Proofs Definition and 671 Threads

Homework Statement I need to prove two things about the Catalan numbers. The first is that Cn is odd iff n=(2^k)-1 for some positive integer k. The second is that given the matrix A defined by the rule a(i,j)=C(i+j), prove that det A=1. I have not covered determinants in my linear class...

Homework Statement ......A..... ....../\..... ....../..\... ...../...\.... ..../...\... .../...\... .../...\... ...../...\..... ..../...\.../.\.... .../...\.../...\... .../...\..F./...\... .../...5..x..6...\... .../.../...\...\... .../.../...\...\.. .../.../....\...\...

Homework Statement The problem comes with a diagram but I'll use the wikipedia diagram because it's nice and pretty and I'll just rearrange the letters to suit it. http://upload.wikimedia.org/wikipedia/commons/9/9d/Circle-trig6.svg Just in case the image doesn't load in the page...

Homework Statement 1. A function f(x) is said to be monotonic increasing in A if for all x1, x2 ∈ A, x1≤x2 implies f(x1)≤f(x2). Prove that if f(x) is monotonic increasing in R [f: R→R] and c is a cluster point of R then the limit of f(x) as x→c^{-} exists (might be +∞). 2. s(δ) =...

[SOLVED] Some Compostion Proofs Homework Statement Prove: 1.) The composition of subjective functions is subjective 2.) The composition of injective functions is injective Homework Equations Subjective: A function f: A->B is surjective iff For all members of B, there exists a...

Hi there everyone, I have the basic idea of what to do, its just trying to show the cases work is where the problems occurs. Anyways for the first one: Homework Statement Prove that if x is any positive integer, then ⌈x/2⌉ ≤ (x + 1)/2. (Here, for any real number r, ⌈r⌉ is the smallest...

hello, i am having some trouble with a few induction problems. 1) Prove by induction that for all natural numbers n, n^2 + 3 < 2^n + 5. 2) Prove by induction that for all natural numbers n, (1-1/2)(1-1/4)...(1-1/(2^n)) > or equal to 1/4 + 1/(2^(n+1)). i got started on these but ran into...

Homework Statement Prove 1^3 + 2^3 + ... + n^3 = (1 + 2 + ... + n)^2 for all natural numbers n. Homework Equations The Attempt at a Solution Well, this seems like the typical induction proof, so I start by testing the hypothesis at 1: 1^3 = 1^2 = 1. Then I assume that the...

More Trig Identity Proofs ... Homework Statement 1. cot^2x - 1 = cot2x ----------- 2cotx2. tanx + cotx = 2csc2x3. cos(A+B) = 1-tanAtanB --------- ---------- cos(A-B) 1+tanAtanB Homework Equations The Attempt at a Solution...

im just starting to write proofs and it's going well but some things aren't immediately obvious to me. for example it is not immediately obvious to me why \forall_i ~ p_i \vee q_i \Leftrightarrow (\forall_i p_i ) \vee (\forall_i q_i) isn't a tautology and it wasn't immediately obvious...

Hey guys, I've got two sets of questions here both requiring proofs. Here is a little progress I made with Question Two part c) f=({a,1},{b,1},{c,2},{d,2}) , A={a,c} & B={b,d} f(A)/f(B) = emptyset & f(A/B)={1,2} Any help with the other two parts to Question Two/Three would be great...

Question 1 Let u, v1,v2 ... vn be vectors in R^{n}. Show that if u is orthogonal to v1,v2 ...vn then u is orthogonal to every vector in span{v1,v2...vn} My attempt if u is orthogonal to v1,v2 ...vn then (u.v1)+(u.v2)+...+(u.vn)=0 Let w be a vector in span{v1,v2...vn} therefore...

Homework Statement Question 1: A) Show that if A is diagonalizable then A^{T} is also diagonalizable. The Attempt at a Solution We know that A is diagonalizable if it's similar to a diagonal matrix. So A=PDP^{-1} A^{T}=(PDP^{-1})^{T} which gives A^{T}=(P^{-1})^{T}DP^{T} as...

Question 1: Proove that if λ is an eigenvalue of [A], then 1/λ is an eigenvalue of [A]{T} Question 2 Proove that a square matrices [A] and [A]T have the same Eigenvalues. Question 3: Show that |det(A)| is the product of the absolute values of the eigenvalues of [A]...

I am looking for a method to semi-automate mathematical proofs. Precisely, what I would like is that, if for example I define (sorry, I have never used latex in this forum, and I still do not know how to do it yet): f[A] := {y|\existsx\inA (y=f(x))} then, if I have f[A\cupB] what I...

given x<y and x,y,z are elements of R prove there exists at least one z such that x<z<y. proof: x<z<y -> z>x and y>z by the fact that the reals are unbounded there is definitely at least one z such that z>x now either z>y,z<y, or z=y by the order axioms. so... do i just let z<y...

im just starting to work through vol 1 of apostol and these questions are kind of dumbfounding? #3 on page 15 Let A={1}, B={1,2} Discuss the validity of the following statements (prove the ones that are true) (a) A\subset B (b) A\subseteq B (c) A \in B (d) 1 \in A (e) 1 \subseteq B (f) 1...

What does QED stand for behind every mathematical proof? i can't seem to find out what it stands for thanks.

http://www.geocities.com/asdfasdf23135/advcal15.JPG Well...I have no idea about this question. I don't even know where to start, and I am having terrible panic on these types of proof. Can someone please explain and guide me through? Believe it or not, my textbook (which is horrible) has...

Homework Statement L is the midpoint of line JN, line PJ congruent line QN, line PL congrent to LINe ql, angel pkj and angle omn are ryte angels. prove: triangle PKJ congruent to TRiangle QMN Homework Equations it mite be line segment, because it has a line on top of it.. no arrows...

Homework Statement prove: sin3x = sinx (3-4sin^2x) tanx+sinx/2tanx = cos^2(x/2) cot2x = (cot^2 x-1)/(2cotx)

We all know the standard proof that the square root of two is irrational, and it's easily extended to all integers that are not perfect squares, but It just striked me yesterday that I have only seen one proof (which really is enough, but still =]). One of the lecturers at the University of...

say function f is continuous on (-\infty,\infty). show that f can be written as f = g + h, where g is an even function and h is an odd function. help pleaseee!

All my classes are just profs doing proofs. Great. Too bad the tests requires us to use what we prove to calculate something. For example, my prof spent the first two lectures proving cross and dot product identities (the ones found on the inside covers of many math or physics books). Why...

Is there any experiment that shows the increase in mass ( or kinetic energy ) of a moving body when seen from an observer at rest ? I know that sincrotons ( particles accelerators ) must change the frequency bla bla .. But once that these particles have been accelerated and hit a target...

(a)Let u be a nonzero vector in R^{n}. For all v\epsilonR^{n}, show that proj_{u}(proj_{u}(v)) = proj_{u}(v) and proj_{u}(v - proj_{u}(v)) = \vec{0} (b) An alternate proof of the Cauchy-Schwarz inequality. For v,w \epsilonR^{n}, consider the function q: R -> R defined by q(t) =...

Homework Statement Proof 1: Show that S= {v1, v2, ... vp} is a linearly independent set iff Ax = 0 has only the trivial solution, where the columns of A are composed of the vectors in S. Be sure to state the relationship of the vector x to the vectors in S 2. The attempt at a solution As far...

Homework Statement I'm currently in first year linear algebra... I'm doing quite well, there's just one area of trouble-- proofs. For example: Suppose u.v = u.w, does it follow that v = w? Prove your generalization. Prove that u is orthogonal to v - proju(v) for all vectors u and v in R^n...

Hello. I have an upcoming exam for my math course and I am aware that much of it will revolve around Epsilon-Delta proofs. My understanding of them is good enough to prove most limits, but I would be more comfortable being able to answer anything that is thrown at me on this test :confused:. I...

First, thanks for all the help so far everyone! vectors a and b exist in the x,y plane and make angles (alpha) and (beta) with x. (Ill use A as alpha and B as beta) prove: cos (A-B) = cos(A)cos(B)+sin(A)sin(B) prove: sin (A-B) = sin(A)cos(B) - cos(A)sin(B) I think there is some...

Hello all, I'm having a hard time trying to prove a few things. I'm looking for a little help because I cannot seem to grasp the concept of proofs and what constitutes a valid proof and if my proof is wrong, correcting it. I have a proof done and if anyone could "critique" it I would be...

...involving Matrix Multiplication... I think it is mainly the notation that is killing me here...but it is killing me. Problem: Check parts (2) and (3) of theorem (1.3.18) which says: 1. A(BC)=(AB)C 2. A(B+C)=AB+AC 3. (A+B)C=AC+BC The author led the way on part one with this proof: Let AB=D...

Homework Statement Prove or give a counterexample to each statement. (S ∩ T) ∪ U = S ∩ (T ∪ U) The Attempt at a Solution If I proved by the contrapositive S (T ∩ U) ≠ (S ∩ T) ∪ U where would I go from there. How do I find the contrapositive with the unions and...

I have no luck with proofs... Prove that B_{r} ((x_{0}, y_{0})) = {(x,y) : || (x,y) - (x_{0}, y_{0})|| < r} is an open set in R. Now I know that to be an open set if and only if each of its points is an interior point and if it contains no boundary points. I would consider trying to prove...

It's hard to find the proofs of these theorems. Please help me... Thanks! Theorem 1: Let V be a vector space over GF(q). If dim(V)=k, then V has \frac{1}{k!} \prod^{k-1}_{i=0} (q^{k}-q^{i}) different bases. Theorem 2: Let S be a subset of F^{n}_{q}, then we have dim(<S>)+dim(S^{\bot})=n.

Can the epsilon associated with f(x) be a function of x? i.e epsilon = delta * (x-2)^2 valid?

I'm planning on taking a limits and infinite series course soon and was wondering what book(s) I could get off Amazon that would make the process a little less painless when it comes to proofs? Quantifiers, those sort of things, I have no idea about any of it and I'm taking a pre-limits class...

im looking for a book that shows the fundamentals of proofs cause I am starting and i want to start with the basics. thanks

L1-, L2-, Linfty-Norm Proofs - Please Help! Homework Statement Show that ||x||1 < or = n||x||infinity and ||x||2 < or = sqrt(n)*||x||infinity for x exists in the set of all real numbers. Homework Equations ||x||2 is defined here: http://mathworld.wolfram.com/L2-Norm.html ||x||1 is...

1) Prove that f defined by f(x)= e^(-1/|x|), x=/=0, f(x)= 0, x=0 is differentiable at 0. [I used the definition of derivative f'(0)=lim [f(0+h)-f(0)] / h = lim [e^(-1/|h|) / h] h->0 h->0 and I am stuck here and unable to proceed...] 2) Suppose lim...

I just stare at difficult proofs. I truly do not understand induction. Like if I was to prove Fermat's Little Theorem, I wouldn't know where to start. And I have my final exam next week and I don't know how to study since its all proofs. And if you say do a lot of problems , what happens if I'm...

Below is a list of notes on mathematical proofs. The notes are directed at beginners who want to learn how to write mathematical proofs.PROOF TECHNIQUES 1) Introduction to mathematical arguments (by Michael Hutchings) http://math.berkeley.edu/~hutching/teach/113/proofs.pdf 2) How to Write...

i need to prove that if f and and g are analytic functions in (-a,a) then so is fg. well basically i need to find the radius of convergence of fg, which its coefficient is: c_n=\sum_{i=0}^{n} b_i*a_{n-i}, by using cauchy hadamard theorom for finding the radius of convergence, and to show that...

Homework Statement Let D be a subset of C and D is open. Suppose a is in D and f:D\{a} -> C is analytic and injective. Prove the following statements: a) f has in a, a non-essential singularity. b) If f has a pole in a, then it is a pole of order 1. c) If f has a removable singularity...

Homework Statement Prove or disprove the following: (A is a nxn square matrix) a) The vector b is in R^n and all its elements are even integers. If all the elements of the A are integers and det(A) = 2, then the equation Ax = b has a solution with only integer elements b) If n is odd and...

When solving a problem, the last thing you want to do is look at the solution. When you're trying to prove a theorem, axiom or whatever, is looking at a proof something that would impede your learning? To me it seems that the answer is yes. Looking at a proof removes the thinking process so...

Homework Statement Prove that if n is an odd positive integer, then n^2\,\equiv\,1\,\left(mod\,8\right). Homework Equations Theorem: a\,\equiv\,b\left(mod\,m\right) if and only if a\,mod\,m\,=\,b\,mod\,m The Attempt at a Solution Using the theorem above: a\,=\,n^2 b\,=\,1,\,m\,=\,8...

How do I go about finding alternative proofs? I wrote an alternative proof to a theorem including its converse, so I'd like to publish it if it does not yet exist. So far, I just looked into 10 different textbooks that had the theorem. I don't really know much else to do. So far so good...

The following theorems are usually left unproved in calculus, for no good reason. See what you think. 2250: Elementary proofs of big theorems The first theoretical result is the Intermediate Value Theorem (IVT) for continuous functions on an interval. Theorem: If f is continuous on then...

Any suggestion on how to improve your reading fluency with proofs of theorems? It's frustrating to spend over 1 hour to read a proof of a theorem that is under 1 page long (or not understanding the proof altogether). Even when every subtopic within a proof is already known, I find that...