Let A be an uncountable set, B a countable subset of A, and C the complement of B in A. Prove that there exists a one to one correspondence between A

Ava-May Nelson 2021-05-19 Answered
Let A be an uncountable set, B a countable subset of A, and C the complement of B in A. Prove that there exists a one to one correspondence between A and C.

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Plainmath recommends

  • Ask your own question for free.
  • Get a detailed answer even on the hardest topics.
  • Ask an expert for a step-by-step guidance to learn to do it yourself.
Ask Question

Expert Answer

Nicole Conner
Answered 2021-05-20 Author has 18995 answers

Let \(B={b0,b1,b2,...}\)
If C were finite, we would have that \(\displaystyle{A}={B}∪С\) is countable, which is a contradiction. Thus C is infinite, thus it has a countable suset \(\displaystyle{D}⊂{C}\).
Let: \(D={d0,d1,d2,...}\)
Then we can define a function f: \(A \rightarrow C\) \(\displaystyle{f{{\left({x}\right)}}}={\left\lbrace{d}{2},{x}={b};{d}{2}+{1},{x}={d};{x},{x}∈{A}{\left({B}∪{D}\right)}\right.}\)

Have a similar question?
Ask An Expert
30
 

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Relevant Questions

asked 2021-08-02
Using cardinatility of sets in discrete mathematics the value of N is real numbers
Currently using elements of discrete mathematics by Richard Hammack chapter 18
Let A be a collection of sets such that \(\displaystyle{X}\in{A}\) if and only if \(\displaystyle{X}\subset{N}\) and \(\displaystyle{\left|{X}\right|}={n}\) for some \(\displaystyle{n}\in{N}\)
Prove that \(\displaystyle{\left|{A}\right|}={\left|{N}\right|}.\)
asked 2021-08-01

Let U \(= \left\{ 1,​2, 3,​ ...,2400 ​\right\}\)
Let S be the subset of the numbers in U that are multiples of 3​, and let T be the subset of U that are multiples of 7.
Since \(\displaystyle{2400}\div{3}={800}\)​, it follows that \(n(S)=n(\left\{3 \cdot 1, 3 \cdot 2, \cdots, 3 \cdot 800\right\})=800\).
​(a) Find​ n(T) using a method similar to the one that showed that \(\displaystyle{n}{\left({S}\right)}={800}\).
(b) Find \(\displaystyle{n}{\left({S}\cap{T}\right)}\).
(c) Label the number of elements in each region of a​ two-loop Venn diagram with the universe U and subsets S and T.
Questions:Find n(T) ? Find n(SnT)

asked 2021-01-25

Let D be the diagonal subset \(\displaystyle{D}={\left\lbrace{\left({x},{x}\right)}{\mid}{x}∈{S}_{{3}}\right\rbrace}\) of the direct product \(S_3 \times S_3\). Prove that D is a subgroup of \(S_3 \times S_3\) but not a normal subgroup.

asked 2021-07-04

Let \(AX = B\) be a system of linear equations, where A is an \(m\times nm\times n\) matrix, X is an n-vector, and \(BB\) is an m-vector. Assume that there is one solution \(X=X0\). Show that every solution is of the form \(X0+Y\), where Y is a solution of the homogeneous system \(AY = 0\), and conversely any vector of the form \(X0+Y\) is a solution.

asked 2020-11-06

(i)Prove that if\({v_1,v_2}\)is linearly dependent, then are multiple of each other, that is, there exists a constant c such that \(v_1 = c v_2\ or\ v_2=cv_1\).
(ii)Prove that the converse of(i) is also true.That is to say, if there exists a constant c such that \(v_1 = c v_2\ or\ v_2 = c v_1\), 1. then\({v_1,v_2}\)is linearly dependent.

asked 2020-12-16

Let \(\times\) be a binary operation on set of rational number \(\displaystyle\mathbb{Q}\) defined as follows: \(a\cdot b=a+b+2ab\), where \(\displaystyle{a},{b}\in\mathbb{Q}\)
a) Prove that \(\times\) is commutative, associate algebraic operation on \(\displaystyle\mathbb{Q}\)

asked 2020-12-24

In there a relationship between confidence intervals and two-tailed hypothesis tests? The answer is yes. Let c be the level of confidence used to construct a confidence interval from sample data. Let * be the level of significance for a two-tailed hypothesis test. The following statement applies to hypothesis tests of the mean: For a two-tailed hypothesis test with level of significance a and null hypothesis \(H_0 : mu = k\) we reject Ho whenever k falls outside the \(c = 1 — \alpha\) confidence interval for mu based on the sample data. When A falls within the \(c = 1 — \alpha\) confidence interval. we do reject \(H_0\). For a one-tailed hypothesis test with level of significance Ho : mu = k and null hypothesiswe reject Ho whenever A falls outsidethe \(c = 1 — 2\alpha\) confidence interval for p based on the sample data. When A falls within the \(c = 1 — 2\alpha\) confidence interval, we do not reject \(H_0\). A corresponding relationship between confidence intervals and two-tailed hypothesis tests is also valid for other parameters, such as \(p,\mu_1 — \mu_2,\) and \(p_1, - p_2\). (b) Consider the hypotheses \(H_0 : p_1 — p_2 = O\) and \(H_1 : p_1 — p_2 =\) Suppose a 98% confidence interval for \(p_1 — p_2\) contains only positive numbers. Should you reject the null hypothesis when alpha = 0.05? Why or why not?

Plainmath recommends

  • Ask your own question for free.
  • Get a detailed answer even on the hardest topics.
  • Ask an expert for a step-by-step guidance to learn to do it yourself.
Ask Question
...