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

learn more… | top users | synonyms

-2
votes
1answer
43 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
71 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
17 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
27 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
37 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
37 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
15 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 $f(\hat{x},\hat{x})=\infty$....
1
vote
1answer
29 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
34 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) = 6^{...
1
vote
2answers
32 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 ...
0
votes
1answer
97 views

Estimating the Riemann integral of $f$ using an upper bound for $f$

Show is that the Riemann integral $\int_a^b f(x)\,dx$ is bounded by $M(b-a)$, where $M$ is a bound of $f(x)$. I was thinking I would show that $M(b-a)$ is larger than every Riemann sum, but my ...
0
votes
2answers
47 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
56 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 z=f(x+\...
1
vote
0answers
49 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) = f'(g(x))g'(...
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
19 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
73 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
27 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
41 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
52 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
87 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
45 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
69 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 s \...
1
vote
1answer
60 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
22 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 linear,...
1
vote
2answers
48 views

Proof concerning Fibonacci and recursively defined sequences

The series $(a_n)_{n\in\mathbb{N}}$ is given through $$a_1=1,\quad a_2=\frac{1}{2},\quad a_{n+2}=a_na_{n+1} \quad\text{ for } n\geq1.$$ I want to show that $$a_n = 2^{-f_{n-1}}$$ whereas $\,f_n$ ...
1
vote
1answer
64 views

Proof of prime numbers in the form..

There exists a unique prime in the form of p^2 -1, p is just some integer with the restriction of p being greater than or equal to 2. Prove this. I understand that I am first suppose show a prime p ...
1
vote
2answers
91 views

Prove $3+ 5 \sqrt {2}$ is irrational

Prove $3+ 5 \sqrt{2}$ is irrational. I have some ideas about this proof, but I am not quite finished. I understand being irrational means the number would not be in the form of $\frac pq$. I have ...
3
votes
4answers
71 views

Prove that $\lim_{x \to \infty}\big(\frac{x}{x-1}\big)^x$ is also $e$.

Trying to make sense out of the idea that $100\%$ continuous decay is $\frac{1}{e}$, I thought about this: You can express $1+\frac{1}{x}$ as $\frac{x+1}{x}$, such that $\big(1+\frac{1}{x}\big)^x = \...
1
vote
1answer
30 views

proof the derivate of gamma function using the limit definition

using $\Gamma(z+1)=z\Gamma(z)$ and $\Gamma(z)=\lim\limits_{n\to+\infty}\frac{n!n^z}{z(z+1)\cdots(z+n)}$ proof that $$\psi(z+1)=-\lim_{n\to\infty}\left(\sum_{m=1}^{n}\frac{1}{m}-\ln n\right)+\sum_{l=...
0
votes
1answer
54 views

The sum which gives $3^n$

So I have the following which I must prove : $$\sum_{(n_1,n_2,n_3)\,:\,n_1+n_2+n_3=n} \binom{n}{n_1, n_2, n_3} = 3^n$$ I'm not sure where I must begin. This is a multinomial.
0
votes
1answer
30 views

Prove a property of a function H based on the definition provided

Define $$H(n) = \begin{cases}{} 0 & n\leq 0\\ 1 & n = 1 \textrm{ or } n = 2\\ H(n-1) + H(n-2) - H(n-3) & n>2\\ \end{cases}$$ Prove $\forall n\geq 1$ that $H(2n) = H(2n-1) = n$. ...
0
votes
2answers
40 views

Proof (a | b and a not divide b) -> a not divide (b+c)

Prove $\forall a\in \mathbb Z, \forall b\in \mathbb Z, \forall c\in \mathbb Z, (a | b \land a\nmid c) \rightarrow a\nmid(b + c)$. Maybe a gentle nudge in the right direction
0
votes
2answers
44 views

Multiplying matrices / corresponding systems of equations

I'm having some trouble with a problem in linear algebra: Let $A$ be a matrix with dimensions $m \times n$ and $B$ also a matrix but with dimensions $n \times m$ which is not a null matrix. (That's ...
1
vote
1answer
34 views

Determining the exact one from all possible Jordan Canonical Forms of a matrix

Here is the example I encountered : A matrix $\ M\ $ $(5\times 5)$ is given and its minimal polynomial is determined to be $(x-2)^3.$ So considering the two possible sets of elementary ...
0
votes
1answer
22 views

Proof of an algebraic statement [duplicate]

Let $V$ be a $n$-dimensional vector space. Let's also say that we have two linear operators: $A,B\in L(V)$ and $AB=0$. Then how do I prove that the sum of the ranks of operators is smaller than $n$, i....
0
votes
2answers
46 views

Two matrix proofs

linear algebra problem I'm having some trouble wrapping my head around: Given two square matrices $A,B$ with dimensions $n\times n$ and that $A=I-AB$ : I've already proved with relative ease that $A$...
0
votes
0answers
40 views

If $\Sigma$ is the splitting field for $f$ over $K$ and $K\subseteq L \subseteq \Sigma$, show that $\Sigma$ is the splitting field for $f$ over $L$.

If $\Sigma$ is the splitting field for $f$ over $K$ and $K\subseteq L \subseteq \Sigma$, show that $\Sigma$ is the splitting field for $f$ over $L$. I believe the general idea of this proof is as ...
0
votes
2answers
47 views

How to prove a statement with two “ if and only if”

If $H$ and $K$ are subgroups of $G$, show that $HK$ is a subgroup if and only if $HK \subseteq KH$, if and only if $KH \subseteq HK$. This statement confuses me. Does mean I need to prove that $HK$ ...
3
votes
1answer
53 views

If $f$ is one-to-one and continuous on the closed interval $[a,b]$ then prove that $f$ is strictly monotone on $[a,b]$

If $f$ is one-to-one and continuous on the closed interval $[a,b]$ then prove that $f$ is strictly monotone on $[a,b]$. So my plan was to prove this by contradiction. I'm wondering if there is a ...
0
votes
1answer
32 views

if $f$ is integrable on $[a,b]$ , show that $\lim_{s \to a^+} \int _{s}^{b}f=\int _{a}^{b}f$

If $f$ is integrable on $[a,b]$ , show that $\lim_{s \to a^+} \int _{s}^{b}f=\int _{a}^{b}f$ I proved that if $f$ is integrable on $[s,b]$ then $f$ is integrable on $[a,b]$ But how to prove the ...
3
votes
0answers
60 views

Is there any mistake in my proof?

My little brother started fiddling around with his calculator, and noticed something curious: $$ \Large \sqrt{a\cdot\sqrt{a\cdot\sqrt{a\cdot\sqrt{a \cdot \ldots}}}} = a $$ So I went ahead and wrote a ...
-1
votes
5answers
151 views

Prove number of handshakes between $n$ people is $\tfrac{n(n−1)}{2}$ by induction [closed]

How do we calculate the number of handshakes between $n$ people? And where do I apply the inductive step?
4
votes
2answers
114 views

Number of Taxicab routes in a triangular city

I am assuming a triangle that is "almost" half a rectangular city with taxicab geometry. I am trying to find the number of paths in this triangular city. Assuming that the ride starts from the corner ...
1
vote
1answer
59 views

Suppose my progress is in Baby Rudin's chapter 4. Is it possible to discuss the uniform continuity of $x^t$ without using facts in later chapters?

Let $f: [0, \infty) \rightarrow \mathbb{R}$ defined by $f(x)=x^t$. Prove that If $t \in (0,1]$ then $f(x)=x^t$ is uniformly continuous on $[0, \infty)$. If $t \in (1, \infty)$ then $f(x)=x^...
1
vote
3answers
56 views

How to write a rigorous proof for this statement?

Prove that for finite set $X$, the function $f:X \to X$ is surjective if and only if it is injective I have the idea of proof in my mind but find it difficult to translate it into mathematical ...
1
vote
4answers
116 views

prove that $n(n+1)$ is even using induction

the base case of $n=1$ gives us $2$ which is even. assuming $n=k$ is true, $n=(k+1)$ gives us $ k^2 +2k +k +2$ while $k(k+1) + (k+1)$ gives us $k^2+2k+1$ whats is the next step to prove this by ...