pairwise disjoint subset question

A) Show that contains infinitely many pairwise disjoint infinite subsets. [Hint: It was shown that ~ x

what is a pairwise disjoint infinite subset, can someone provide the definition of it?

B) Prove that any infinite set can be written as the countably infinite union of pairwise disjoint infinite subsets.

Re: pairwise disjoint subset question

Re: pairwise disjoint subset question

to answer the original poster, a pair-wise disjoint collection of sets is a family of sets such that:

Re: pairwise disjoint subset question

Quote:

Originally Posted by

**Deveno** to answer the original poster, a pair-wise disjoint collection of sets is a family

of sets such that:

so is this just mean a union of infinite many mutually exclusive sets?

Re: pairwise disjoint subset question

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Originally Posted by

**Drexel28** The basic idea is as was stated to you. If

is infinite then

is equipotent to

, say that

is the bijection. We then have that

is an infinite collection of pairwise disjoint sets in

, thus

is an infinite collection of pairwise disjoint subsets of

.

this approach seems to work for both question, I do not understand the difference of part A and B? Need some explainations please!

Re: pairwise disjoint subset question

Quote:

Originally Posted by

**wopashui** this approach seems to work for both question, I do not understand the difference of part A and B? Need some explainations please!

In my example there is no need for the union to be countable. Give us an idea how to the second part, a little effort would be nice.

Re: pairwise disjoint subset question

Quote:

Originally Posted by

**Drexel28** In my example there is no need for the union to be countable. Give us an idea how to the second part, a little effort would be nice.

for B) if S is an infinite set, we have 2 cases, S can be uncountable or countably infinite, if S uncountable, this is not possible, so we need to show that S is countable

since S is infinite, then we have , f is injective, so if we show that f is also a surjection, we have f is a bijectiion, i.e. . i.e we need to show that for all there exisis

If we have f is a bijection, then by part A, we have part B

so how can I do this?

Re: pairwise disjoint subset question

Re: pairwise disjoint subset question

Hi, Plato, I don't understand what comes from, and can a uncountable set has a countable subset, so if S is uncountable, can T be countable? Do I need part A to show part B?