Proofs Definition and 699 Threads

1. Hi! I was wondering if anyone could help me to solve the following problem! Let L : [R][n] ->[R][m] and M :[R][m]-> [R][m] be linear mappings. Prove that if M is invertible, then rank (M o L) = rank (L) thanks! :)

My prof gave us twelve basic properties of numbers, and I think I'm supposed to use those in my proof, but I'm not sure how to incorporate them. The properties are: P1 Associative law for addition P2 Additive identity P3 Additive inverse P4 Commutative law for addition P5 Associative law...

Hi, I have four similar problems that I am not sure how to do: Given: A1 and A2 are in X, B1 and B2 are in Y f: X->Y, g - inverse of f I have to either prove or if false find counterargument 1. f(A1 U A2) = f(A1) U f(A2) 2. f(A1 n A2) = f(A1) n f(A2) 3. g(-1)(B1 U B2) = g(B1) U g(B2) 4...

O-notation – O(g(n)) = { f (n) : there exist positive constants c and n0 such that 0 ≤ f (n) ≤ cg(n) for all n ≥ n0} – g(n) is an upper bound of f(n), may not be tight Ω-notation – Ω(g(n)) = { f (n) : there exist positive constants c and n0such that 0 ≤ cg(n) ≤ f(n) for all n ≥ n0}...

Here's an example that helps illustrate my question: Prove: A sequence in R can have at most one limit. Proof: Assume a sequence {xn}n\inN has two limits a and b. By definition: -For any \epsilon>0, there exists an N\inN such that n\geqN implies that |xn-a| < \epsilon/2. -A...

So how should I approach more difficult math? On a lower math level it is possible to really understand a certain property, relation, formula, ... you can see where it's coming from: intuitive and/or by quickly derivating it in your head. But as the math becomes more difficult, this...

Like many people on this forum, i am seemingly having a lot of trouble grasping the concepts of Epsilon Delta proofs and the logic behind them. I have read the definition and i realize for e>0 there is a d>0 such that... 0<sqrt((x-1)^2 - (y-b)^2) < d then f(x,y) - L <e (excuse my use of...

I really, really suck at proving things. Do you know of any free online resource one could use to master proving things? I know enough math up to about one semester of calculus, a bit about sentential logic and a little bit of set theory, but even still, past finding a trig identity or...

What are some classic, good books on learning mathematical language and writing proofs? (to gain facility with mathematical language and method of conjecture, proof and counter example, with emphasis on proofs. Topics: logic, sets, functions and others.)

Homework Statement Statements: 1. Line GB congruent to Line GD 2. Angle BGE congruent to Angle DGE 3. Line GE congruent to Line GE 4. Triangle BGE congruent to Triangle DGE Homework Equations The Attempt at a Solution...

looking for proofs of the following integrals integral ( e^(au)sin(bu)du ) and integral(e^(au)cos(bu)du) and integral (sec^3u du)

Homework Statement The maximum of 2 numbers x and y is denoted by max(x,y) and the minimum of 2 numbers x and y is denoted by min(x,y). Prove that max(x,y) = (x + y + l y - x l) / 2 and min(x,y) = (x + y - ly - xl ) / 2. Homework Equations The Attempt at a Solution Theorem...

Amateur question ahead, be warned. I really dislike proofs that something cannot be done. My first gripe is that they limit the areas we are "allowed to think about", so to speak. But more importantly, I have this feeling that any proof of impossibility is unavoidably flawed because it cannot...

formal proofs, where...? In which fields of maths are formal proofs used often?

I understand most of the logic behind the formal definition of a limit, but I don't understand the the logic behind an epsilon delta proof. The parts I'm having trouble with are these: 1. How does proving that, the distance between the function and the limit is less than epsilon whenever the...

Is there a point to geometric proofs? The ones that I have encountered are so obvious that it almost seems useless to have to use them for anything.

I've noticed in my physics textbooks that every time the author wishes to prove something, he'll go for a direct proof/derivation. Is there any particular reason for this? I think that I have yet to see any proofs by contradiction/proving the contrapositive/mathematical induction in any of my...

Does anyone know of any good precaluclus-level books that are good for learning how to write basic proofs? I have a lot of very nice 1960-early 70s era texts that ask for proofs in the majority of the exercises but the problem is that not only do I not know how to write proofs, but there are no...

1. Let O be an open cover for [a,b] and let x=sup{d|[a,d] can be covered by finitely many elements of O}. Clearly x>a. If x<b or x=b, then there is an element O_1 of O which is a neighborhood of x, and there is an \epsilon>0 such that x-\epsilon is an element of O_1, and since there is a...

I am giving a short presentation on Fermat's polygonal number theorem (any number may be written as the sum of n n-gonal numbers). I need books that provide some exposition/history on the theorem as well as a proof. I acquired Nathanson's Additive Number Theory from my university's library, but...

Homework Statement 1) (AB)^-1=A^-1B^-1 A and B are nonsingular nxn 2) If A is nonsingular then A^-1 is nonsingular also and then (A^-1)^-1=A The Attempt at a Solution 1) I do know that I have to multiply but I don´t know why. Can you tell me why?? This is how I do it...

If m and n are positive integers, (mn)!=m!n! Prove or disprove. its so obviously true i can't prove it. anyone help? -also- Prove: The square root of a prime integer is an irrational number. any help?

Homework Statement F and G and H are vectors in D-3 a and b are real numbers Proof that F+G=G+F Proof that (F+G)+H=F+(G+H) Homework Equations The Attempt at a Solution I did put F=a,b,c and G=a1,b1,c1 and H=a2,b2,c2 and put that in. I just don´t know if that´s enough...

If a|(b+c) and gcd(b,c)=1, then gcd(a,b)=1=gcd(a,c) Suppose a|(b+c) and the gcd(a,b)=d. al=b+c and d|a and d|b. This implies a=dr and b=ds. drl=ds+c => drl-ds=c => d(rl-s)=c => d|c Since d|b, d|c and 1|b and 1|c, d must divide 1. Therefore d=1. By the same reasoning gcd(a,c)=1...

Hey all, I'm an absolute noob to number theory stuff and I've got this assignment to do with a few proofs. Homework Statement Proove that: i) if gcd(a,b) = c then gcd(a,a+b) = c ii) if gcd(a,b) = c and a = a'c and b = b'c then gcd(a',b') = 1 iii) if there exists r,s such that rx...

Homework Statement Are all types of mathematical arguments based on the following types of proofs? Types of proofs 1. Direct proof, P -> Q 2. Proof by contradiction, \neg Q -> \neg P 3. ~Ad absodium, P and \neg Q -> false statement (such as 0 = 1) I know the following types of...

Algebra Proofs! I have two questions just to help verify what I'm doing: Let R be a commutative ring with identity. Prove that R is an integral domain if and only if cancellation holds in R (that is, a no equal to 0 and ab=ac in R imply b=c) => Suppose cancellation holds: ab=0 -> ab=0a...

I think college is way tooo late to learn how to write mathematical proofs! Proof writing should begin at least either in elementary school or middle school. Proof writing is just as important to a students education as learning how to write sentences and learn how to combined those sentences to...

Homework Statement Consider two events A and B such that p(A) = r and p(B) = s with r,s >0 and r + s > 1. Show that P(A|B) >or= 1- ( (1-r)/s) Homework Equations The Attempt at a Solution By definition P(A|B) = P(A &...

Hey people. I find myself getting through my course but currently with not as much understanding as I would like. We've got to some proofs and i either vaguely understand them or do not know how to prove them.Homework Statement The first would be to prove the Dimension theorem that. dimU +...

Hello all, I am stuck on some homework, basically I am stuck on the problems dealing with proofs. I am not asking for complete answers just any direction would be helpful. 1) I have to prove the Grötzsch graph is not 3-colorable (vertex can be colored in such a way that no edge shares 2...

Hi, I am doing some self-studying on the topic of functional analysis, specifically set theory at the moment. Suppose that we want to show that some set is denumerable. Is it required that we directly show the one-to-one correspondence between the elements of the set and the set of natural...

I am interesting for mathematical background od fast algorithms for computing number \pi with complete proofs only. More specific, I am interesting for Gauss Legendre algorithm, Borwein algorithm, Ramanujan formulas and Chudnovsky formula.

I don't know where to fit this question, but here goes. As an engineer should I be concerned a lot (or some) about proofs? Sure I know certain basic proofs anyone involved in math should know however, I have come across certain advanced proofs such as why certain methods of solving differential...

Two fairly simple proof problems. . . why aren't they simpler? :( Homework Statement Let A be an nxn matrix... If A is not invertible then there exists an nxn matrix B such that AB = 0, B != 0. (not equal to) Homework Equations None really. The Attempt at a Solution Obviously...

Cambridge press says available as of around 1 March, so in a couple of months. Samples are available of page proofs as they will, I gather, appear in the book. Here's the 3-page table of contents http://assets.cambridge.org/97805218/60451/toc/9780521860451_toc.pdf Here's an 11-page exerpt...

Hey all, I'm looking for a decent, (and preferably cheap) book, or books, on trigonometry. Something that proves some or all the trig equations we're expected to remember in high school stuff (most of which I've forgotten), but it should also leave room for my curiosity so I can prepare for...

Well, I have a take-home quiz and I need help with 6 geometry proofs. It is due tomorrow (Monday) and I honestly have no clue about any of it... here is the first question, please help me! Homework Statement Given: Angle ABG is congruent to Angle DEH Angle GBC is congruent to...

Got no clue .. need some clues

My situation: I did good in high-school, learned algebra, functions, exp/log functions, limits/continuity, calculus, vectors, trigonometry, diff equations of 1st and 2nd order, and perhaps a few others things I left out. Came out with an A in math and physics, so far so good. Now in college I...

Homework Statement Suppose f >= 0, f is continuous on [a,b], and {integral from a to b} f(x)dx = 0. Prove that f(x) = 0 for all x in [a,b] Homework Equations The Attempt at a Solution Suppose there exists p in [a,b] s.t. f(p) > 0. Let epsilon = f(p) / 2 > 0. The...

Hey, I'm sure this must have been asked before, but I couldn't really find anything specific using the search tool; I'm a second year maths major and I love maths and would really like to pursue a career in mathematics. My problem is, often I can understand a proof (whether easily or not...

Homework Statement question 1: Define ~ on Z by a ~ b if and only if 3a + b is multiple of 4. question 2: Let A = {1,2,3,4,5,6} and let S = P(A) (the power set of A). For a,b \in S define a ~ b if a and b have the same number of elements. Prove that ~ defines an equivalence...

Must a valid rule of inference always lead to a true conclusion?

Homework Statement 1. Prove that the sequence sqrt(n+1) - sqrt(n) converges to 0. 2. If sequence {an} is composed of real numbers and if lim as n goes to infinity of {a2n} = A and the limit as n goes to infinity of {a(2n-1)} = A, prove that {an} converges to 1. Is converse true? 3. Consider...

Let A,B,C,X,Y be subsets of E,and A' MEAN the compliment of A in E i.e A'=E-A,and A^B = A \cap B Then prove the following: a) (A^B^X)U(A^B^C^X^Y)U(A^X^A') = A^B^X b) (A^B^C)U(A' ^ B^C)U B' U C' = E Thanks

Homework Statement 1) Show by the use of vectors that the three altitudes of a triangle pass through the same point. 2) Show using vectos that the bisectors of the angles of a triangle pass through thr same point. 3)Find the distance from the point (1,0,-2) to the plane 3x-2y+z+1=0...

1) proe that for all non-negative real numbers x and y: xy(<or=)((x+y)/2)^2 2) prove that the sum of 2 prime numbers strictly greater than 2 is even 3) If n is a multiple of 3 then either n is odd or it is a multiple of six. I don't know how to start any of them. any hints would be v...

the function f(x) = 1 if x is rational f(x) = 0 if x is irrational is not continuous for all real numbers, c the function f(x) = x if x is rational f(x) = 0 if x is irrationa is continuous at x=0 and not continuous for all other real...

Geometry is arguably my weakest link in mathematics. The answers just don't "hit me" in geometry like some other sections of math do. When trying to prove something in a polygon, such as congruence of triangles made by segments etc. I find it difficult since the equal sides/angles aren't...