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.
In the wikipedia page and on every book they proof the transformation by equaling the the equivalent resistance between any pair of terminals while disconnecting the other node.https://en.wikipedia.org/wiki/Y-%CE%94_transform
Why this should make the two circuits equal? How can we apply...
Hi all,
There's this proof that I've been trying to wrap my head around but it just doesn't seem to sink in. I've attached a screenshot below. Many thanks in advance!
Consider Case 1. There is a step that goes
$$\text{Then} \ |r| = r$$
$$Then -|r| \leq |r| \ \text{and} \ r \leq |r|$$
Why is...
I'm trying to figure out how to prove this, but I'm unsure how to approach it.
Let G and H be groups, let ϕ: G → H be an isomorphism, and let ψ be the inverse function of ϕ. Prove that ψ is an isomorphism from H to G.
any help? thanks
I am trying to understand how to derive equations for the inductance of an x-conductor line. Any number really. But I want to understand the proof for a two-conductor line first. So to start any of these proofs you first need the equation for the per unit length inductance:
I know R is...
Hi,
I am questioning about this specific proof -https://quantummechanics.ucsd.edu/ph130a/130_notes/node134.html.
Why to do this proof is needed to compute the complex conjugate of the expectation value of a physical variable? Why can't we just start with < H\psi \mid \psi > ?
I'm trying to proof an identity from Munkres' Topology
A \ ( A \ B ) = B
By definition A \ B = {x : x in A and x not in B}
A \( A \ B) = A \ (A ∩ Bc) = A ∩ (A ∩ Bc)c = A ∩ (Ac ∪ B) = (A ∩ Ac) ∪ (A ∩ B) = ∅ ∪ (A ∩ B) = A ∩ B
What did I miss?
In texts treating Hilbert spaces, it's usually given as an example that "any finite dimensional unitary space is complete", but I've found no proof so far and failed prove it myself.
Homework Statement
Why does ##\lim_{n \rightarrow \infty} f(x_n) = f(c)## contradict ##\lim_{n \rightarrow \infty} \vert f(x_n) \vert = +\infty##?
edit: where ##c## is in ##[a,b]##
Homework Equations
Here's the proof I'm reading from Ross page 133.
18.1 Theorem
Let ##f## be a continuous real...
Why is the time it takes for a vertically thrown ball to reach max height the same as the time it takes for the same ball to fall from max height to ground level?
I agree with this logically but I can’t prove it mathematically...
Can you please show me the mathematical proof for this fact...
Homework Statement
Wherein α, β are strings, λ = ∅ = empty string, βr is the shortest suffix of the string β, βl is the longest prefix of the string β, and T* is the set of all strings in the Alphabet T, |α| denotes the length of a string α, and the operator ⋅ (dot) denotes concatenation of...
Homework Statement
Wherein α is a string, λ = ∅ = the empty string, and T* is the set of all strings in the Alphabet T.
Homework Equations
(exp-Recursive-Clause 1) : α0 = λ
(exp-Recursive-Clause 2) : αn+1 = (αn) ⋅ α
The Attempt at a Solution
[/B]
This one is proving difficult for me. I...
Hello everyone,
Is there a proof that takes us from the sum idea of the volume:
$$\sum_{i=1}^m \sum_{j=1}^n f(x_i,y_j) \Delta x \Delta y$$
To the integral idea:
$$\iint_R f(x,y) dxdy$$
Or something that relates the volume to the integral just like The Fundamental Theorem of Calculus?
There are plenty of proofs of Schur's lemma on the internet, but I find them hard to follow. Then I came up with my own result, but I'm not sure if it's good enough.
Consider ##A v = \kappa v## and ##A v=\kappa v ##. Operating with ##D(g)## the equation then becomes ##D(g)A v = \kappa D(g) v##...
Homework Statement
[/B]
Let X be a vector space over ##\mathbb{R}## and ## f: X \rightarrow \mathbb{R} ## be a convex function and ##g: X \rightarrow \mathbb{R}## be a concave function. Show: The set {##x \in X: f(x) \leq g(x)##} is convex.
Homework Equations
[/B]
If f is convex...
Homework Statement
For any integer n, let A(n) be the statement:
“If n 2 = 4k + 1 for some k ∈ Z, then n = 4q + 1 or 4q + 3 for some q ∈ Z.”
Use proof by contradiction to show that A(n) is true for all n ∈ Z.The Attempt at a Solution
[/B]
the answer sheet says that since n !=4q+1 and n !=...
Homework Statement
“If n = 3q + 1 or n = 3q + 2 for some q ∈ Z, then n 2 = 3t + 1 for some t ∈ Z.”
Use proof by contradiction to show that the converse of A(n) is true for all n ∈ Z.
For the proof by contradiction, on the answer sheet provided they have assumed n^2 = 3t+1 but n != 3q+1 and n...
I'm trying to prove that a^0 is = 1
So if I define a^1 to be = (a)(1)
and a^n to be = (1)(a)(a)...(a) with the product being taken n times
and a^m to be = (1)(a)(a)...(a) with the product being taken m times
a^n * a^m would then = (1)[(a)(a)...(a) with the product being taken n times * and...
Homework Statement
Use mathematical induction to prove that (8n − 7n − 1) is divisible by 49 for any n ∈ N.
Correction by mentor for better readability: ##49\,|\,(8^n-7n-1)##
The Attempt at a Solution
We can see that the base case is satisfied here:
n = 1,
8^1-7*1-1 = 0 and 49 | 0 is true...
<Moderator's note: Moved from a technical forum and thus no template.>
Not sure this should be under Linear and Abstract Algebra, but regardless I need help with a question in my mathematical proofs course.
Here it is:
Let ∼ be a relation defined on Z by x ∼ y if and only if 5 | (2x + 3y).
(a)...
I am reading Paul E. Bland's book, "Rings and Their Modules".
I am focused on Section 4.3: Modules Over Principal Ideal Domains ... and I need yet further help in order to fully understand the proof of Proposition 4.3.14 ... ...
Proposition 4.3.14 reads as follows:
In the above proof by...
Homework Statement
Let a, b be natural numbers then there exists a unique pair (q,r) that are elements of the non-negative integers such that b=aq+r and 0 is less than or equal to r which is less than a
I have a question regarding the existence part of the proof, now if I assumed a is less...
Homework Statement
Show that ##\arcsin 2x \sqrt{1-x^2} = 2 \arccos{x}## when 1/√2 < x < 1
Homework Equations
All trigonometric and inverse trigonometric identities, special usage of double angle identities here
The Attempt at a Solution
I can get the answer by puting x=cosy, the term inside...
Homework Statement
Given a continuous non-periodic function, its Fourier transform is defined as:
$$f(x) = \int_{-\infty}^\infty c(k) e^{ikx} dk, \ \ \ \ \ \ \ \ \ \ \ \ \ c(k) = \frac{1}{2\pi} \int_{-\infty}^\infty f(x) e^{-ikx} dx$$
The problem is proving this is true by evaluating the...
Homework Statement
Let G be a group. Assume a to be an element of the group. Then the set <a> = {ak I k∈ℤ} is a subgroup of G.
I am confused as to why the proof makes the assumption that <a> is a subset of the set G.
Homework EquationsThe Attempt at a Solution
The proof I think is like the...
I have a Dover edition of Louis Brand's Advanced Calculus: An Introduction to Classical Analysis. I really like this book, but find his proof of limit laws for sequences questionable. He first proves the sum of null sequences is null and that the product of a bounded sequence with a null...
I am reading Kaplansky's text on metric spaces and this part seems redundant to me. It was stated below (purple highlight) that we need to show that the convergence of ##(f(a_n))## to ##c## is independent of what sequence ##(a_n)## converges to ##b##, when trying to prove the claim ##f(b)=c##...
Homework Statement
1. Show that for all real numbers x and y:
a) |x-y| ≤ |x| + |y|
Homework Equations
Possibly -|x| ≤ x ≤ |x|,
and -|y| ≤ y ≤ |y|?
The Attempt at a Solution
I tried using a direct proof here, but I keep getting stuck, especially since this is my first time ever coming...
I was browsing through Spivak's Calculus book and found in a problem a very simple way to prove the cauchy schwarz inequality.
Basically he tells to substitute x=xᵢ/[√(x₁²+x₂²)] and similarly for y (i=1 and 2), put into x^2 + y^2 >= 2xy. Add the two cases and we get the result.
The problem is...
Hi, I’m going to be entering my first year of University this fall to study physics. In my second semester I will have to take a linear algebra course; however, my school has two different lower level linear algebra courses, and I must choose one. One course is focused more on applications of...
Proof by contradiction that cube root of 2 is irrational:
Assume cube root of 2 is equal to a/b where a, b are integers of an improper fraction in its lowest terns. So the can be even/odd, odd/even or odd/odd.
The only one that can make mathematical sense is even/odd. That is...
If I am given a function
f( x , y , z , ...) = C
then the normal direction to it is simply the (unit vector of the) divergence of the function. How has this been proven?
Hey, I have been told to study calculus following Spivak's book.
I was in an Engineering program and I have moved to a Physics one, and I want to retake calculus to really get good at it.
The problem is, Spivak's seems to me like it's very proof based, and I'm having a hard time even with the...
So I have been having a bit of trouble trying to derive the moment of inertia of a solid sphere through its center of mass. Here is my working as shown in the attached file.
The problem is, I end up getting a solution of I = (3/5)MR^2, whereas, in any textbook, it says that the inertia should...
What is the difference between Constructive Proof of existence and Existential generalization?
Logically they seem to be the same because, for a given predicate and specific member of the predicate's domain, you are concluding the general statement about the predicate.
Homework Statement
Let ##V## and ##W## be vector spaces, ##T : V \rightarrow W## a linear transformation and ##B \subset Im(T)## a subspace.
(a) Prove that ##A = T^{-1}(B)## is the only subspace of ##V## such that ##Ker(T) \subseteq A## and ##T(A) = B##
(b) Let ##C \subseteq V## be a...
I am struggling to understand the induction proof of the pigeonhole principle in my textbook. The theorem and the proof, from Biggs Discrete Mathematics, is pasted below, and I will explain further (see bold text) what I am having trouble with.
Theorem. Let m be a natural number. Then the...
Proposition(Zorn's Lemma): Let ##X\neq\emptyset## be of partial order with the property that ##\forall Y\subseteq X## such that ##Y## is of total-order then ##Y## has an upperbound, then ##X## contains a maximal element.
Proof:
Case 1: ##B\neq\emptyset## such that ##B##=##\{####b\in X##: ##b##...
Hi since U.S. education is shite, I've decided that I'm going to learn math from the ground up by myself. My goal is to reach graduate level mathematics in 2-3 years.
I'm currently reading Book of Proof, what should I read after this? My end goal is to be proficient in applied math/ physics.
In Grosso's Solid State Physics, chapter 1, page 2, The author said that:
Therefore, I plug (4) into (1), and I expect that I can get the following relationship, which proves that ##H\left|W_{k}(x)\right\rangle## belongs to the subspace ##\mathbf{S}_{k}## of plane waves of wavenumbers...
Hello all, I have only seen this paper brought up here once before based on the search function 2 years ago, and the thread devolved into something off topic within the first page.
I am asking in reference to this paper:
https://arxiv.org/pdf/1604.07422.pdf
Which claims to show that single...
Homework Statement
I am looking at the wikipedia proof of uniqueness of laurent series:
https://en.wikipedia.org/wiki/Laurent_seriesHomework Equations
look above or belowThe Attempt at a Solution
I just don't know what the indentity used before the bottom line is, I've never seen it before...
Homework Statement
Hi
I am looking at the attached proof for this property.
I agree with the first line due to periodicity, but unsure about the next- see below 3)attempt
Homework Equations
To me, I deemed the integration substituion rule as relevant to this question, but perhaps...
So I am planning on launching a Satellite to promote the Dogecoin cryptocurrency. One of the main points is printing/painting (Or whatever) the logo on the side of a metal panel. How can I make it so it doesn't melt off or turn white from radiation so quickly?
Homework Statement
The problem is question 2(a) in the attached pdf. I seem to find myself at a dead end and am not sure where to go from here - I will attach my working in a separate file, but basically I need to show that the oscillator passes/crosses over the x = 0 boundary at a positive...
I'm trying to really get a grasp on proofs of uniqueness.
Here is a model problem: Prove that ##x=-b/a## is the unique solution to ##ax+b=0##.
First method:
First we show existence of a solution: If ##x = -b/a##, then ##a(-b/a)+b = -b+b = 0##.
Now, we show uniqueness: If ##ax+b=0##, then...
Prove that if a simple graph G has 6 vertices then G or its complement has a subgraph isomorphic to ##K_3##.
The proof begins by noting that is must be the case that G or its complement as a vertex with degree at least 3. Why is this the case?
Homework Statement
Homework EquationsThe Attempt at a Solution
I don't understand why for the first part where the series goes up until arn-1, it cannot just go up until arn.. why will that first series always go up until arn-1 until it is multiplied by r?