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.

Re: Infinite sets and Countable sets

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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?

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

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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 $\displaystyle \left\{ {\frac{\pi }{{{2^n}}}:~n \in \mathbb{N}} \right\}~?$

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

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**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

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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.

Re: Infinite sets and Countable sets

Ok then,

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

Re: Infinite sets and Countable sets

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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.

Re: Infinite sets and Countable sets

What about $\displaystyle \{f\mapsto\(f(1),f(2),f(3),...\)\}$???(Crying)

Re: Infinite sets and Countable sets

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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?