1. L

    Count members of all sets of interchanged reals

    What is the cardinality of members of unique sets representing all bijective interchanges among the real numbers?
  2. S

    Induction of reals

    Can be some cases of induction used on real numbers? Why? or Why not?
  3. L

    Limits' and reals' cardinality

    Does the limit function relate to a maximum cardinality? Does the set of real numbers, as they are an "absolute" continuum? Can one map the set of real numbers onto a finite surface? By bijection?
  4. J

    Find a basis for the vectors over the reals:

    the vectors are (1+i) (1-i) (2+3i) where the vectors are complex numbers that span the reals. Do i just have to set up a system of equations and row reduce it to find the basis vectors and find the rank of the system? Or could i just put coefficients infront of the vectors and set it equal to 0...
  5. A

    Non-computable Reals

    Question 1: Can a non-computable real $x$ be created using the following method? Let $s \in ( \,0,1) \,$ be a computable number that is irrational. A turing machine can therefore encode the binary expansion of $s$ to any desired number of bits: $$s = s_1s_2s_3s_4\dots\text{, where each }...
  6. L


    Let $R$ be a finite commutative ring with no zero divisors then A) $R$ is a field B) $R$ has unity C) Characteristic of $R$ is prime number D) None of the above Option A & B are true and I got confused with Option C. I know that every field has characteristic of either $0$ or prime and...
  7. M

    The Completeness Axiom Of the Reals

    Hi, I've been taught the completeness axiom for the real numbers in my calculus lectures, but whilst reading an analysis book I discovered an axiom with the same name, but different in its presentation: (lecture definition) every convex subset of the real numbers is an interval. (book) every...
  8. Z

    Countability of the Reals without "Infinity"

    Associate with each n-place decimal in [0,1) a natural number by removing the decimal (.035->35). Let Sn be the set of natural numbers so created. The real numbers correspond to the case S has no maximum, ie, Sn=N, the set of natural numbers. By definition, the reals are countable.
  9. M

    Reals are uncountable - without Cantor diagonal argument

    Gist of proof: length of unit interval = 1, if the reals are countable, the length = 0, contradiction. Proof: Arrange reals into countable list. Let x be an arbitrarily number > 0. Cover first point on list by interval of length x/2, second point by interval of length x/4, third point by...
  10. Z

    Power Set of the Reals is Countable

    1) Reals in [0,1) correspond to the natural numbers.* 2) Power Set of natural numbers is countable.** * **
  11. Z

    Countability of the Rationals and Reals

    List the binary fractions 0 \leq x < 1 and assign natural numbers as follows: .1 -> 1 .01 -> 2 .11 -> 3 .001 -> 4 .101 -> 5 .011 -> 6 .111 -> 7 .0001 -> 8 ........ ........ Every real number appears in the list because the list includes all possible combinations of 0,1 (ending in 1) for any...
  12. Z

    Cantors Diag Argument Proves Reals Countable

    Let T be the set of all infinite binary series. List all the elements of T and assume the list is uncountable. Use Cantor's Diagonal Argument to show there is a member of T not in the list. Contradiction. Therefore T is countable.
  13. Z

    Cantor: The reals are uncountable

    The following proof is incorrect From: -------------------------------------------------------------------------------------- In his 1891 article, Cantor considered the set T of all infinite sequences of binary digits (i.e...
  14. Z

    The reals can be placed in countable order

    This post is meant to replace the OP of which is more intuitive than precise. It also provides a precise simple clear proof to comment on. I also include the proof that the reals are countable for easy reference and...
  15. A

    Messing Around - Relating Sets of Dyadics to Reals

    I haven't seen any new threads here for a while. As always, I promise this is Fields Medal material ;), but perhaps someone will find it interesting anyways. To start, in my hopeless quest to find a discrete uniform distribution over $\mathbb{N}$, I was thinking in terms of base 2 on the...
  16. B

    Cardinality of integers equals cardinality of reals

    pi squared is an integer because its last digit in base 2 is 0. The surface hypervolumes (or hyperareas/hypervolumes on alternating indexs) are an integer multiple of eachother while having sometimes different and sometimes same exponent of pi. So theres really no important difference between a...
  17. M

    Approach reals by 3-smooth numbers

    Hello everyone, I am not a mathematician but I am looking for a theorem. I want to know if it is possible to approximate any real number by the quotient of two 3-smooth numbers. In practice that means: can any real number be approached arbitrary close by 2^m3^n where m and n are in \mathbb{Z}...
  18. mathbalarka

    Independence of Reals

    Well, I think this worths asking, that's why I thought to post it here : In a recent sequence (not yet approved) on OEIS, I defined independence of two reals by the fact that if x and y are independent over a subset of R if there is no polynomial with coefficients in the specified subset of R of...
  19. H

    Prove a set is a subring of the reals

    Prove that \{x +\sqrt{3}y : x,y \in \mathbb{Z}\} is a subring of the ring \mathbb{R} I know a nonempty subset S of a ring R is a subring of R if and only if S is closed under subtraction and multiplication. Is this all I need to show that is closure under subt. and mult.? Or do I have to...
  20. C

    Integration over the reals? finding R(dx)

    I am trying to find a function R(dx) in a paper by Rosinski "Tempering Stable Processes" which has the following theorem Theorem 2.3. The Levy measure M of a tempered alpha stable distribution can be written in the form M(A) = \int_{R^d} \hspace{7mm} \int_0^{\infty}...