a,b)) measurable Looks fairly easy if f^{-1}([a,b)) is one piece. Suppose f^{-1}([a,b)... 2answers 44 views ### How to prove that the square matrix A_{n} matrix is nilpotent such that A^{(n-1)}=0 The matrix A looks like this:$$A=\begin{bmatrix} 0 & 1 & 0 & 0 & .&.&. &0\\ 0 & 0 & 2 & 0 & .&.&. &0\\ 0 & 0 & 0 & 3 &... 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$ain terms of prime factors $$a = p_1^{\... 3answers 45 views ### Proof help: Prove that x^2+y^2+z^2 \geq xy+xz+yz [duplicate] x^2+y^2+z^2 \geq xy+xz+yz for all real numbers, x, y, and z. I'm not very good with working inequality proofs. Can someone help me prove this? The technique doesn't really matter. 0answers 18 views ### Existence of convergent sequence in a closed set? Is it true that if C is a closed set, and x \in {C} then there exist a sequence \{x_i\} which tends to x? I'm not sure about the correctness of this statement, and whether it is true for ... 3answers 51 views ### Prove that graph with odd number of odd degree vertices does not exist I need to prove that it is impossible to have a graph in which there are an odd number of odd degree vertices. What is the easiest way to formally prove this? I feel that I can prove it just by ... 2answers 37 views ### Prove that if A and B are sets such that A \cup B \neq \emptyset, then A \neq \emptyset or B \neq \emptyset Prove that if A and B are sets such that A \cup B \neq \emptyset, then A \neq \emptyset or B \neq \emptyset. It was suggested to me that the easiest way to approach this was with a proof by ... 2answers 29 views ### Proof outer measure satisfies monotonicity: A \subseteq B \implies m^*(A) \leq m^*(B) Theorem:$$A \subseteq B \implies m^*(A) \leq m^*(B)$$Proof Attempt: By definition, m^*(B) = \inf\{\sum\limits_{k=1}^\infty |J_k||\{J_k\} \text{ is a cover of B }\}, m^*(A) = \inf\{\sum\limits_{... 2answers 30 views ### Moments of a continuous random variable (Exercise 4.3.3 from Grimmett and Stirzaker) Let X be a non-negative random variable with density function f. Show that$$ E(X^r) = \int_0^\infty r x^{r-1} P(X > r)\,dx. I tried using integration by parts to obtain \begin{align} \... 0answers 17 views ### 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 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 ... 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]. ... 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 ... 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 ... 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 ... 0answers 46 views ### 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 ... 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&... 0answers 40 views ### Why is this proof that a circular cone is not a surface not rigorous? In example4.1.5$, 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.... 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 ... 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 ... 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 ... 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\... 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)&...
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 $k$ ...