The proof-strategy tag has no wiki summary.
4
votes
1answer
65 views
How does one DERIVE the formula for the maximum of two numbers
I want to derive (not prove that this is true) the formula
$\max (x,y) = \dfrac{x + y + |y-x|}{2}$
I was reading a proof (which they have the result ahead of time already) that we do cases and then ...
1
vote
2answers
94 views
A typo in Spivak's solution?
Problem
Solution
I honestly cannot figure out what he is doing. On one hand, I think Spivak wants to write $|\phi(b)/b^n| > 1/2$ instead of $|\phi(b)/b^2| < 1/2$. On the other ...
1
vote
2answers
47 views
Induction on the Fibonacci sequence?
Prove by induction that the $i$th Fibonacci number satisfies the equality:
$$F_i = \frac {\phi^i - \hat\phi{}^i}{\sqrt5}$$
where $\phi$ is the golden ratio and $\hat\phi$ is its conjugate.
...
3
votes
2answers
42 views
Show that $\exists A \subset \mathbb{R}$ such that $\forall x$ $\in \mathbb{R}$, we may write $x$ uniquely as $x=a+q$, where $a\in A,q\in\mathbb{Q}$.
Not sure where to go with this one. Clearly will have to use the axiom of choice at some point. I haven't been able to think of a good example for the set A. Once we've got that, it'd be a matter of ...
1
vote
0answers
19 views
Farthest vector pair in subset of unit circle.
This question is extended from this question
Given a set $S$ and a pair of vector $x,y\in S$
In this version the set $S$ is a subset of unit circle.
That is for all $s \in S$, $||s||=1$
Does the ...
1
vote
1answer
19 views
Showing a pair of vector is the farthest vector pair in certain set
Given a set $S$ and a pair of vector $x,y\in S$
I would like to show $x$ and $y$ are the farthest vector pair in the set $S$
I start with showing there doesn't exist a vector $a \in S$ s.t.
...
1
vote
2answers
59 views
Prove that $(S \cap T = \varnothing) \land (S \cup T = T) \rightarrow S = \varnothing$.
Logically, the following proposition makes sense:
$(S \cap T = \varnothing) \land (S \cup T = T) \rightarrow S = \varnothing$
Or, in english, if sets $S$ and $T$ share no elements, and the union of ...
5
votes
2answers
126 views
How can a matrix be Hermitian, unitary, and diagonal all at once
I was given the following problem in class, and I'm not really sure how to begin this proof.
Describe all 3 by 3 matrices that are simultaneously Hermitian, unitary, and diagonal. How many are ...
13
votes
1answer
206 views
Spivak's proof on “hard theorems”
I have the second edition of Spivak. Consider
Can someone tell me why he considers $2n|a_{n-1}| \dots$? Later he shows everything is squeezed between -1/2 and 1/2 and he gets the desired result. I ...
1
vote
1answer
30 views
Let $X$ has countable extent. Does $X^2$ have countable extent?
Definition 1: A space $X$ has countable extent if every uncountable subset of $X$ has a limit point in $X$.
I'm struggling with this question:
Question 2: Let $X$ has countable extent. Does ...
0
votes
1answer
47 views
Lie derivative: Leibniz rule proof
How can I prove $\mathcal{L}_v(\omega\wedge\alpha) = (\mathcal{L}_v\omega)\wedge\alpha + \omega\wedge(\mathcal{L}_v\alpha)$ ?
4
votes
0answers
59 views
Infinite “String” of Implication Statements
This question is inspired by the conversations at
Does this require transfinite induction?
First of all, does an infinite string of implication statements have a conclusion? I don't think so, but I ...
-1
votes
1answer
59 views
Proof of divisibility by 2 and 3 if and only if divisible by 6
I can't find a way of proving that:
For integer a, a is divisible by 2 and divisible by 3 if and only if a is divisible by 6.
I’m not sure where to go from here. Any help would be great!
3
votes
1answer
121 views
Prove that there exist infinitely many squares $a$ such that $\sqrt{\sqrt{a}}$ is a square
I was just thinking about squares while randomly punched numbers into my calculator and I was wondering do there exist infinitely many squares such that $\sqrt{\sqrt{a}}$ is a square and $a$ is also a ...
8
votes
0answers
77 views
Find all functions $F(x,y)$ such as $\frac{\sqrt{3}}{2}\frac{\partial{f}}{\partial{x}}+\frac{1}{2}\frac{\partial{f}}{\partial{y}}=0$
How to find all possible functions $f(x,y)$ such as:
$$ \frac{\sqrt{3}}{2}f_x+\frac{1}{2}f_y=0$$
(with $f_x = \frac{\partial{f}}{\partial{x}}$ )
Here's everything I tried:
1) I can guess the ...
2
votes
2answers
59 views
Proving if $a_1\equiv b_1\pmod{n}$ and $a_2\equiv b_2\pmod{n}$ then we have $a_1+a_2\equiv b_1+b_2\pmod{n}$.
How can I prove if $a_1\equiv b_1\pmod{n}$ and $a_2\equiv b_2\pmod{n}$ then we have $a_1+a_2\equiv b_1+b_2\pmod{n}$? I have tried several ideas I've found online but don't really understand them. Is ...
1
vote
2answers
30 views
$T\circ T=0:V\rightarrow V \implies R(T) \subset N(T)$
Question
Let $T:V \rightarrow V$ be a linear map. How do I prove that $T \circ T = T_0$ ( the zero linear map) iff $R(T) \subset N(T)$?
Attempt
\begin{eqnarray}
T\circ ...
5
votes
1answer
64 views
The genus of the Fermat curve $x^d+y^d+z^d=0$
I need to calculate the genus of the Fermat Curve, and I'd like to be reviewed on what I have done so far; I'm not secure of my argumentation.
Such curve is defined as the zero locus
...
0
votes
1answer
45 views
Existence Proof: $T(v_i)=w_i$ for all $i=1,2,3,\dots,n$
Theorem to prove:
Let $\{v_1,\dots,v_n\}$ be a linearly independent set in a finite-dimensional vector space $V$ and let $w_1,\dots,w_n$ be arbitrary vectors in a vector space $W$. Then there exists ...
2
votes
1answer
74 views
Hard-wiring a proof method in my head
There's s a kind of proof regularly used in linear algebra ( proving facts about Transformations, direct sums, basis, ... ) that i have definitely agreed with but still couldn't connect my intuitive ...
0
votes
1answer
53 views
Proof: Two circles have a most 2 intersections
I already prooved the statement here in general, but know I tried to proove it in an other way:
I put $M_1$ on $(0/0)$ and the x-axis through $M_1$ and $M_2$.
That simplifies the equatons for ...
0
votes
1answer
50 views
Proving recurrence relations
So, I initially proved the theorem that if $a != b^d$ and $n$ is a power of $b$, then $f(n) =
C_1n^d + C_2n^{log_b a}$, where $C_1 = b^dc/(b^d − a)$ and $C_2 =
f(1) + b^dc/(a − b^d )$.
This is seen ...
5
votes
2answers
103 views
If $f$ and $g$ are continuous, prove $f\circ g$ is continuous.
Suppose that $(X,T)$, $(Y,U)$ and $(Z,V)$ are three topological spaces and that
$g\colon X\to Y$ and $h\colon Y \to Z$ are continuous. Prove that $h\circ g\colon X \to Z$ is a continuous ...
0
votes
2answers
40 views
Proof pythagoras theorem with dot product + distance
I want to proof $d(A.B)^2=d(A,C)^2+d(B,C)^2$ for
with $(\vec a-\vec c) \bullet (\vec b - \vec c)=0$. I applied the definitions of distance and got
$d(A,B)^2=d(A,C)^2+d(B,C)^2 \Leftrightarrow ...
-4
votes
1answer
54 views
Transformation Existence Proof: A Call for Critique [duplicate]
QUESTION
Prove that there exists a $T:V\rightarrow W$ such that $N(T)=V'\subset V$ and $R(T)=W'\subset W$
ATTEMPTED ANSWER
Let $V$ and $W$ be finite-dimensional vector spaces over $F$. Let ...
3
votes
2answers
58 views
Repeating Square Root Closed Form [duplicate]
I've been thinking about repeating square roots:
$\sqrt{x+\sqrt{x+\sqrt{x+\cdots}}}$. I wrote a program on my calculator to do it $n$ times and I found that, if $x = y^2 - y$ then ...
2
votes
1answer
40 views
Proofs regarding diagonalization of matrices
I'm trying to prove that a unitary matrix can be diagonalized using an orthonormal basis of eigenvectors, and that the eigenvalues are on the unit circle. So far, I have been able to show that if A is ...
3
votes
2answers
43 views
Solving two simultaneous recurrence relations
If we have the two recurrence relations $$a_n = 3a_{n-1} + 2b_{n-1}$$ $$b_n = a_{n-1} + 2b_{n-1}$$ with $a_0 = 1$ and $b_0 = 2$.
My solution is that we first add two equations and assume that $f_n = ...
3
votes
1answer
24 views
Using a particular image to justify a (specific) trig integral equality.
I would like to include the following string of equalities in a paper:
$$\sin ^2(x) + \cos ^2(x) = 1$$
$$\int _0^{\dfrac{\pi}{2}} \sin ^2 (x)dx + \int_0^{\dfrac{\pi}{2}} \cos ^2 (x)dx = ...
3
votes
1answer
103 views
Is my proof correct? $p_1p_2p_3\cdots p_n+1)$ cannot be the square of an integer
Prove that $p_1p_2p_3\cdots p_n+1$, where $p_n$ is the $n^{th}$ prime, cannot be the square of an integer.
Let $p_1p_2p_3\cdots p_n+1=Q$ and assume it is the square of an integer, so ...
4
votes
1answer
89 views
Discreteness of eigenvalues for certain operators - can this approach be made rigorous?
I was idly thinking about why one might naïvely expect a discrete spectrum of eigenvalues for a linear operator $L$ when I dreamt up the following argument (which I expect isn't new instead - ...
1
vote
0answers
50 views
Visual/intuitive proof of why $\sum k^3 = (\sum k)^2$, where $k$ goes from 1 to $n$? [duplicate]
I understand that one could prove this by first proving the analytic expressions of the sigma terms through induction, and then square the $\displaystyle\sum_{k=1}^n k$ term to show LHS = RHS.
Are ...
2
votes
3answers
190 views
is this a foolish way to do proofs?
When I'm asked something like "show X is equal to Y", I first try to manipulate what I know (X) into the result (Y). A lot of the time, I do not investigate the result I'm trying to conclude with. I ...
1
vote
2answers
108 views
A one-to-one function from a finite set to itself is onto - how to prove by induction?
I'm not sure if I can do this without knowing what f actually is?
Let $X$ be a finite set with $n$ elements and $f: X \rightarrow X$ a one-to-one function. Prove by induction that $f$ is an onto ...
1
vote
2answers
53 views
Showing a contraction without a fixed point
Suppose $f: [1, \infty) \to [1, \infty]$ defined by $f(x) = x + \frac{1}{x}$ for all $x \geq 1$. I want to prove that:
\begin{equation}
|f(x)-f(y)| < |x-y|
\end{equation}
except when $x=y$, but ...
2
votes
0answers
46 views
Show that exists $x$ and $y$ such that $P(x)*P(y) < 0 \ ; x,y\in \Bbb C$
I would like to ask how could I perform following proof:
Prove(show) that exists such $x$ and $y$ that $P(x)*P(y) < 0$. Where $P(x)$ and $P(y)$ are polynomials.
$\forall\ x,y\in \Bbb C$
1
vote
2answers
41 views
Prove that a greedy algorithm selects the maximum number of programs
This is a homework problem. Intuitively, I know it to be true, because the largest group of programs (say, $j$ programs) must be composed of the smallest $j$ programs. But how to go about formally ...
3
votes
4answers
58 views
Combinatorial Proof
I have trouble coming up with combinatorial proofs. How would you justify this equality?
$$
n\binom {n-1}{k-1} = k \binom nk
$$
1
vote
1answer
134 views
$T:P_n(F) \rightarrow F$ PROOF OUTLINE
I'd like some heavy critique if you don't mind. See here for more details.
Let $S=\{f \in P_n(F) : f(1)=0\}$. Clearly, the polynomial $f(x)=0 \in S$ because $f(c)=0$ for any choice of $c\in F$. To ...
1
vote
0answers
44 views
Determining Complete Metric Spaces
I need to determine if $((0, 1), d)$ where $d(x, y) = |x^2 - y^2| \forall_{x,y}\in (0,1)$
My argument is as follows, take the sequence defined by $\dfrac{1}{n}$, then we know by $d(x, y)$ that ...
1
vote
3answers
45 views
Can someone check the solution to this recurrence relation?
Here's the recurrence relation: $a_n = 4a_{n−1} − 3a_{n−2} + 2^n + n + 3$ with $a_0 = 1$ and $a_1 = 4$
Here's the solution:Write:
$$
a_{n + 2} = 4 a_{n + 1} - 3 a_n + 2^n + n + 3 \quad a_0 = 1, a_1 = ...
0
votes
2answers
43 views
Finding this solution to a recurrence relation
So, I know that the recurrence relation $a_n = 4a_{n−1} − 3a_{n−2} + 2^n + n + 3$ with $a_0 = 1$ and $a_1 = 4$ has the solution of $a_n = -4(2^n) - n^2 / 4 - 5n / 2 + 1/8 + (39/8)(3^n)$. I just ...
1
vote
3answers
50 views
Prove that $\frac{1}{\sqrt{2n-1}}-\frac{1}{2n}\geq \frac{1}{2n}$ for $n = 1, 2, 3,…$
Prove that $\frac{1}{\sqrt{2n-1}}-\frac{1}{2n}\geq \frac{1}{2n}$ for $n = 1, 2, 3,...$
This is required to prove that the series $1-\frac{1}{2}+\frac{1}{\sqrt{3}}-\frac{1}{4}$ is divergent, but I ...
2
votes
1answer
51 views
Limit of Binomial distribution
In showing us that Binomial distribution: $$B_{N,p}(n) := \binom {N}{n} p^n(1-p)^{N-n}$$ tends to Poisson's: $$P_ \lambda (n) = \dfrac {\lambda ^n}{n!}e^{-\lambda}$$where I guess lambda should be ...
0
votes
1answer
22 views
How to show all solutions for a particular recurrence solution
I've found that the recurrence relation $a_n = 4_{an−1} − 4a_{n−2} + (n + 1)2^n$ has the solution of $an = 2^n(p_0 + p_1n + n^2 + n^3/6)$. I'm just trying to understand the steps necessary to solve ...
1
vote
1answer
48 views
Diagrammatic Representations: $\dim(Skew_{n\times n}(\mathbb{R}))+\dim(Sym_{n\times n}(\mathbb{R})) = \dim(M_{n\times n}(\mathbb{R}))$
SEE AUTHOR'S ANSWER BELOW
So I'm trying to derive the dimensions of both $Skew_{n\times n}(\mathbb{R})$ and $Sym_{n\times n}(\mathbb{R})$. I know that $\dim(M_{n\times n}(\mathbb{R}))=n^2$, but I ...
1
vote
2answers
48 views
Find the recurrence solution of this relation
How would we find the solution of the recurrence relation: $a_n = 2a_{n−1} + 3 · 2^n$ ?
After trying it, I've found it to be $a_n = 2^{n-1} (c_1 + 6n)$
Not sure if this is right..
Thanks!
1
vote
1answer
34 views
On the Dimensionality of Space: An Elementary Analysis
The below theorem I am to prove. Perhaps you have a critique...
Theorem 2.4 Let $W_1$ and $W_2$ be two subspaces of a vector space $V$. Then $\dim(W_1 \cap W_2)=\dim(W_1)$ if and only if $W_1 ...
2
votes
4answers
66 views
Orthogonal matrices proof
Let $\upsilon _{n} $ be the set of all $n \times n$ orthogonal matrices(for all fixed $n$).
Show that $\upsilon _{n} $ is not a subspace of $\ M _{n\times n} $. Thank you !
Additional:
Suppose $A ...
1
vote
1answer
26 views
Prove that the Iwata function is Submodular
The Submodularity property for $f: 2^V \rightarrow \mathbb{R}$ is defined as:
$f(X) + f(Y) \geq f(X \cup Y) + f(X \cap Y)$ where $X, Y \subseteq V$
While the Iwata function is defined as:
...
