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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.
1,294 Questions
Every uniformly convergent sequence of bounded function is uniformly bounded?
The answer is yes is and only if da limit of the sequence is a bounded function.The suficiency derives directly from the definition of the uniform convergence. The necesity follows from making n tend to infinity in |fn(x)|
Let me rephrase it. You mean take a bounded subset of real numbers, S, and find a subset of all the upperbounds of S, say D, such that sup S is not in D?
If I get you right, then yes.
Take D := {a : a = sup s + n, n is natural and n < 4} so the first element is sup s + 1 > sup s, and the remaining two terms are even larger than the first one.
But I think I got you wrong, go through the Completeness Axiom.
That is for any two set A and B such that for all a in A, b in B, a <= b, and they share a maximum of one element, then there exist at least one number x such that a <= x <= b
In particular, A have a supremum, sup A <= x and B have an infimum inf B > = x
sup A <= inf B if they are equal then they must be x.
Axioms and postulates are accepted as true without proof True or false?
True. Axioms and postulates do not require proof to be used.
Can someone prove that 2k 2k 1-1?
You can prove that 2K,2K,1-1 by first determining that the integer N can be written in the form of N=2K.
Can a discontinuous function be developed in a Fourier series?
Yes, a Fourier series can be used to approximate a function with some discontinuities. This can be proved easily.
How we can convert 396 into binary number in mathematical form?
To convert any number from one base to another base, iteratively divide by the second base, using the rules of arithmetic for the first base, and write down the remainders in reverse order, until the quotient is zero. In this example, converting 39610 into 1100011002, the process is as follows...
396 / 2 = 198 remainder 0
198 / 2 = 099 remainder 0 (Keeping the leading zeros to align the columns.)
099 / 2 = 049 remainder 1
049 / 2 = 024 remainder 1
024 / 2 = 012 remainder 0
012 / 2 = 006 remainder 0
006 / 2 = 003 remainder 0
003 / 2 = 001 remainder 1
001 / 2 = 000 remainder 1 (Stopping because the quotient is now zero.)
Look at the remainders above in reverse order, and you will see 110001100.
In mathematical form, it is far more complex. You need to solve for the coefficients of a polynomial equation of variable degree, in this case...
A28 + B27 + C26 + D25 +E24 + F23 +G22 + H21 +I20 = 396
... limiting the values of the coefficients to the valid integers in the base, 0 and 1, and then writing the coefficients in a string, ABCDEFGHI, giving 110001100.
The diophantine solution of this degree of polynomial equation seems very hard, by any other method than iteration. If someone has a better method, other than the first method stated above, please feel free to enhance this answer.
You will have to bear with the angle being represented by x because this browser will not allow characters from other alphabets!
sin^2x + cos^2x = 1=> sin^2x = 1 - cos^x = (1 + cosx)(1 - cosx)
Divide both sides by sinx (assuming that sinx is not zero).
=> sinx = (1 + cosx)(1 - cosx)/sinx
Divide both sides by (1 - cosx)
=> sinx/(1 - cosx) = (1 + cosx)/sinx
=> sinx/(1 - cosx) - (1 + cosx)/sinx = 0
Is every group whose order is less than or equal to 4 a cyclic group?
Yes.
The only group of order 1 is the trivial group containing only the identity element. All groups of orders 2 or 3 are cyclic since 2 and 3 are both prime numbers. Therefore, any group of order less than or equal to four must be a cyclic group.
How do you mean the center point of the world is in chidamparam?
It is claimed that the Chidamparam temple is the center point of the Earth's "magnetic equator", the point where the magnetic field is parallel to the Earth's surface. However, as of 2010, the magnetic equator passes off the southern end of India, through Sri Lanka. The Earth's magnetic field is slowly changing, so perhaps historically, this claim may have been true. <br /><br /> The "population center" of the Earth is approximately in Kashmir, about 2600 miles away.
However, if you would rather trust the National Oceaonographic and Atmospheric Adminstration (the entity responsible for accurate magnetic readings that can be used to land an airplane in the dark), then the magnetic equator misses India entirely. A detailed map can be found at the NOAA website.
The dancing pose of Lord Shiva is already revealed the formation of
the Universe and a recent research about God particle proved it.
But a Tamil Siddhar (Sage) named 'Thirumoolar' already written this
fact in his book through his poetic lines 5000-6000 years ago.
How do you prove decreasing function theorem?
this is the increasing function theorem, hope it helps
"If F'(x) >= 0 , and all x's are and element of [a,b], Then F is increasing on [a,b]"
use Mean Value Theorem (M.V.T)
Let F'(x)>=0 on some interval
Let x1< x2 (points from that interval)
by M.V.T there is a point C which is an element of [x1,x2] such that
F(x2)-F(x1) / X2- X1 = F'(C)
this implies: F(x2)-F(x1) = F'(C) X [x2-x1]
F'(C)>=0
[x2-x1]>0
therefore: F(x2)>=F(x1)
Therefore: F is increasing on that interval.
How do you prove the diagonals of an isosceles triangle congruent?
You can't because triangles do not have diagonals but an isosceles triangle has 2 equal sides
Example of work simplification?
An example of work simplification is a farmer which purchases a tractor to plow his fields. The tractor greatly simplifies the process when compared to hand tools or horse pulled plows.
If P is a positive integer, then let 2n be the largest power of two that divides P. Then P = Q2n, where Q is the quotient of this division. Clearly Q is odd - for otherwise, 2 would divide Q, which would mean 2n + 1 also divides P, a contradiction.
Is f continuous on the interval a b if f prime is continuous on the interval a b why or why not?
Yes. You can make sense of this by referring to the definition of a derivative:
f'(x) = lim h goes to 0 of (f(x+h)-(fx))/h
As long as f(x+h) (as h goes to 0), and f(x) are defined so is f'(x). In fact, the only way f' is defined is if f(x) is defined.
How many edges must a simple graph with n vertices have in order to guarantee that it is connected?
The non-connected graph on n vertices with the most edges is a complete graph on n-1 vertices and one isolated vertex. So you must have one more than (n-1)n/2 edges to guarantee connectedness. It is easy to see that the extremal graph must be the union of two disjoint cliques (complete graphs). (Proof:In a non-connected graph with parts that are not cliques, add edges to each part until all are cliques. You will not have changed the number of parts. If there are more than two disjoint cliques, you can join cliques [add all edges between them] until there are only two.) It is straightforward to create a quadratic expression for the number of edges in two disjoint cliques (say k vertices in one clique, n-k in the other). Basic algebra will show that the maximum occurs when k=1 or n-1. (We're not allowing values outside that range.)
What is the difference between a theorem and postulate?
Postulates are assumed to be true and we need not prove them. They provide the starting point for the proof of a theorem. A theorem is a proposition that can be deduced from postulates. We make a series of logical arguments using these postulates to prove a theorem. For example, visualize two angles, two parallel lines and a single slanted line through the parallel lines. Angle one, on the top, above the first parallel line is an obtuse angle. Angle two below the second parallel line is acute. These two angles are called Exterior angles. They are proved and is therefore a theorem.
How do you make woking models of mathematics?
Make sure you follow all of the rules of the field of mathematics for which you are making the model.
How do you prove by contradiction that square root of 7 is an irrational number?
If √7 is rational, then it can be expressed by some number a/b (in lowest terms). This would mean: (a/b)² = 7. Squaring, a² / b² = 7. Multiplying by b², a² = 7b². If a and b are in lowest terms (as supposed), their squares would each have an even number of prime factors. 7b² has one more prime factor than b², meaning it would have an odd number of prime factors. Every composite has a unique prime factorization and can't have both an even and odd number of prime factors. This contradiction forces the supposition wrong, so √7 cannot be rational. It is therefore irrational.
Is there a direct way to determine the number of positive factors that a given number like 1800 has?
The easiest way is through easy websites such as this: http://wiki.answers.com/Q/What_are_the_factors_of_math
Alternatively, you can use prime factorization. In this case, 1800 = 2*2*2*3*3*5*5. Thus any divisor contains at most three factors of 2;
that means there are four possibilities: 0, 1, 2, or 3 factors of 2. Similarly there are three possibilities for the number of factors of 3 in a divisor, and the same for factors of 5. Multiplying these together, we find there are 4*3*3 = 36 positive integer divisors of 1800.
In general, take the exponents in the prime factorization, add one to each of them, and multiply the resulting numbers together.
Is it easier to prove something false than true?
Well, they are both the same hardness, because you need the same proof for both, ex. If somone says ''I am 21.'' This could be true or it could be false. You could find their birthday and see if they are right or wrong.
