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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.

###### Asked in Algebra, Pirates, Proofs

### 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.
...

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###### Asked in Math and Arithmetic, Proofs, Irrational Numbers

### 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...

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###### Asked in Proofs, Statistics, Probability

### 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.]
...

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###### Asked in Math and Arithmetic, Proofs

### 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...

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###### Asked in Algebra, Proofs

### 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
=...

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###### Asked in Math and Arithmetic, Proofs, Topology

### 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...

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###### Asked in Calculus, Proofs

### 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...

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###### Asked in Math and Arithmetic, Algebra, Proofs, Linear Algebra

### 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....

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###### Asked in Physics, Proofs, Flag of the United States

### 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...

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###### Asked in Math and Arithmetic, Geometry, Proofs

### Which postulate or theorem can be used to prove that XYZ equals LYZ?

AAS (apex)

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###### Asked in Algebra, Proofs, Abstract Algebra

### 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...

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###### Asked in Motorcycles, Computer Terminology, Math and Arithmetic, Proofs

### What does cc mean when measuring speed?

cubic centimeters

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###### Asked in Proofs

### 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
...

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###### Asked in Proofs, Math and Arithmetic, Prime Numbers

### 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

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###### Asked in Algebra, Proofs, Abstract Algebra

### 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...

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###### Asked in Math and Arithmetic, Geometry, Proofs

### 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.
...

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###### Asked in Math and Arithmetic, Geometry, Proofs

### Definition for irregular quadrilateral?

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.
...

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###### Asked in Math and Arithmetic, Proofs, Abstract Algebra

### 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...

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###### Asked in Math and Arithmetic, Proofs, Numbers , Irrational Numbers

### 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...

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###### Asked in Math and Arithmetic, Algebra, Geometry, Proofs

### 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...

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###### Asked in Math and Arithmetic, Algebra, Proofs

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

1, 3, 5, 9, 15, 45

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###### Asked in Jobs & Education, Algebra, Keyboarding, Proofs

### 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)
...

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###### Asked in Numerical Analysis and Simulation, Math and Arithmetic, Algebra, Proofs

### 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...

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###### Asked in Math and Arithmetic, Mathematical Finance, Roman Numerals, Proofs

### What does the measurement DM stand for?

decimeter

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###### Asked in Math and Arithmetic, Algebra, Proofs, Factoring and Multiples

### 814 is divisible by 4?

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

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