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

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4
votes
2answers
88 views

Prove by induction that $21 | 4^{n+1} + 5^{2n-1}$

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} + ...
0
votes
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 ...
8
votes
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 ...
0
votes
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 ...
0
votes
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 ...
1
vote
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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
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 ...
2
votes
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, $p(n) \implies p(n+1)$ ...
1
vote
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 ...
1
vote
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? ...
0
votes
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 ...
0
votes
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 ...
2
votes
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 $ ...
1
vote
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 ...
1
vote
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 ...
1
vote
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 $R_f \subset D_g$. ...
1
vote
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 ...
1
vote
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 ...
1
vote
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 ...
4
votes
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 ...
0
votes
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 ...
1
vote
1answer
44 views

SO(n) homeomorphic to cartesian product of spheres

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 ...
0
votes
1answer
27 views

Prove that function is *not* injective $g : \mathbb{Z} \to \mu_{102}$

I have this math problem, that I can't seem to figure out. Let $\zeta = e^{\frac{2 \pi i}{102}}.$ Define $g : \mathbb{Z} \to \mu_{102}$ with the formula $g(n)= \zeta^{3n}$ for $ n \in ...
1
vote
1answer
77 views

Derive the dual function $g(\lambda, \nu)$ for the least-norm problem

I am trying to find the dual function $g(\lambda, \nu)$ to this problem $$\min\limits_{Ax = b} \|x\|$$ Step 1. Form the Lagrangian $$L(x, \lambda, \nu) = \|x\| + \nu^T(Ax-b) = \|x\| + \nu^TAx - ...
1
vote
1answer
36 views

Is the following operation legal within the $\sup$?

Suppose I have two vectors $z, x$ of equal size, and $|x| \leq 1$ I know the following is true: $$\sum_iz_ix_i \leq |\sum_i z_ix_i| \leq \sum_i|z_ix_i|$$ is it legal to write the following: ...
1
vote
0answers
19 views

Proof of Antisymmetry [duplicate]

I am stuck with proving antisymmetry under the natural numbers. Prove "For each x,y E N, if x <= y and y <= x, then x=y. I am not really sure where to start with this proof. Im assuming I will ...
1
vote
1answer
60 views

Proof of “if $x<y$, then $xz<yz$”

I need help proving "For each $x,y,z$ in $\mathbb{N}$, if $x<y$, then $xz<yz$." This must be done using Peano's axioms and the definitions of addition, multiplication, and ordering. I have ...
3
votes
1answer
47 views

Proof of existing degree $n$ binomial

Let $P(x)$ be a polynomial with real coefficients such that $P(x) > 0$ for all $x \ge 0$. Prove that there exists a positive integer $n$ such that $(x + 1)^n P(x)$ is a polynomial with ...
0
votes
4answers
106 views

Why can't a triangular matrix with only zeros in its diagonal be invertible?

Why can't a triangular matrix with only zeros in its diagonal be invertible? I know that it is not invertible but I don't know well the reasons, perhaps. Actually, I read that can't have any zero in ...
0
votes
2answers
73 views

Show that for any natural number $n>24$ there exist natural numbers $p$ and $q$ such that $ n=5p+7q$

Show that for any natural number n>24 we have : $n=5p+7q$ such that $p$ and $q$ are natural. I tried using induction 1) for $n=24$ we have $n=(7 \cdot 2)+(5 \cdot 2)$ 2) we suppose that ...
-1
votes
1answer
43 views

Prove by contrapostivie; how do you not a set?

The question is Proof by contrapositive: show that (A - B) ∩ (B - A) = ∅. So the contrapositive is ¬(∅) = ¬((A - B) ∩ (B - A)). What I've got is, ¬(∅) = (A - B) ∪ (B - A). If that is correct, ...
2
votes
2answers
43 views

Proving a function is continuous with respect to a metric

Definition: Let $f:(M,d)\rightarrow(N,d')$ be a mapping between metric spaces. Then $f$ is continuous at $x\in M$ if for all $\epsilon > 0 $ there exists $\delta$ such that $y\in M$, ...
1
vote
1answer
46 views

Help on Structural Induction?

Don't really understand how to use structural induction The question is: Use structural induction to prove that if (x,y)∈S then x+y is multiple of 7. Let S be the set of ordered pairs of integers ...
2
votes
2answers
73 views

How to write down proof that if $\lim_{x\to \infty}f(x)=\alpha$ then $\lim_{x\to \infty}f'(x)=0$?

Let $a, \alpha \in \Bbb{R}$; let $f: (a,+\infty)\to \mathbb{R}$ be differentiable; let $\lim_{x\to \infty}f(x)=\alpha$; let $\beta := \lim_{x\to \infty}f'(x)$. I want to show that $\beta = 0$. Now, ...
0
votes
0answers
36 views

Writing proofs properly.

I have a problem. I am finding it hard to express my ideas in concise written form. That makes me think something is not clear in my head and my understanding is vague. I present a question here and I ...
3
votes
0answers
23 views

Looking for references for learning the words and sentences used in proofs

I'm familiar with textbooks on logic, proof techniques, and sets. But I have yet to encounter a textbook that dives into the language used w/ definitions and sentence structure used in proofs, for ...
1
vote
1answer
27 views

How can I prove this statement?

I wanted to prove this using induction, but since the RHS is a sum, I can't use the assumption. Therefore I'm stuck in the middle. Is there maybe a way proving that without induction? If induction ...
3
votes
2answers
41 views

How can I prove this inequality for $n\geq 2$?

How can I prove this inequality for all natural numbers $n\geq 2$? ${2n\choose n}>\frac{4^n}{n+1}$ I've tried induction but that was a dead end.
1
vote
0answers
33 views

equivalence relations example

determine if the relation on $\mathbb{Z} \times \mathbb{Z}$ is an equivalence relation. $$R=\{((a,b),(c,d)) \in A \times A: a+3b=c+3d\}$$ What I have thus far I need to show that R is reflexive, ...
0
votes
0answers
21 views

Is every Complex Square Matrix similar to its transpose? [duplicate]

I am aware that every complex square matrix is similar to its transpose but I am having a hard time proving this. Should I try to use the previously asked question listed at $A matrix is similar to ...
1
vote
1answer
41 views

Can I mix direct proof with inductive proof?

Let's say I want to prove with induction that $3|n$ implies $3|n^2$ Let $n = 3k$. The statement is true for $k=1$ since $3|3$ and $3|9$ We assume the statement is true for $k=z$ so $3|3z$ ...
1
vote
1answer
33 views

An approximation for the Lambert W-function

Proposition Let $f(x) = k^{x}x$, where the values of both $f(x)$ and $k$ are known. Let $x_{0} = f(x)$, and: $$x_{n + 1} = \frac{1}{2}\log_{k}{\left(\frac{k^{x_{n}}x_{0}}{x_{n}}\right)}$$ ...
1
vote
2answers
63 views

Prove that the set [a,b] is not well ordered. where a,b are real numbers.

My Proof: Assume towards contradiction that [a,b] is well ordered. (a,b) is a subset of [a,b]. Thus (a,b) has a least element. Let's call this element m. We know that: $m>a \\ m-a>0 \\ ...
0
votes
1answer
28 views

Let A be an n × n real diagonalizable matrix. Show that A + αIn is also real diagonalizable.

Let A be an n × n real diagonalizable matrix. Show that A + αIn is also real diagonalizable. I am having trouble figuring out where to start. I know that if I show that A + αIn has n distinct real ...
1
vote
3answers
44 views

Prove or disprove that $(a_n)_{n=1}^{\infty}$ is Cauchy $\iff$ $\displaystyle\inf_{n \ge 1}{\sup_{k,l \ge n}{|a_k - a_l|}} = 0$

(How to state that a sequence is Cauchy in terms of $\limsup$ and $\liminf$? For the above post by me, I have this new claim and its unfinished proof, but I am not sure should I edit that old question ...
3
votes
1answer
53 views

Writing proofs with modular arithmetic

I am enrolled in Discrete Mathematics 2 and I am having trouble understand a lot of the material. For the particular problems I need help with I need to: Prove each of the given statements, assuming ...
0
votes
1answer
23 views

Question about a proof of $ g(A\cap B)\subset g(A)\cap g(B) $

I was trying to prove: $$ g(A\cap B)\subset g(A)\cap g(B) $$ which has been answered lots of time on here but I had a question about a part of my attempted proof (which is hopefully correct, I'm ...
2
votes
2answers
76 views

Demonstrating the image of the inverse image of a subset

I need to demonstrate the following: Let $E, F$ be sets, $Y \subset F$ and $f : E\longrightarrow F$. Prove that $f(f^{-1}(Y)) = Y \cap f(E)$ I tried do prove that using double inclusion with ...