A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation". Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in all possible cases. An unproven proposition that is believed to be true is known as a conjecture, or a hypothesis if frequently used as an assumption for further mathematical work.Proofs employ logic expressed in mathematical symbols, along with natural language which usually admits some ambiguity. In most mathematical literature, proofs are written in terms of rigorous informal logic. Purely formal proofs, written fully in symbolic language without the involvement of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, and so-called folk mathematics, oral traditions in the mainstream mathematical community or in other cultures. The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language.
I am trying to solve (a) and (b) of this tutorial question.
(a) Attempt:
Since ##c'## is at ##c'(0) = 1##, then from the definition of continuity at a point:
Let ##\epsilon > 0##, then there exists ##d > 0## such that ##|x - 0| < d \implies |c'(x) - c'(0)| < \epsilon## which is equivalent to...
Consider this proof:
Is it a valid proof?
When we divide by ##z##, we assume that ##z \neq 0##. So, we cannot put ##z=0## on the next step. IOW, after dividing by ##z## we only know that $$c_1+c_2z+c_3z^2+...=d_1+d_2z+d_3z^2+...$$ in a neighborhood of ##0## excluding ##0##.
For this problem,
The solution is,
However, does someone please know why this did not use ##2n ≤ 2n^2 + 2n + 1## which would give
##\frac{3n - 1}{2n^2 + 2n + 1} ≤ \frac{3n}{2n} = \frac{3}{2}##?
In general, after solving many problems, it seems that when proving the convergence of a rational...
The problem and solution are,
However, I am confused how the separation vector between the two masses is
##\vec x = x \hat{k} = x_2 \hat{x_2} - x_1 \hat{x_1}= l\theta_2 \hat{x_2} - l\theta_1 \hat{x_1 } = l(\theta_2 - \theta_1) \hat{k}##. where I define the unit vector from mass 2 to mass 1...
I am trying to understand the proof given in Ethan Bloch's book "The real numbers and real analysis". I am posting snapshot of the proof in the book.
I am also posting theorem 1.2.9 given in the book.
Here author is trying proof by contradiction. First, I don't understand why specific...
Hello everyone,
I've been trying to understand the proof for the binomial theorem and have been using this inductive proof for understanding.
So far the proof seems consistent everywhere it's explicit with the pattern it states, but I've started wondering if I actually fully grock it because I...
Hello, found this proof online, I was wondering why they defined r_2=r_1-(r_1^2-2)/(r_1+2)? i understand the numerator, because if i did r_1^2-4 then there might be a chance that this becomes negative. But for the denominator, instead of plus 2, can i do plus 10 as well? or some other number...
My first solution is
Let
##S = \{x_1, x_2, x_3, ..., x_n\}##
##T = \{2x_1, 2x_2, 2x_3, ... 2x_n\}##
##T = 2S##
Therefore, ##inf T = inf 2S = 2inf S = 2M##
May someone please know whether this counts as a proof?
My second solution is,
##x ≥ M##
##2x ≥ 2M##
##y ≥ 2M## (Letting y = 2M)
Let...
For this problem,
My solution:
Using definition of Supremum,
(a) ##M ≥ s## for all s
(b) ## K ≥ s## for all s implying ##K ≥ M##
##M ≥ s##
##M + \epsilon ≥ s + \epsilon##
##K ≥ s + \epsilon## (Defintion of upper bound)
##K ≥ M ≥ s + \epsilon## (b) in definition of Supremum
##M ≥ s +...
Hello everyone, I've been brushing up on some calculus and had some new questions come to mind.
I notice that most proofs of the fundamental theorem of calculus (the one stating the derivative of the accumulation function of f is equal to f itself) only use a limit where the derivative is...
I've read a proof from Complex Made Simple (David C. Ullrich)
Proposition 4.3. Suppose that ##V## is an open subset of the plane. There exists a branch of the logarithm in ##V## if and only if there exists ##f \in H(V)## with ##f'(z) = \frac{1}{z}## for all ##z \in V##.
Proof: One direction is...
Hello everyone,
maybe some of you know the formula for the number of multiplications in the FFT algorithm. This is again given as ##N/2 \cdot log(N)##. Why is that so? Can you really "prove" this?
I can only deduce this from what I know, because we have ##log(N)## levels and ##N/2##...
Hello everyone,
I found a good proof for the area of a circle being ##{\pi}r^2## but I was recently working on my own proof and I used a change of variables and was wondering if I did it correctly and why a change of variables seems to work.
I start with the equation of a circle ##r^2 = x^2 +...
Hi people,
It's been years I wanted to post this question here. I would like to build a zero knowledge proof that a given chess position contains at least one checkmate. I know that anything provable admits a zero k proof. I know about...
Professor showed this result in the lecture without giving any proof (after proving the existence of the interpolating polynomial in two variables). I've been trying to prove it myself or find a book where is proved but I failed. This is the theorem:
Let
$$ x_0 < x_1 < \cdots < x_n \in [a, b]...
I'm looking for theorems related to using modulo arithmetic.
As an example, if I apply a sequence of arithmetic operations to a given number to get an answer and then apply a modulo operation on the result to get a remainder in a given base. Wiil that be the same if I apply the modulo operation...
I've tried this problem so, so, so so so many times. Given the equations above, the proof starts easily enough:
$$\partial_\mu T^{\mu\nu}=\partial_\mu (∂^μ ϕ∂^ν ϕ)-\eta^{\mu\nu}\partial_\mu[\frac{1}{2}∂^2ϕ−\frac{1}{2}m^2ϕ^2]$$
apply product rule to all terms
$$=\partial^\nu \phi \cdot...
I tried to prove this but I fall into a loop when I try to apply integration by factors, that is I prove that the integral is equal to itself.
Any helpfull tips?
Want to understand how set C contains ##N## x H. H is only defined to be a set with element e and as the domain/range of function k. Is this enough information to conclude that the second set in the cartesian product W is H and not a subset of H?
My thinking is to show that ##N## and H satisfy...
"bubbles are ball" is called isoperimetric problem in serious mathematic. In this topic, many essay were written. Here's my serious essay about "why earth ball", which has been rejected by arxiv and my mentors...... I would want to know if physicists are interest?
I really think that is...
Hi,
I have problems proving task d
I then started with task c and rewrote it as follows ##\lim_{n\to\infty}\sum\limits_{k=0}^{N}\Bigl( \frac{z^k}{k!} - \binom{n}{k} \frac{z^k}{n^k} \Bigr)=0 \quad \rightarrow \quad \lim_{n\to\infty}\sum\limits_{k=0}^{N} \frac{z^k}{k!} =...
with this background, we proceed to the proof. Let us define a set
$$ G = \{ z \in \mathbb{N} | \; x, y \in \mathbb{N}\; (x \cdot y) \cdot z = x \cdot (y \cdot z) \} $$
We want to prove that ##G = \mathbb{N} ##. For this purpose, we will use part 3) of Peano postulates given above...
with this background, we proceed to the proof. Let us define a set
$$ G = \{ z \in \mathbb{N} | \mbox{ if } y \in \mathbb{N}, y\cdot z = z \cdot y \} $$
We want to prove that ##G = \mathbb{N} ##. For this purpose, we will use part 3) of Peano postulates given above. Obviously, ## G...
I want to prove that ##(a+b)\cdot c=a\cdot c+b\cdot c## using Peano postulates where ##a,b,c \in \mathbb{N}##.
The book I am using ("The real numbers and real analysis" by Ethan Bloch ) defines Peano postulates little differently.
Following is a set of Peano postulates I am using. (Axiom 1.2.1...
I have to prove ##a \cdot 1 = a = 1 \cdot a## for ##a \in \mathbb{N}##.
The book I am using ("The real numbers and real analysis" by Ethan Bloch) defines Peano postulates little differently.
Following is a set of Peano postulates I am using. (Axiom 1.2.1 in Bloch's book)
There exists a set...
I have to prove that ##1 + a = s(a) = a + 1## using Peano postulates if ##a \in \mathbb{N}##. The book I am using ("The real numbers and real analysis" by Ethan Bloch) defines Peano postulates little differently.
Following is a set of Peano postulates I am using. (Axiom 1.2.1 in Bloch's book)...
Hi
I have to prove the following three tasks
I now wanted to prove three tasks with a direct proof, e.g. for task a)$$\sqrt[n]{n} = n^{\frac{1}{n}}= e^{ln(n^{\frac{1}{n}})}=e^{\frac{1}{n}ln(n)}$$
$$\displaystyle{\lim_{n \to \infty}} \sqrt[n]{n}= \displaystyle{\lim_{n \to \infty}}...
Hi,
I'm having problems with the proof for the induction of the following problem: ##\sum\limits_{k=0}^{n} \frac{(-1)^k}{k+1} \binom{n}{k}=\frac{1}{n+1}## with ##n \in \mathbb{N}##
I proceeded as follows:
$$\sum\limits_{k=0}^{n+1} \frac{(-1)^k}{k+1} \binom{n+1}{k}=\frac{1}{n+2}$$...
Hi,
I have problems with the proof for task a
I started with the supremum first, but the proof for the infimum would go the same way. I used an epsilon neighborhood for the proof
I then argued as follows that for ##b- \epsilon## the following holds ##b- \epsilon < b## and ##b- \epsilon \in...
I let,
## 4\tan^{-1}\left[\dfrac{1}{5}\right]- \tan^{-1}\left[\dfrac{1}{239}\right]= \dfrac{π}{4}##
##\tan^{-1}\left[\dfrac{1}{5}\right]- \dfrac{1}{4}\tan^{-1}\left[\dfrac{1}{239}\right]= \dfrac{π}{16}##
Then i let, ##\tan^{-1}\left[\dfrac{1}{5}\right] = α ...
Have I proved Fermat last theorem?
X^4 + Y^4 != Z^4 has been proved by Fermat that if X,Y,Z = integer numbers, the formular is fine. Set x=X^2, y=Y^2, z=Z^2, so x, y, z are (some) integer numbers based on X,Y,Z.
x^4 + y^4 != z^4 //x, y, z are still integer, would be obey to Fermat's Fermat...
Ok in my approach i have,
##2 \tan^{-1} \left(\dfrac{1}{5}\right)= \sin^{-1} \left(\dfrac{3}{5}\right) - \cos^{-1} \left(\dfrac{63}{65}\right)##Consider the rhs,
Let
##\sin^{-1} \left(\dfrac{3}{5}\right)= m## then ##\tan m =\dfrac{3}{4}##
also
let
##\cos^{-1} \left(\dfrac{63}{65}\right)=...
Please check for errors my proof of P=NP:
PDF file
It is based on set theory and logic (incompleteness of ZFC). It uses also inversions of bijections, algorithms as arguments of other algorithms, reduction of SAT to another NP problem.
[Moderator's note: link removed.]
My interest is on the highlighted part ...
Now to my question,
in what cases do we have ##mn<(m,n)[m,n]?##
I was able to use my example say,
Let ##m=24## and ##n=30## for example, then
##[m,n]=120## and ##(m,n)=6## in this case we can verify that,
##720=6⋅120## implying that, ##mn≤...
i want to prove that if ##F:\mathbb{R}^n\to\mathbb{R}## is a differentiable function, then
$$F(x)=F(a)+\sum_{i=1}^n(x^i-a^i)H_i(x)$$
where ##H_i(a)=\frac{\partial F}{\partial x^i}\bigg|_{x=a}##. the hint is that with the 1-dimensional case, convert the integral into one with limits from ##0## to...
My interest is on the associative property; is there anything wrong of showing and concluding proof by;
##c(\vec u⋅\vec v)=(c⋅\vec v)⋅\vec u.##
or are we restricted in the prose?
My question is motivated by the proof of TH 5.13 on p 84 in the 2nd edition of Linear Algebra Done Right. (This proof differs from that in the 4th ed - online at: https://linear.axler.net/index.html chapter 5 )
In the proof we arrive at the following situation:
##T## is a linear operator on a...
Let ##\Lambda## be a lattice and ##a, b \in \mathbb{R}^n##, then
$$a \equiv b \text{ mod } \Lambda \Leftrightarrow a- b \in \Lambda$$
I want to prove the statement.
For the left to right direction I would say, ##a \equiv b \text{ mod } \Lambda \Leftarrow a = b +k\Lambda##, where ##k \in...
So I've thought of an admittedly crude proof that the process of pattern recognition i.e. finding patterns, be they linguistic, mathematical, artistic, whatever, is a process that can never end.
It goes like this: Imagine we find all patterns, and I mean ALL of them, and we list them on a...
in the Proof of Engel's Theorem. (3.3), p. 13:
please, how we get this step:
##L / Z(L)## evidently consists of ad-nilpotent elements and has smaller dimension than ##L##.
Using induction on ##\operatorname{dim} L##, we find that ##L / Z(L)## is nilpotent.
Thanks in advance,
Assume that players A and B play a match where the probability that A will win each point is p, for B its 1-p and a player wins when he reach 11 points by a margin of >= 2The outcome of the match is specified by $$P(y|p, A_{wins})$$
If we know that A wins, his score is specified by B's score; he...
I have been trying to understand this proof from the book 'Introduction to classical mechanics' by David Morin. This proof comes up in the first chapter of statics and is a proof for the definition of torque.
I don't understand why the assumption taken in the beginning of the proof is...
For,
Does anybody please know why they did not change the order in the second line of the proof? For example, why did they not rearrange the order to be ##M^n = (DP^{-1}P)(DP^{-1}P)(DP^{-1}P)(DP^{-1}P)---(DP^{-1}P)## for to get ##M^n = (DI)(DI)(DI)(DI)---(DI) = D^n##
Many thanks!
I know that when giving an algorithm to prove something we need to prove two things about the algorithm ( there’s another option which is to show time-complexity but that’s optional since it’s irrelevant to the proof):
1. Correctness
2. That it halts
But there are also algorithms/procedures...
Let ##\Omega## here be ##\Omega=\sqrt{-u}##, in which it is not difficult to realize that ##\Omega ## is real if ##u<0##; imaginary, if ##u>0##. Now, suppose further that ##u=(a-b)^2## with ##a<0## and ##b>0## real numbers. Bearing this in mind, I want to demonstrate that ##\Omega## is real. To...
For this
I am trying to prove that angle theta between PQ and QO is equal to theta highlighted so that I know I can use theta is the path difference formula. I assume that the rays ##r_1## and ##r_2## are parallel since ##L >> d##
My proof gives that the two thetas are equal, however I am...