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

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8
votes
2answers
129 views

Prove that $\sqrt{2} + \sqrt[3]{3}$ is irrational [duplicate]

$\sqrt{2} + \sqrt[3]{3}$ is irrational ? These are my steps - $\sqrt{2} + \sqrt[3]{3} = a$ $3 = (a-\sqrt{2})^{3}$ $3 = a^{3} -3a^{2}\sqrt{2} + 6a -2\sqrt{2}$ $3a^{2}\sqrt{2}+2\sqrt{2} = ...
0
votes
1answer
38 views

How to negate this statement for a proof by contradiction

I want to try and construct a proof by contradiction but am having a hard time negating this statement. The statement that I am working with is There are only a finite number of points accepted ...
1
vote
2answers
58 views

Cardinality of all rational points in $R^3$

Question: Find the cardinality of the set of all points in $R^3$ all of whose coordinates are rational, and justify the answer. Idea: Call the set of all points in $R^3$ all of whose coordinates are ...
2
votes
1answer
37 views

cardinality of the set of points with one irrational coordinate, and one rational.

Find the cardinality of the set of all points in the plane which have one rational and one irrational coordinate. Justify you answer. My thoughts so far. We know that $\mathbb Q$, the set of all ...
0
votes
1answer
57 views

Proving Euler's formula using infinity sums

I want to prove $e^{i x} = \cos x + i \sin x$. Proof: $$e^{i x } = \sum \frac{x^{n} i^{n} }{n!} = -i\sum_{\textrm{odd}} (-1)^{n} \frac{x^{2n+1} }{(2n+1)!} + \sum_{\textrm{even}} (-1)^{n} ...
1
vote
1answer
50 views

$f$ and $g$ are two functions from $[0,1]$ to $[0,1]$ with $f$ strictly increasing .Then…

$f$ and $g$ are two functions from $[0,1]$ to $[0,1]$ with $f$ strictly increasing .Then which of the following is true $?$ $A)$ If $g$ is continuous then so is $f\circ g$ counterexample : ...
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.