# Infinite set as a countably infinite union of infinite disjoint subsets

Mathematics Asked by XuUserAC on December 19, 2020

I am struggliglng with an exercise as the title.

I have been able to get to a proof, if the set is infinitely countable, but I am having problems on infinite uncountable sets.

Any thoughts on how to think about it?

EDITED: Added the word ‘disjoint’ and "countably"

Given an infinite set $$S$$, for any element $$xin S$$ the subset $$Ssetminus{x}$$ is also infinite. By assumption, $$S$$ has infinitely many elements, therefore the union $$bigcup_{xin S}(Ssetminus{x})=S$$ is an infinite union of infinite subsets of $$S$$.

Edit:

With the added requirement of countably many disjoint sets, the following needs the axiom of choice (via the well-ordering theorem).

According to the well-ordering theorem, the set $$S$$ is in bijection with a von-Neumann ordinal. Be $$phi:alphato S$$ such a bijection.

Since you say you already have the proof for countable $$S$$, let's further assume that $$S$$ (and therefore $$alpha$$) is uncountable.

Now it is possible to write each ordinal uniquely as sum $$lambda + n$$ of a non-successor (i.e. zero or limit) ordinal $$lambda$$ and a finite ordinal $$n$$.

Define $$alpha_n = {rhoinalpha | rho = lambda + n text{ for some non-successor ordinal lambda}}$$

Clearly there are countably many such $$alpha_n$$, one for each $$ninomega$$. Moreover, quite obviously the different $$alpha_n$$ are disjoint, their union is $$alpha$$, and each of them is clearly infinite (because otherwise $$alpha$$ would be countable).

Now we get the desired union for $$S$$ by simply using the images of $$alpha_n$$ under the bijection $$phi$$: $$S = bigcup_{ninomega} {phi(x)|xin alpha_n}$$

Answered by celtschk on December 19, 2020

## Related Questions

### Why are all linear maps that sends one basis to another basis or to 0 still linear?

3  Asked on December 9, 2020

### Total variation regularization superior to classical quadratic choice

1  Asked on December 9, 2020 by pazu

### Why can we not expand $(a+b)^n$ directly when $n$ is a fractional or negative index?

1  Asked on December 8, 2020

### Let $A, B$ be skew-symmetric matrices such that $AB = -BA$. Show that $AB = 0$

1  Asked on December 8, 2020 by carlos-andres-henao-acevedo

### A question on locally integrable function on $mathbb{R}^n$

1  Asked on December 8, 2020 by kevin_h

### How to show that the integral is convergent without actually evaluating it?

4  Asked on December 8, 2020 by sandro-gakharia

### Saddle Point Approximation for Multiple Contour Integrals

0  Asked on December 8, 2020 by motherboard

### Finding minimum value of observation for a given power in hypothesis testing.

2  Asked on December 8, 2020 by hachiman-hikigaya

### What is sum of the Bernoulli numbers?

2  Asked on December 8, 2020 by zerosofthezeta

### Composition of two power series. We can compose two power series and get one power series. But why?

1  Asked on December 8, 2020 by tchappy-ha

### How much land space is required to meet the energy needs of the United States with solar?

0  Asked on December 7, 2020 by dev-dhruv

### Outer measure question (disjoint set st $lambda^*(A cup B) < lambda^*(A) + lambda^*(B)$)

1  Asked on December 7, 2020 by julian-ven

### Show that $mathbb{Z}[sqrt{3}]$ is dense in $mathbb{R}.$

1  Asked on December 7, 2020 by confusion

### Row Rank and Column vectors of Matrix

1  Asked on December 7, 2020 by james-black

### Applying the M-L estimate

1  Asked on December 7, 2020

### Stuck on Mathematical Induction Proof

2  Asked on December 7, 2020 by e__

### Depleted batteries

0  Asked on December 7, 2020 by francesco-totti

### Sphere’s surface area element using differential forms

2  Asked on December 7, 2020 by cryo

### Why the orthogonal complement of 0 is V?

2  Asked on December 6, 2020 by ivan-bravo