# Tagged Questions

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

88 views

The problem that I have is: Prove by induction that 21 divides 4n+1 + 52n-1 So far I have: Base Case: $n = 1$ $4^{1+1} + 5^{2-1} = 4^2 + 5^1 = 16 + 5 = 21$ Inductive Step: Assume: $4^{k+1} + ... 0answers 38 views ### Find the closure of a subset in the sequence space.$\newcommand{\R}{\mathbb{R}}$Problem Let$\R^{\infty}$be the subset of$\R^{\omega}$, which is defined to be the countable infinite cartesian product of$\R$and$\R^{\infty}$consists all ... 1answer 81 views ### Is there some sort of trick to show naturality? This is about natural transformations in category theory. Almost always, I somewhat know why some defined maps or homomorphisms behave naturally, but I am almost never entirely sure (if things get ... 2answers 34 views ### For all sets$A,B$and$C$, if$B \cap C \subseteq A$, then$( A\smallsetminus B) \cap (A\smallsetminus C) = \varnothing$I'm trying to prove that this is true. This is my current process. If$B$AND$C$are subsets of$A$, then everything in$A$has elements of$B$and$C$. So, if we were to remove all of$B$AND all ... 2answers 54 views ### Are these types of proofs always valid? Task: prove that if c|ab and gcd(c,a)=d, then c|db. Is this proof correct? If c|db then ck=db for some integer k. We know that because c|ab, ch=ab, for some integer h, and that ... 0answers 25 views ### Smallest Rational Number Proof [duplicate] Can anybody help me out with solving this mathematical proof? Prove the statement “There is no smallest rational number greater than 2” by contradiction. Contradiction: There is a smallest ... 1answer 18 views ### Dihedral Group Inner Automorphism I have to prove$D_n\cong\textrm{Inn}(D_n)$, for$n\geqslant 3$. I know that$R_{180}$is not in any odd dihedral group and so the centers of the groups are all trivial. I'm just not sure where to ... 2answers 23 views ### The rigorousness of this proof about greatest common divisors. My task is to prove that gcd(n, n+1)=1 for all n>0. It is obvious that 1 is a common divisor of both n and n+1 since $$1|n → 1x=n$$ if x=n, and $$1|n+1 → 1y=n+1$$ if y=n+1. To prove that 1 is the ... 1answer 98 views ### Comments on my proof of the transitive property of subsets? Because I am in an advanced Calc III course that is quite proof-based, my university does not require that I take the "introduction to formal proofs" course before proceeding to higher level math ... 2answers 45 views ### Use induction to show that$\sum_{i=1}^k 2^{i-1} (k-i) = 2^k -k -1, k\ge1$This is an exercise question in Fundamentals of Computer Algorithms by Horowitz and Sahni. The base case for this is trivial. However for the inductive case, we need to verify,$p(n) \implies p(n+1)$... 1answer 64 views ### Learning how to prove How to learn writing proofs? Could you give me some advice and sources, please? I have to learn how to prove, if I want to continue to study science. For now, the fields I am studying are real ... 0answers 16 views ### Prove that ∀d ∈ N − {0, 1} ∃a, b, u, v ∈ Z − {0} (ua + vb = d ∧ gcd(a, b) ≠ d) I have to prove this particular statement:$\forall d \in \mathbb{N}-\{0,1\}\hspace{1em}\exists a,b,u,v \in \mathbb{Z}-\{0\}\hspace{1em}(ua+vb=d~\wedge~gcd(a,b)≠d)$What's the best way to start off? ... 1answer 15 views ### Triangle inequality for specific vector distance function Define the metric d on$\mathbb{R}_n$by d$(\vec{v},\vec{u})$=max{${|v_1−u_1|, |v_2−u_2|,...,|v_n−u_n|}$}. Show that this metric satisfies the property the "triangle inequality property" that ... 0answers 23 views ### Triangle inequality for uniquely defined vector distance function Define the metric d on$\mathbb{R}_n$by d$(\vec{v},\vec{u})$=max{${|v_1−u_1|, |v_2−u_2|,...,|v_n−u_n|}$}. Show that this metric satisfies the property the "triangle inequality property" that ... 1answer 33 views ### proving an iff statement about the definition of a limit using epsilon delta Let$A$and$x_0$be real numbers, and let$f(x)$be a real-valued function defined in a deleted neighborhood of$x_0$. Use the definition of limit to prove that$\lim \limits_{x \to x_0} f(x) = A $... 0answers 58 views ### Using 'let' too many times? When I begin a proof, I need to define certain things. for example the classic: $$\text{Let } \epsilon > 0.$$ However, it seems that there are times where I need to define a lot of things in this ... 1answer 154 views ### Proof that {xor,not} is not functionally complete I am trying to figure out the formal proof that set of {xor, not} is not a functional complete system. Firstly I tried to build it by structural induction and show that any formula created by using ... 1answer 24 views ### If$\displaystyle \lim_{z \to z_o} f(z)=w_o$and$\displaystyle \lim_{w \to w_o} g(w)=L$then$\displaystyle \lim_{z \to z_o} g(f(z))=L$. If$\displaystyle \lim_{z \to z_o} f(z)=w_o$and$\displaystyle \lim_{w \to w_o} g(w)=L$then$\displaystyle \lim_{z \to z_o} g(f(z))=L$.$f$and$g$are complex functions with$R_f \subset D_g$. ... 1answer 31 views ### Understanding a proof$f(\bigcup \mathfrak{C})\subseteq \bigcup f(\mathfrak{C})$I want to show you an example what my book says If$f:X\to Y$,$\mathfrak{C}$is a collection of subsets of$X$, then$f(\bigcup \mathfrak{C})=\bigcup f(\mathfrak{C})$. Proof: (..) Let ... 1answer 34 views ### euclidian algorithm proof I am having difficulty with the following proof: The Euclidean algorithm can be used to express$x := gcd(a, b)$in the form$x = ma + nb$with$m, n \in Z$. Use this fact to prove the ... 2answers 76 views ### Trivialness of Center of Odd Dihedral Group [duplicate] So I'm trying to prove the center of Dn is trivial for odd n greater than or equal to 3. I know that since Dn for n greater than 2 is non-Abelian, the flips and the rotations do not commute in ... 2answers 125 views ### Help to find all different cases need for proof about homomorphism from Z to R I am a bit confused about why my professor approached the following a certain way, and also why it cannot be done differently. The question is to prove that for any ring R we there is a unique ... 1answer 47 views ### Maximum sum of polynomial - choice of coefficient and variables There are two given arrays:$[x_1,...,x_n] $and$[y_1, ..., y_n]$. Our task is to make pair such that: $$\sum_{i=1}^{n} x_iy_j$$ is maximum. I know that we should sort these arrays:$x_1 \le ...
I suspect that $O(n)$ is homeomorphic to a product of spheres $S^m$ (equipped with the product metric) for various $m$ like so: $$O(n) \cong S^{n-1} \times S^{n-2} \times \dots \times S^0$$ I need ...