1. ## Proofs

Hi friends. I have some exercises. If you can help me to solve them I will be greatfull.
1. To prove that every subset from a countable set is countable
2. To prove the following senstences:
a. A is denumerable
b. Exists the function f: A--> N injective
c. Exists the function f ´ :N--> A onto
Thanks Friends

2. Originally Posted by user
Hi friends. I have some exercises. If you can help me to solve them I will be greatfull.
1. To prove that every subset from a countable set is countable
2. To prove the following senstences:
a. A is denumerable
b. Exists the function f: A--> N injective
c. Exists the function f ´ :N--> A onto
Thanks Friends
How does your text define finite and denumerable?

Can you show some work?

3. Originally Posted by user
Hi friends. I have some exercises. If you can help me to solve them I will be greatfull.
1. To prove that every subset from a countable set is countable
2. To prove the following senstences:
a. A is denumerable
b. Exists the function f: A--> N injective
c. Exists the function f ´ :N--> A onto
Thanks Friends
Originally Posted by Plato
How does your text define finite and denumerable?

Can you show some work?
I agree with Plato, some work would be nice and the terms finite and deumerable are a tad sketchy.

For though notice that for some countable set $\mathcal{T}$ there exists some mapping $f:\mathcal{T}\mapsto\mathbb{N}$ such that $f$ is bijective. Thus for some subset $\mathcal{I}\subseteq\mathcal{T}$ clearly the restriction $f_{\mathcal{I}}:\mathcal{I}\mapsto\mathbb{N}$ given by $f_{\mathcal{I}}(n)=f(n)\quad\forall n\in\mathbb{I}$ is an injection....soo

4. Originally Posted by Drexel28
IFor though notice that for some countable set $\mathcal{T}$ there exists some mapping $f:\mathcal{T}\mapsto\mathbb{N}$ such that $f$ is bijective.
I am sure that some author somewhere has done that.
However, most often $f:\mathcal{T}\mapsto\mathbb{N}$ such that $f$ is bijective means the set $\mathcal{T}$ is deumerable.
Countable sets are finite or deumerable (countablely infinite).

5. Originally Posted by Plato
I am sure that some author somewhere has done that.
However, most often $f:\mathcal{T}\mapsto\mathbb{N}$ such that $f$ is bijective means the set $\mathcal{T}$ is deumerable.
Countable sets are finite or deumerable (countablely infinite).
That's true. In that case, replace bijection with injection.