Set Theory

Question

How to distinguish between countably finite , countably infinite , uncountably infinite set?

for reference see this ques:https://gateoverflow.in/36654/why-set-of-all-functions-f-n-0-1-is-uncountably-infinite

Comments

@sreshta. I answered everything about the cardinality of sets below. Now in reference to your question, there is a theorem-
Power Set of an infinite set is uncountable. 
I selected the best answer to this question. Refer the proof given in it.  
You can also refer to this and this.

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and also (a+b)* is a countably infinite set

Answer 1

Consider a set $A.$
It is-
1. Finite : Either it is empty or if there exist a bijective function $f :$ {$0,1,2...n-1$}$\rightarrow A$ where {$0,1,2...n-1$} is subset of $\mathbb{N}$. 
Suppose $A$ = {$a_1,a_2,a_3,a_4...a_n$}
We can find a mapping(not necessarily this one) like $f(A)=${$(a_1,0),(a_2,1)...., (a_n,n-1)$}. It's bijective so A is finite.

2. Infinite -  if there exists an injective but not surjective function $ f: A \rightarrow A$  i.e range of f is a proper subset of set A.
Ex. - $A=$ {$1,2,3,4....$}  and let A= x+1 where $x \epsilon A$ . It's one to one but not onto(no pre-image for 1 ) so A is infinite. 

3.Countable (Either finite or infinite) - if there exists an injective function $ f: A \rightarrow \mathbb{N} $ where $\mathbb{N}$.  is set of natural number.
Note that condition for one to one function is $|A| \leq |B|$  .Above function ensures that |A| $\leq|\mathbb{N}|$.
Or equivalently we can say that a set is countable if exists a surjective function $f :\mathbb{N}  \rightarrow A$ which ensures that set of natural numbers cover this set A entirely.

4. Countably infinite - if there exists a bijective function $ f: A \rightarrow \mathbb{N}$. It shows $|A| = |\mathbb{N}|$. So set A will be countably infinite.

5 . Uncountable-  if there doesn't exist an injective function $ f: A \rightarrow \mathbb{N}$ .
or equivalently we can say that a set is uncountable if there doesn't exist a surjective function $f :\mathbb{N} \rightarrow A$ .

Comments

Countably Finite and Finite are same things. So, Yes, First point itself is enough (just small modification to cover Empty set)

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Edited :)

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@Soumya

@Deepak

plz chk this

https://gateoverflow.in/226433/countable-uncountable-functions