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

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2
votes
1answer
45 views

Fibonacci identity $F_{n+1}^2 - (F_{n+1}F_n) - F_n^2 = (-1)^n$

I am trying to prove Let $F_n$ be the $n$th Fibonacci number. Then $F_{n+1}^2 - F_{n+1}F_n - F_n^2 = (-1)^n$ I am not sure where to start with this.
1
vote
1answer
50 views

Proof of arithmetic properties in $\mathbb{Z}$

In general, in elementary numbers theory when we prove properties we begin with natural numbers then I was wondering how you can extend the proof to $\mathbb{Z}$ clearly and properly. For instance, ...
2
votes
3answers
66 views

Is there a way of making “guess the next number in the sequence” rigorous?

This is maybe more of a question for matheducators.SE than math.SE but I'm more interested in the math than the education. A common problem given to middle and high school kids (at least in America) ...
-1
votes
1answer
34 views

Prove that if a relation R on a set A is reflexive, symmetric and antisymmetric, then $R=I_A$ [closed]

Prove that if a relation $R$ on a set $A$ is reflexive, symmetric and antisymmetric, then $R=I_A$ I know a relation is a set of ordered pairs and that $I_A = (x,x)$ but I have no idea how to do this ...
3
votes
1answer
37 views

Successful studying for a proof-based course final exam?

I'm currently taking a Transition to Advanced Mathematics course, which is entirely proof-based, so it's pretty new territory. Up until now, all the classes I've taken were fairly computational, so ...
-1
votes
4answers
54 views

How can one show that $f(n)>n$

Given a function $f\colon \Bbb N^\ast\to \Bbb N^\ast$ such that we have $f(f(n))=f(n+1)-f(n)~\forall~n\in\Bbb N^\ast$. Show that: If $f(n+1)-f(n)>1$ then $f(n)>n$. Since $f(n+1)>f(n)+1$ I ...
1
vote
1answer
24 views

Difference between “Let x be a …” and “Let x … be arbitrary”

What is the difference between "Let x be a ..." and "Let x ... be arbitrary"? Consider the following example: Let $R$ be an equivalence Relation on $A$. Then $\forall x, y\in A.\ [x]_R=[y]_R.$ ...
1
vote
1answer
17 views

Let T(n) be defined by the following recurrence relation

Let T(n) be defined by the following recurrence relation $\begin{equation} \begin{cases} T(0) = T(1) = 1 \\ T(n) = T(n−1) + T(n−2) + 1 \quad for \quad n ≥ 2 \end{cases} \end{equation}$ Show that ...
1
vote
0answers
16 views

Is this proof about the “Maximum Chunk Product Problem” (my own name for it) sufficient and clear?

- Question: For $L \in \mathbb{N}$, how do you choose $[x_i]$ so as to maximize $\prod x_i$ where $\sum x_i = L$ and $x_i \in \mathbb{N}$? (The number of "chunks" $x_i$ can be whatever you want.) - ...
1
vote
1answer
31 views

infimum of a sequence > 0, if the sequence converges proof

Let $(x_n)_{n=1}^{\infty}$ be a sequence of real numbers such that $x_n \neq 0$ for all $n \in \Bbb{N}$ and $x_n \rightarrow x$ as $n \rightarrow \infty$, where $x \neq 0$. Prove that $inf({|x_n|: n ...
2
votes
4answers
36 views

Geometry inequality proof

I started off with the given and by using the triangle inequality theorem but I don't know what to do next. Can someone please help? Thank you very much. I greatly appreciate it!
2
votes
1answer
13 views

How to Prove Triangle Centers in Tetrahedra

How would you prove the existence of triangle centers in tetrahedra, for example, the incenter, circumcenter, or centroid?
0
votes
1answer
33 views

What does it mean to 'show that' coefficients are a solution of this system of linear equations?

I don't really know what counts as a proof and haven't been taught maths since 16yo (29yo PhD now). I've got working knowledge of e.g. basic linear algebra, geometry, and statistics, but this feels ...
2
votes
4answers
89 views

(Proof) If $f$ and $g$ are continuous, then $\max\{f(x),g(x)\}$ is continuous

Consider the continuous functions $f,g:\mathbb{R}\rightarrow\mathbb{R}$. Show that $F:\mathbb{R}\rightarrow\mathbb{R}$ with $x\mapsto \max\{f(x),g(x)\}$ is continuous using the $\epsilon - \delta$ ...
0
votes
1answer
18 views

If $A$ is an open set of the taxicab metric $(R^2,d_1)$ then $A$ is a union of open balls of the Euclidian $(R^2,d)$ metric.)$

Is is true that if $A$ is an open set of the taxicab metric $(R^2,d_1)$ then $A$ is a union of open balls of the Euclidian $(R^2,d)$ metric. Where as taxicab metric I mean $R^2$ equipped with the ...
1
vote
1answer
41 views

Proof that $\|Ax\|<\|x\|$ for $A$ with absolute value of eigenvalues $<1$

I'm trying to prove that if all eigenvalues of the square matrix $A$ are $|\lambda|<1$ then $\|Ax\|≤\|x\|$. This seems like a simple proof but I fail to understand how to relate the basic ...
0
votes
2answers
26 views

The intersection of an infinite number of subspaces is a subspace

Let $V$ be a finite dimensional vectorspace over a field $\mathbb{ F}$. It's easy to show that if $U$ and $V$ are subspaces of $V$ then $U \cap V$ is a subspace. But what if there are an infinite ...
0
votes
1answer
26 views

Cardinality of set difference of finite sets

Is $|A \setminus B| = |A| - |A \cap B|$, where $A$ and $B$ are finite sets, true? I have been unable to prove this or find a good reference on cardinality of set differences. The only reference I ...
0
votes
0answers
35 views

problem from Stoll's introduction to real analysis [duplicate]

Suppose f: [a,b]$\rightarrow \Re $ is continuous. Let $M=max\{|f(x)| : x \in [a,b]\}$ Show that $$\lim_{n\rightarrow \infty}(\int_a^b |f(x)|^n)^{\frac{1}{n}}=M$$ My attempt: Suppose for ...
1
vote
2answers
41 views

How to prove that a number is composite?

How can one prove that a number is a composite number? I'm trying to prove that $6n + 1$ and $6n - 1$ are both composite for an infinite amount of integers $n$, with $n$ greater than or equal to $1$. ...
1
vote
3answers
55 views

Why must an inverse function be bijective?

Explain why $f^{-1}$ is a function if and only if $f$ is a bijective function. My attempt: $f^{1}$ is the inverse relation from B to A $\equiv$ function from B to A By definition of a function ...
1
vote
2answers
28 views

Ways to deal with generating sets of groups

I find that when I come across questions to do with generating sets of groups, I'm never quite sure how to go about the problem. It's difficult to deal with them purely set-theoretically, as you can't ...
-2
votes
1answer
40 views

How to write a proof of $ A\setminus B = \emptyset \leftrightarrow A \subseteq B$

I think the best way to prove this is by contradiction, but I'm struggling with the concept of how to write it properly. $$ A\setminus B = \emptyset \leftrightarrow A \subseteq B$$
0
votes
2answers
69 views

Proving $\mathbb{Z × N}$ is countable. [closed]

How would I prove that $\mathbb{Z × N}$ is countable? The hint given was to follow to indicated order. Thanks!
0
votes
0answers
14 views

Proving the treewidth of a graph is the maximum treewidth of the connected components

We have a graph $G = (V,E)$ and $C$ is the set of connected components of G. I want to prove that $tw(G) \geq $ Max $tw(C)$ where tw is the treewidth. I know the out sketch of the proof is to take ...
2
votes
2answers
25 views

Proving a matrix is triangular

linear algebra proof I'm having trouble with: Let A be a square matrix. Prove that there exists a matrix $B$ so that $BA$ is a triangular matrix. I tried turning it into a homogeneous system of ...
3
votes
0answers
31 views

Proving a Particular Coefficient of a Power Series Equals $0$

Suppose I have a particular function $$F(x,z) = \sum_{n=0}^\infty{A_n(x)\frac{z^n}{n!}}$$ and suppose, through the use of a particular computer algebra system, that the particular polynomial ...
1
vote
1answer
33 views

Vector Proof Involving Triangle

I'm stuck on the following homework question: Given the triangle $PQR$, with $X$ placed on $PR$ dividing it into a ratio of $2:3$, and $Y$ the midpoint of $PQ$, prove that if $Z$ is the ...
0
votes
1answer
14 views

Equality between limit and integral whose integrand diverges at some point.[Edited]

Let $f:[0,1]\times[0,1]\to\mathbb{R}\cup\{\pm\infty\}$ be a function such that, for a given point $\hat{x}\in(0,1)$, $f$ is continuous in $[0,\hat{x})\times[0,\hat{x})$ and ...
1
vote
1answer
22 views

Proving a lower bound and upper bound?

I understand why the empty set is a lower bound and A is an upper bound. The only problem I am having is putting my thoughts into a mathematical solution. Can anyone help out? Thanks. Let A be a set ...
2
votes
2answers
33 views

Show that $f: G \to H $ is a homomorphism.

This is my first encounter with homomorphisms and I'd like to have my proof verified. Question: Let $G = (\mathbb{Z}, +)$ and $H = \{6^{n} \mid n \in \mathbb{Z} \}$. Define $f: G \to H$ by $f(x) = ...
1
vote
2answers
25 views

Proof on modular congruence

Prove that for n in the set of natural numbers, n is greater thean or equal to 2: For all a belonging to the set of natural numbers, For all b belonging to the set of natural numbers, a is modular ...
1
vote
1answer
55 views

If f is a Riemann integrable function on [a,b], is there always a Riemann sum whose value is greater than or equal to the the value of the integral?

Can you give me a hint as to how to show this? I need it for a homework problem I'm working on, and I'm technically not supposed to know about the Upper Riemann sum of a function, although I've ...
0
votes
2answers
44 views

If $f(a) < f(p)$ and $f(p) > f(b)$ then there is a $d$ such that $f'(d)=0$

If $f: [a,b] → R$ is a continuous function which is differentiable on $(a,b)$, And if $f(a) < f(p)$ and $f(p) > f(b)$ for some $p ∈ (a, b)$. Show that there exists $d ∈ (a, b)$ such that $f'(d) ...
1
vote
2answers
47 views

Deriving $\Delta z=\frac{\partial y}{\partial x}\Delta x+\frac{\partial f}{\partial y}\Delta y+\alpha\sqrt{\Delta x^2+\Delta y^2}$

I was reading a math book, which contained. "Let us consider a function $$z=f(x,y)$$ of two variables. If it has continuous partial derivatives, we can prove that its increment $$\Delta ...
1
vote
0answers
38 views

How to give an alternative proof of the chain rule using the little-o notation?

The chain rule. If $g$ is a function that is differentiable at $x$ and $f$ is a function that is differentiable at $g(x)$, then $f \circ g$ is differentiable at $x$, and $(f \circ g)'(x) = ...
1
vote
0answers
56 views

Prove $3 \cdot 5 \cdot 7 \cdot 11 \cdot prime_n = 2k + 1$ [duplicate]

It is known that any prime greater than 2 is odd. How do I show the combinations of all primes greater than 2 is also odd, $2k+1$? I tried using induction, but what is appropriate for $prime_n$? ...
0
votes
0answers
18 views

Frobenius complement in semidirect product

This is problem 1.D.4 in Isaacs, Finite Group Theory. I think I have a proof, but it's a rather grungy element-pushing argument (very un-Isaacs in style). My questions are: Is there a cleaner, more ...
0
votes
1answer
17 views

Prove $f : A\rightarrow B, g: B\rightarrow C$ , and $g\circ f: A \overset{1-1}{\rightarrow}C$, then $g:B\overset{1-1}{\rightarrow}C$

I am completely stuck on this, I want to say it's true and do a proof by contrapositive, since if g is not surjective, then $\exists b \in B $ such that for $c \in C, f(b)\neq f(c)$, but I'm not sure ...
0
votes
1answer
49 views

Show that in a group of seventeen people, there exists a trio who are either three mutual friends, three mutual enemies, or three mutual strangers.

Suppose that in a group of people that any two people are either friends, enemies of strangers. Show that in a group of seventeen people, there exists a trio who are either three mutual friends, three ...
1
vote
1answer
23 views

multiplication of consecutive prime numbers in the form $4k +3$

How can I prove that prime numbers beginning with $2$, multiplied with the next consecutive prime plus $1$, $2\times3\times5\times7\times\cdots+1$, will give the form $4K+3$?
2
votes
1answer
36 views

Prove that there exists only 1 prime number of the form $p^2−1$ where $p≥2$ is an integer.

by factoring $p^2−1$, we have $(p+1)(p-1)$. I know that p=2 which gives 3 is the only solution, however how do I prove that p=2 is the only integer which gives a prime?
1
vote
2answers
49 views

How can I prove that there is a bijective function?

Let $A$ be a nonempty set. Prove that there is a bijective function $$ F \colon \{ \text{Equivalence relations on } A\} \rightarrow \{\text{Partitions of }A\}. $$ I am completely lost on where to ...
-1
votes
4answers
81 views

Prove by Mathematical Induction $3^{2n}\equiv 1\pmod 4$ for every natural number n. [closed]

Prove by Mathematical Induction $3^{2n}\equiv 1 \pmod 4$ for every natural number n.
0
votes
0answers
41 views

Proving $A_5$ Has No Subgroup of Order 30 [duplicate]

$H$ is a subgroup of $A_5$ that has order 30. From this I know that $|A_5 : H|$ = 2. From this I'm supposed to prove that $H$ contains all 3 and 5 cycles and then use that to prove that there cannot ...
0
votes
1answer
61 views

difference between “let” and “for all”

What is the difference between "let" and "for all"? Consider the following example For all natural numbers n, if n is even, then n squared is even. Let n be a natural number. If n is even, ...
1
vote
1answer
29 views

How to write a rigorous proof for normalisers $N_{G}(H)$ being the largest subgroups of $G$ such that $H \unlhd N_{G}(H)$

Prove that $N_G(H)=\{g \in G| gHg^{-1}=H\}$ is the largest subgroup of $G$ such that $H \unlhd N_G(H)$. I have an idea of the proof that, if we assume $S \leq G$ with $H \unlhd S$ then $$\forall ...
1
vote
1answer
59 views

Is the following proof correct?

Is the following proof correct? Let’s say we find integers $x$ and $y$ such that $x^2 ≡ y^2($mod $n)$ and $n$ has at least $2$ distinct factors not equal to $0$ or $n$. I intend to show that there is ...
0
votes
1answer
28 views

Rational Power Question

Show that if $a ∈ Q$ is positive and if $0 < x < y$ then $x^a < y^a$. I was told to use the difference theorem for this question, but the difference theorem is only for natural numbers.
0
votes
1answer
21 views

Prof of Reflexive, symmetric, or transitive relations

Consider the relation R on Z as: ∀m,n ∈Z, mRn ⇔ m − n is odd . Is R reflexive, symmetric, or transitive? What would the proof or counter proof be? Since R is a reflexive since m-n is ...