What is Mathematical proof: Definition and 61 Discussions
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.
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...
##A ∖ B## can't include any elements that are not in ##A##, so it is the same as saying ##A∖(A∩B)##; it's exactly the elements of ##A## except those in ##A∩B##.
##A∖(A∖(A∩B))## is exactly the elements of ##A## except those in (exactly the elements of ##A## except those in ##A∩B##). This is the...
Homework Statement
For all natural numbers n, prove whether the following is true or false:
n3-n is always divisible by 6.
From SQA Advanced Higher Mathematics 2006 Exam Paper
Homework Equations
I can choose from the following types of proof:
Direct proof
Proof by contradiction
Proof by...
is this a practical way of proving math theorems? i asked because when i tried, it seemed difficult for me to decide as to how exactly i should translate theorems and given statements into logical forms and since there are so many different ways, i do not know which one is correct.
For example...
hi guys, I'm supposed to write a paper and do some research on all aspects of drilling regarding torque and force need to drill an oil well and everything from start to finish and to provide mathematical proof and calculations. I don't know where to start and what are the equations used to...
Hello friends (I hope :biggrin:),
For a maths project I am working on, I need to be able to prove the equation for an elliptical orbit, related to Kepler's first law:
and p = a(1-e2) (or should be as p can be replaced by that value)
Where:
r = distance from sun to any point on the orbit
p =...
Homework Statement
I'm doing quite a strict proof in school. Where we should proof something and use mathematical language and symbols.
Homework Equations
The Attempt at a Solution
To proof what I have to proof I need to draw some help lines. As for instance the "red" one I did from A to B...
Homework Statement
Let a,b be in the positive reals. Prove a/b+b/a is >=2
Homework EquationsThe Attempt at a Solution
I have no idea. Maybe add the two ratios: (a^2+b^2)/a*b and then try to analyze separately the numerator and denominator?
Using these equations I am about to prove that photons have a rest mass of zero (mathematically)
________________________________________________________________________________________
E=hc/λ Photon Energy Equation
E2=(pc+mc2)2 Mass-Energy Equivalence with Momentum Equation
p=h/λ Momentum...
Prof. S Lakshmi Bala from Department of Physics, Madras, India writes a blackboard of equations which show how beamsplitters used alone affect the wavefunctions of input photons. It seems that it depends on the number of photons you use and in which input port to get you a different entangled...
There is an article on Wikipedia gravity assist and in the explanation of the term, author gives an analogy:
Isn't this violating the law of conservation of energy and momentum? Can someone make a mathematical proof of this? I tried, but the variables like the mass of the train and the tennis...
As I understand it, the usual method for proving uniqueness of a mathematical object (for example the identity element of a group) is to use a proof by contradiction.
Now, for example, if we have ##a## such that ##ax=b## and we want to prove this is unique, we start by assuming the contrary...
The book I am using for my Introduction to Topology course is Principles of Topology by Fred H. Croom.
Problem: Prove that if ##X=X_1\times X_2## is a product space, then the first coordinate projection is a quotient map.
What I understand:
Let ##X## be a finite product space and ##...
Homework Statement
A step in this process of proving Sn: 1+4+7+...+(3n-2) = n(3n-1)/2
confuses me. I hope someone can clarify this for me.
I do not require the work done, I need clarification on a step only. Thanks!
Homework Equations
After assuming n=k, we say Sk: 1+4+7+...+(3k-2) =...
I have a problem when trying to proof orthogonality of associated Laguerre polynomial. I substitute Rodrigue's form of associated Laguerre polynomial :
to mutual orthogonality equation :
and set, first for and second for .
But after some step, I get trouble with this stuff :
I've...
Homework Statement
Consider the continuous-time processing system in figure 1, which has two inputs and one output. The linear sub-system H is characterised by the impulse response h(t) = e −2t , where t denotes time.
Block diagram is the product of x1(t) and x2(t) going through a block step...
Hello all,
I know that perhaps this question has cropped frequently in a lot of places, but I would like to be quite specific in my case. I've just finished High School with UK A level syllabus. The syllabus inside Math A level is mostly straight solving equation type of questions, such as for...
Dear all,
I am new to this forum and this is my first post.
I hope this is the right place to open this topic.
I have a homework for my mathematical methods for physics course and i am kind of stuck into a question:
v=ψ∇β
show that ∇xv is perpendicular to v.
Here v is a vector
x is cross...
Specifically I plan on specializing as a Mechatronics engineer. I recently bought the book "Mathematical Proofs: A Transition to Advanced Mathematics" and I plan to study it on my own due to curiosity and interest towards mathematics. I would like to listen on one's opinion on how useful it...
"Mathematical Proof That The Media is Sexist and Bad at Math"
Here's is a link to my favorite comedy website: Cracked.com (The website itself may be NSFW, but this particular article doesn't get vulgar).
I thought that this was an interesting article that points out what is too obvious for the...
I need to design a voltage doubler that takes in 115V-50Hz a.c. and produces a doubled voltage output of 230V dc.
I am using this circuit:
I need help in the mathematics part:
i) How do I find the optimum value of capacitance? (Using high capacitance values reduces ripples; but can...
Hi, all
Going to take the Intro to mathematical proof class this coming quarter. It says the prerequisite is Cal III and linear algebra. I have taken those prereq years ago and forgot most of the materials. I don't know if the Intro to mathematical proof class has a lot of materials from...
It is easily derived from trigonometric identities that
cos(2*pi*f1*t)+cos(2*pi*f2*t)=2cos(2*pi*(f2+f1)*t)*cos(2*pi*(abs(f2-f1)*t))
which proves that superposition of two cosine wave will generate a beat frequency, but what about about a universal proof that applies to any kind of...
I am having difficulty with students understanding mathematical proofs. Even if they do understand them in the class they cannot reproduce them in the final examination. Majority of the students avoid doing the proof questions and concentrate on the numerical or technique application of...
I have found that for a number of students doing proofs is an unnecessary evil. For some of them it is like a painful visit to the local dentist. I thought I had explained the need for proves but it still puts a number of students off.
What motivating reasons can I give students for them to at...
"100% Mathematical Proof" vs Vellmen's "How to Prove it"
Hello, I'm looking for books that teach proof methods and techniques. I know Vellemen's book is a popular choice but a few dissenting reviews among unanimous praise, on its amazon page, caught my attention. Like this one:
And here are...
[b]1. Assume that f : A -> B and g : B -> C and that f is not one
to one. Prove that g(f) is not one to one.
[b]2. one-to-one: x1 != x2 and f(x1) != f(x2)
not one-to-one: f(x1)=f(x2) and x1 != x2
[b]3. The start of the proof should go as follows:
Assume f: A->B, g: B -> C, f is not...
I was planning to take differential equations with Linear Algebra, but there is a conflict so I can't take both.
Finally there is a course called Introduction to Linear Programming
In my Physics/Math Calendar, it doesn't say I have to take it, so I am wondering what is it really...
I am looking for a rather simple proof that if the matrix A has eigenvalues>0, then (x^t)Ax>0 for any vector x not 0.
My first tought was if the eigenvalues are bigger than 0, then (x^t)Ax=(x^t)"eigenvaulue"x="eigenvalue"(x^t)x>0, if x is nonzero and eigenvalues is bigger than 0.
Is this...
Hi,
There is a 2x2 matrix lock with one key in each cell. A key is nothing but a straight lever which can either be either in vertical or horizontal state. Please see the attached picture. The lock opens only when all the levers are horizontal. However, when you change the state of the lever...
I have only taken a few college physics courses, and I have watched some videos about string theory. I would like to get excited about M-theory, but I have always wondered, why there wouldn't be an infinite number of ways to unite the forces, in which case M-theory is just one possibility...
I have already asked the question of piston diameter size and efficiency in an earlier thread but i'll recap.
My question was that is it true that the larger diameter piston have a larger propensity to overcome the mechanical loss of fricition due to the piston rings contact with the mating...
Homework Statement
Suppose r and s are two positive real numbers. Let Dr and Ds be defined as in part 3 of Example 4.3.1. What is D_r \circ D_s? Justify your answer with a proof. (Hint: In your proof, you may find it helpful to use the triangle inequality.)
Homework Equations
Example 4.3.1...
Homework Statement
I originally made this thread for something else, but I have another problem that I need help with.
Suppose a and b are nonzero real numbers. Prove that if a < 1/a < b < 1/b then a < -1.
Homework Equations
A hint was given for the problem: Assume a < 1/a < b <...
Let's say all things are known in the Universe and magic doesn't exist, then:
K is the set of all things known and Magic, M, doesn't exist.
M \notin {K}
Let {\neg K} be all things not known.
{\neg K} \notin {K}
Since Magic can not be defined by {K} Then
M \in {\neg K} by...
I saw a rather easy proof of the Heisenberg Uncertainty Principle in a PDE textbook the other day, but I'm not sure if it's correct. The proof goes as following:
Note that \left| \int xf(x) f'(x) \right| \le \left[ \int |xf(x)|^2 dx \right]^{1/2} \left[ \int |f'(x)|^2 dx \right]^{1/2} , by the...
Hi there,
I am giving a talk to my class entitled "Beyond the standard model" - guess where I got that name from :p
And I was wondering if there are any interesting mathematical proofs I can give to a group of Year 11's (GCSE) who don't have a huge amount of mathematical knowledge. I'm going...
Can anyone tell where I can find the mathematical proof of the Big Bang theory and by whom, I'm interest in the calculation only.
If you have a scientific paper available with you please send it to me(yousifhot@hotmail.com) I really need it.
Thanks a lot,
Homework Statement
The thing is that there is a question which needs me to prove something. I have done it already but the thing that troubles me is that it wants me to prove this:
a_{1}+a_{2}+a_{3}+....a_{K}>K
a_{i} is a positive integer.
I know this is always true. but how should I...