# Infinite sets and Countable sets

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• Apr 5th 2012, 11:20 AM
jfk
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)$\displaystyle \mathbb{Q}\backslash\mathbb{Z}$
b)$\displaystyle \mathbb{R}\backslash\mathbb{Q}$
c)$\displaystyle \{0,1\}^\mathbb{N}$
d)$\displaystyle \mathbb{N}^{\{0,1\}}$

How can I do this?
I was wondering if the following would be correct:
for example in (a): $\displaystyle \mathbb{A}\subset\mathbb{Q}\backslash\mathbb{Z}$,
where $\displaystyle \mathbb{A}= \{ x | x = \frac{1}{n}, & \forall n\in \mathbb{N} \}$.
Thanks in advance for the help.
• Apr 5th 2012, 11:29 AM
emakarov
Re: Infinite sets and Countable sets
Quote:

Originally Posted by jfk
I was wondering if the following would be correct:
for example in (a): $\displaystyle \mathbb{A}\subset\mathbb{Q}\backslash\mathbb{Z}$,
where $\displaystyle \mathbb{A}= \{ x | x = \frac{1}{n}, & \forall n\in \mathbb{N} \}$.

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 $\displaystyle \{x\mid\exists n\in\mathbb{N}\;x=1/n\}$ or $\displaystyle \{1/n\mid n\in\mathbb{N}\}$.

Why don't you try other parts and post the results for verification?
• Apr 5th 2012, 11:35 AM
jfk
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... :)
• Apr 6th 2012, 03:51 AM
jfk
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?
• Apr 6th 2012, 04:21 AM
Plato
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)$\displaystyle \mathbb{R}\backslash\mathbb{Q}$.

Have you thought about $\displaystyle \left\{ {\frac{\pi }{{{2^n}}}:~n \in \mathbb{N}} \right\}~?$
• Apr 6th 2012, 06:48 AM
jfk
Re: Infinite sets and Countable sets
:) ouch! Nope I forgot about that. Thanks
• Apr 7th 2012, 09:54 AM
DrSteve
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?
• Apr 7th 2012, 09:42 PM
jfk
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).
• Apr 8th 2012, 01:59 AM
emakarov
Re: Infinite sets and Countable sets
• Apr 8th 2012, 05:01 AM
jfk
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?
• Apr 8th 2012, 05:27 AM
Plato
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?

$\displaystyle {\left\{ {0,1} \right\}^\mathbb{N}}$ is the set of all functions mapping $\displaystyle \mathbb{N}\to\{0,1\}$.
Think characteristic functions

$\displaystyle {\mathbb{N}^{\left\{ {0,1} \right\}}}$ is the reverse of that.
• Apr 8th 2012, 06:01 AM
jfk
Re: Infinite sets and Countable sets
Ok then,

(c) could be $\displaystyle \left\{f|f(x) = \left\{ \begin{array}{rcl}1 & \mbox{for x is Even} \\ 0 & \mbox{for x is Odd}\end{array}\right.\right\}$ ?
• Apr 8th 2012, 06:51 AM
HallsofIvy
Re: Infinite sets and Countable sets
Quote:

Originally Posted by jfk
Ok then,

(c) could be $\displaystyle \left\{f|f(x) = \left\{ \begin{array}{rcl}1 & \mbox{for x is Even} \\ 0 & \mbox{for x is Odd}\end{array}\right.\right\}$ ?

That would be a single function in the set, not a countabe collection of them.
• Apr 8th 2012, 06:55 AM
jfk
Re: Infinite sets and Countable sets
What about $\displaystyle \{f\mapsto$$f(1),f(2),f(3),...$$\}$???(Crying)
• Apr 8th 2012, 07:13 AM
Deveno
Re: Infinite sets and Countable sets
Quote:

Originally Posted by jfk
What about $\displaystyle \{f\mapsto$$f(1),f(2),f(3),...$$\}$???(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?
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