How to prove

[muiz205]

Suppose that S and T are sets and that [MATH] T\subseteq S[/MATH].
(a) If S is a countable set, then T is a countable set.
(b) If T is an uncountable set, then S is an uncountable set.

 

[muiz205]

Please also give an example for me to understand. Thanks.

 

[pka]

Suppose that S and T are sets and that [MATH] T\subseteq S[/MATH].
(a) If S is a countable set, then T is a countable set.
(b) If T is an uncountable set, then S is an uncountable set.

One huge problem with this is the matter of variation on definitions.
In the text material that I have written a set \(S\) is countably infinite if it is equipotent with \(\mathbb{Z}^+\)
Thus If there is an injection \(F:S\to \mathbb{Z}^+\) and \(T\subseteq S\) then there is an injection \(G: T\to \mathbb{Z}^+\).
Now please do not panic. I fully realize that may well be so much gibberish to you.
I put it up to show why one cannot ask for help with a set theory question without including all the definitions & theorems in use with the problem. So now you need to tell us what definitions are used in your course and what theorems you can use.

 

[muiz205]

 

[pka]

Well, thank you for those notes. They are fully consistent with those that I wrote.
Moreover, they go beyond what I posted here! So my question to you is: what is your problem?
It seems to me that you have everything you need to answer the questions that you posted.
Is that not the case? If not then post a more detailed question.

 

[muiz205]

i can't prove b.. in theorem 1.3.9.

 

[Dr.Peterson]

i can't prove b.. in theorem 1.3.9.

Do you see that (b) is the contrapositive of (a)? Then if you accept their proof of (a), there is nothing more to say.