Infinite sets and Countable sets

Hi everybody,

I need to prove the following sets are infinite sets by finding a countable in everyone of them:

a)

b)

c)

d)

How can I do this?

I was wondering if the following would be correct:

for example in (a): ,

where .

Thanks in advance for the help.

Re: Infinite sets and Countable sets

Quote:

Originally Posted by

**jfk** I was wondering if the following would be correct:

for example in (a):

,

where

.

This is almost correct except that 1 ∈ A and, depending on the definition of natural numbers, 0 may be a natural number. Also, it is more correct to write this version of A as or .

Why don't you try other parts and post the results for verification?

Re: Infinite sets and Countable sets

Thanks emakarov I just realized that 1 and 0 are problematic in my definition of A.

I'll think about (b)(c)(d) and I'll post them later... :)

Re: Infinite sets and Countable sets

For (b) I'm not sure about that one. If I take out all the rationals from the real numbers, then I'll have only the transcendentals in that remaining set and therefore it cannot be counted... Is that right?

Re: Infinite sets and Countable sets

Quote:

Originally Posted by

**jfk** to prove the following sets are infinite sets by finding a countabl(

y infinite subset) in everyone of them:

b)

.

Have you thought about

Re: Infinite sets and Countable sets

:) ouch! Nope I forgot about that. Thanks

Re: Infinite sets and Countable sets

@jfk Just a correction to something you said:

Not every irrational number is transcendental. For example square roots of non-perfect squares are irrational, but not transcendental.

Hint for (c): functions from the naturals to {0,1} are essentially just countable sequences of 0's and 1's. Can you write down infinitely many such sequences? There are many ways to do this.

Hint for (d): elements of this set are essentially ordered pairs of natural numbers. Can you write down infinitely many such pairs?

Re: Infinite sets and Countable sets

Quote:

Originally Posted by

**DrSteve** Not every irrational number is transcendental. For example square roots of non-perfect squares are irrational, but not transcendental.

10x DrSteve for the correction. Though I'm not sure I understood your example. I'm pretty confused about the relation (e.g: who contains who) between Irrational, Algebraic and Transcendental numbers, I would apreciate it very much if some one can help me make some order with those concepts(Worried).

Re: Infinite sets and Countable sets

Why don't you read Wikipedia about it?

Re: Infinite sets and Countable sets

I'm sorry I still don't understand (c) and (d), what's the meaning of a set to the power of another set?

Re: Infinite sets and Countable sets

Quote:

Originally Posted by

**jfk** I'm sorry I still don't understand (c) and (d), what's the meaning of a set to the power of another set?

is the set of all functions mapping .

Think characteristic functions

is the reverse of that.

Re: Infinite sets and Countable sets

Ok then,

(c) could be ?

Re: Infinite sets and Countable sets

Quote:

Originally Posted by

**jfk** Ok then,

(c) could be

?

That would be a **single** function in the set, not a countabe collection of them.

Re: Infinite sets and Countable sets

What about ???(Crying)

Re: Infinite sets and Countable sets

Quote:

Originally Posted by

**jfk** What about

???(Crying)

that's not good, either. what about a function f for which f(k) = 1, and f(n) = 0, if n ≠ k? how many of THOSE functions are there?