6. (5 pts.) A real number r is called an algebraic number if r is a...
13. An algebraic number is a real number which is the root of a polynomial co + ciz c2n in which all of the coefficients c i 1,2,.,n) are integers. The order of an algebraic number is the smallest natural number n for which z is a root of an n-th degree polynomial with integer coefficients. A real number is transcendental if it is not algebraic. a) Show that the set of algebraic numbers of order n is countable (b)...
We write R+ for the set of positive real numbers. For any positive real number e, we write (-6, 6) = {x a real number : -e < x <e}. Prove that the intersection of all such intervals is the set containing zero, n (-e, e) = {0} EER+
We write R+ for the set of positive real numbers. For any positive real number e, we write (-6, 6) = {x a real number : -e < x <e}. Prove that the intersection of all such intervals is the set containing zero, n (-e, e) = {0} EER+
The Fibonacci Sequence F1, F2, ... of
integers is defined recursively by F1=F2=1
and Fn=Fn-1+Fn-2 for each integer
. Prove
that (picture) Just the top one( not
7.23)
n 3 Chapter 7 Reviewing Proof Techniques 196 an-2 for every integer and an ao, a1, a2,... is a sequence of rational numbers such that ao = n > 2, then for every positive integer n, an- 3F nif n is even 2Fn+1 an = 2 Fn+ 1 if n is odd....
prove that the set of all algebraic number is countable
6. For a positive real number z, the difference 1.-z- is called the fractional part of r. Given arbitrary positive real numbers a and b, state a condition in terms of the fractional parts of and that is necessary and sufficient for la + bl = lal + Ibl Prove that this equation is true if and only if your condition holds. 7. Evaluate 10002+4 23k+5 k- 2 algebraically and simplify as much as possible. You ust show all steps....
Step by step explanation please.
3. If R is the set of all real numbers, use the fact that every cubic equation with real coefficients has a real root, or zero, to show that x → x3-x defines a mapping of R onto R. Is the mapping one to one?
7. Prove that for any positive real number r, if r is not an integer, then [x]+-1= 1
6) If E is any countable subset of real numbers prove that A*(E) = A*(E) = 0. 7) Show that the set of all real numbers IR is measurable with >(IR) = . 8) Prove that If f : [a, b] IR is continuous [a; b]then it is measurable [a, b]. 9) Give an example of a function f : [O, 1] IR which is measurable on [O, 1] but not continuos on [O, 1]. 10) Find the Lebesgue integral...
please post clear picture or solution.
Bonus question: 4 bonus marks] A positive integer r is called powerful if for all prime numbers P, p implies p | r. A positive integer z is called a perfect power if there exist a prime number p and a natural number n such that p". An Achilles number is one that is powerful but is not a perfect power. For example, 72 is an Achilles number. Prove that if a and b...