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# Proofs

## Proof means sufficient evidence to establish the truth of something. It is obtained from deductive reasoning, rather than from empirical arguments. Proof must show that a statement is true in all cases, without a single exception.

### Vjersha per 7 - 8 marsin?

7 Marsi Festën e mësuesit Ne do ta festojmë Lulet më të bukura Ne do ti dhurojmë. Vijmë tek ti mësues Buzëqeshjen ta dhurojmë Kënga krahët reh si flutur Midis jush jam e lumtur. ...

### Give a proof that the square root of 7 is an irrational number?

Proof by contradiction: suppose that root 7 (I'll write sqrt(7)) is a rational number, then we can write sqrt(7)=a/b where a and b are integers in their lowest form (ie they are fully cancelled). Then square both sides, you get 7=(a^2)/(b^2) rearranging gives (a^2)=7(b^2). Now consider the prime factors of a and b. Their squares have an even number of prime factors (eg. every prime factor of a is there twice in a squared). So a^2 and b^2 have an even number...

### A fair coin is continually flipped What is the probability that the first four flips are a. H-H-H-H b. T-H-H-H c. what is the probability that the pattern T-H-H-H occurs before the pattern.?

a) 1/16 b) 1/16 c) 1/256 [this answer was given, but it is unclear what part-c is even asking: The pattern occurs before what pattern? There are many variables which are unspecified and would affect the outcome.] ...

### State and prove Lusins theorem?

Lusin's theorem says that every measurable function f is a continuous function on nearly all its domain. It is given that f measurable. This tells us that it is bounded on the complement of some open set of arbitrarily small measure. Now we redefine ƒ to be 0 on this open set. If needed we can assume that ƒ is bounded and therefore integrable. Now continuous functions are dense in L1([a, b]) so there exists a sequence of continuous functions an tending...

### Gauss Problem 4 A positive integer n is such that number 2n plus 1 and 3n plus 1 are perfect squares. Prove that n is divisible by 8?

This proof uses modular arithmetic. If you are unfamiliar with this, the basic principle is that if we have integers a, b, and a nonzero integer c, then a = b (mod c) if a/c and b/c have the same remainder. For example, 8 = 2 (mod 3), because 8/3 and 2/3 have remainder 2. One property of this relation is that for any integer x and for any nonzero integer y, there exists a unique integer z such that x =...

### How do you prove the Baire Category Theorem?

The Baire Category Theorem is, in my opinion, one of the most incredible, influential, and important results from any field of mathematics, let alone topology. It is known as an existence theorem because it provides the necessary conditions to prove that certain things must exist, even if there aren't any examples of them that can be shown. The theorem was proved by René-Louis Baire in 1899 and is a necessary result to prove, amongst other things, the uniform boundedness principle and the...

### Give exampal of function continuous everywhere but not derivable any where?

I think the following piecewise function satisfies the two criteria: when x is rational: f(x)=x when x is irrational: f(x)=x*, where x* is the largest rational number smaller than x. I think not. When x is irrational, there is no largest rational number less than x. No matter what rational number you pick, there is a larger one that is less than x. For example, between 3.1415926 and pi, there is 3.14159265. The usual answer is the one given by Weierstrass, which is...

### If A is an orthogonal matrix then why is it's inverse also orthogonal?

First let's be clear on the definitions. A matrix M is orthogonal if MT=M-1 Or multiply both sides by M and you have 1) M MT=I or 2) MTM=I Where I is the identity matrix. So our definition tells us a matrix is orthogonal if its transpose equals its inverse or if the product ( left or right) of the the matrix and its transpose is the identity. Now we want to show why the inverse of an orthogonal matrix is also orthogonal. Let A be orthogonal....

### Can time be folded?

In Einstein's theory of gravitation, which is also referred to as General Relativity, space-time is warped (i.e., curved) by the presence of mass. It is not meaningful to speak of folding time alone, but rather one speaks of bending space-time. To make a significant bend in space-time, a very massive object (such as a star) or a very dense object (such as a black hole) is required. In 1915, Einstein succeeded in writing down an equation that describes how much space-time is...

AAS (apex)

### How do you prove Cayley's theorem which states that every group is isomorphic to a permutation group?

Cayley's theorem: Let (G,\$) be a group. For each g Ð„ G, let Jg be a permutation of G such that Jg(x) = g\$x J, then, is a function from g to Jg, J: g --> Jg and is an isomorphism from (G,\$) onto a permutation group on G. Proof: We already know, from another established theorem that I'm not going to prove here, that an element invertible for an associative composition is cancellable for that composition, therefore Jg is a permutation of G. Given...

### What does cc mean when measuring speed?

cubic centimeters

### What are the Math proof letters?

QED from the Latin "quod erat demonstrandum", meaning "that which was to be demonstrated", normally put at the end of a mathematical proof ...

### What is the sum of all the prime numbers less than 50?

2+3+5+7+11+13+17+19+23+29+31+37+41+43+47 = 328

### What is the proof of the ''Fundamental Theorem of Algebra''?

The Fundamental Theorem of Algebra: If P(z) = Σ­­nk=0 akzk where ak Є C, n ≥ 1, and an ≠ 0, then P(z0) = 0 for some z0 Є C. Descriptively, this says that any nonconstant polynomial over the complex number space, C, can be written as a product of linear factors. Proof: First off, we need to apply the Heine-Borel theorem to C. The Heine-Borel theorem states that if S is a closed and bounded set in an m-dimensional Euclidean space (written...

### If a 3D circle is called a Sphere what is a 3D oval called?

A three dimensional oval is simply called an egg, or more mathematically, an ovoid. A three dimensional ellipse (a more symmetric oval) is called a prolate spheroid, or oblate spheroid, depending on how the ellipse is rotated. ...
###### Asked in Math and Arithmetic, Geometry, Proofs

Answer: A regular quadrilateral is one with equal sides and equal angles, so it is a square. To negate this definition, we say an irregular quadrilateral is one where the sides are unequal or the angles are unequal OR BOTH. In simpler terms, we could say it is a quadrilateral which is not a square. ...

### How do you prove Schur's lemma?

Schur's theorem: Let (J,+) be an abelian group with more than one element, and let K be a primitive ring with endomorphisms, E. Then the centralizer, C, of K, in the ring Î¾(E), which is defined as the set of all endomorphisms of (J,+), is a subdivision of Î¾(E). Proof: First off, it needs to be stated that C is a non-zero set because it contains the identity function, I, which obviously fits the definition of a centralizer: CK(J) = {x Ð„ K: jx...

### Is the square root of 14 an irrational number?

Yes, here's the proof. Let's start out with the basic inequality 9 < 14 < 16. Now, we'll take the square root of this inequality: 3 < âˆš14 < 4. If you subtract all numbers by 3, you get: 0 < âˆš14 - 3 < 1. If âˆš14 is rational, then it can be expressed as a fraction of two integers, m/n. This next part is the only remotely tricky part of this proof, so pay attention. We're going to assume that m/n...

### Can you prove that cossquaredx - sinsquaredx equals 2cossquaredx -1?

When proving an identity, you may manipulate only one side of the equation throughout. You may not use normal algebraic techniques to manipulte both sides. Let's begin with the identity you wish to prove. cos2x - sin2x ?=? 2cos2x - 1 We know that sin2x + cos2x = 1 (Pythagorean Identity). Therefore, sin2x = 1 - cos2x. Substituting for sin2x, we may write cos2x - (1 - cos2x) ?=? 2cos2x - 1 cos2x - 1 + cos2x ?=? 2cos2x - 1 2cos2x...

### List all the factors of 45 from least to greatest?

1, 3, 5, 9, 15, 45

### In typing how are net words per minute calculated?

To determine net words a minute: Total words (5 characters = 1 word), divided by number of minutes of the timing, minus 2 for each error = nwpm So, if you typed 75 words in 3 minutes with 2 errors on the timing, you would calculate net words per minute as follows: 75 words, divided by 3 (number of minutes) = 25 gwpm (gross words per minute) minus 4 (2 errors) = 21 nwpm - (net words per minute) ...

### Please show that the absolute value of absolute value of x minus absolute value of y less than or equal to absolute value of x minus y.?

to prove x|-|y≤|x-y| have to look at 4 cases: a) x>0, y>0 b) x<0, y<0 c) x>0, y<0 d) x<0, y>0 (to save typing it out over and over again, i have shortened absolutes to abs) For: a) the values dont matter both sides will always have the same value. b) let x=-a, y=-b. Because of the abs around x and y, the left will be a-b and the right b-a so the left and right will have the same number but opposite signs, so after...

decimeter

### 814 is divisible by 4?

No, it comes out with a decimal, 814 divided by 4 equals 203.5