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The following is a recursive definition of positive real numbers from book "Computer Theory" by I. Cohen.
1 is in positive R
If x and y are in R, then so x+y, xy, and x/y
but the author said that
it does define some set, but it is not the set of positive real numbers
What does it mean as all the positive numbers are in the set defined by the above definition?
Those are all rational operations, so that set is not the positive real numbers because it doesn't include any positive irrational numbers (e.g. sqrt(2)).
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I am studying "FAMILIES OF SETS" in the book "Real analysis for graduate
students(v 3.1)" by Richard F. Bass and I could not figure out this example.
The example
Definition of an algebra and sigma-algebra
They are stating that
Verifying parts (1) and (2) of the definition is easy.
This is exactly the part I do not understand.
I do not understand how we define the complement for a set {0,1,2}. The set {0,1,2} should be in D, as it is countable, but what is its complement? It seems that it is {...,-3-2-1} union {3,4,5,...}. Are these sets both countable?
And what about the set {1.1, 2.5, 3.4}, how do we define the complement of such a set? (and how do we show that it is in fact in D?)
P.S.
I do not know how to write formulas so I'm sorry for the ugly mathematical writing
The complement of {0,1,2} in R is every real number except those three. It's also in the algebra because that was the definition, you defined an algebra of all countable subsets or the complements of countable subsets.
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Suppose the lifetime of an bulb can be modeled with an exponential distribution with parameter 1.
What is the expected value of a bulb’s remaining life if it has already survived 2 hours?
Exponential distribution is memoryless. Therefore, the time that has passed so far is irrelevant, and the expected value of the bulb’s remaining life is 1 (as the expected value of exponential distribution with parameter c is 1/c).
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I want to understand the physical significance of n raise to some decimal power.
Like when i say 2^5. I understand that it means 2 multiplied 5 times. But how do i analyse 2^0.1.
Please suggest.
2^0.1 is the tenth root of 2. For rational powers, x^(p/q)=(x^p)^(1/q) is a combination of powers and roots.
For general real numbers,
x^y = exp(log(x)*y).
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An input sequence is given. Each stage of the iteration finds another sequence by calculating difference between n-i and n-i-1 number. We continue the process and at the end of the last iteration (iteration: n-1) we find only 1 number. What is the mathematical formulation for finding the last number as shown in the image?
Basically, the mathematical formulation is finding the n-1'th derivative of the degree-n-1 polynomial passing through all points (i,arr[i]). That derivative is guaranteed to be a constant. This is equivalent to the coefficient of the term with exponent n-1, divided by (n-1)!.
This method is a special case of what is known as Neville's Algorithm.
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In formal, does there exist such that for all ?
No, $\pi$ and thus $\pi/2$ are irrational, thus the (additive) equivalence classes of the integers modulo $2\pi$ are dense in $\Bbb R$ and thus approach infinitesimally, but never reach $\pi/2$.
The fundamental fact is that for any given number x the set of numbers {mx+n : m,n integer} is either
an arithmetic sequence {mr : r integer} which implies and is equivalent to x as a multiple of r being rational, or
dense in the real numbers, which by the first case happens for all irrational x.