aProve that if A is uncountable and B is countable then C is

(a)Prove that if A is uncountable and B is countable then C is uncountable.

(b)Suppose that C is uncountable. What can you say about A and B (countable or uncountable)? List all possibilities.

(c)Suppose that C is countable. What can you say about A and B (countable or uncountable)? List all possibilities.

I know this question has been asked before, but the answers do not answer what I am confused about. For parts (b) and (c) I don\'t quite understand what is meant by possibilities. I know that if C is uncountable then A and B are uncountable and the same goes for if C is countable, then A and B are countable. What are the possibilities?

Solution

Part a)

Suppose that A-B is countable set. Also A=A-BUB (being union of two countable sets ) is countable thus A is a countable set. But this is not possible as A is an uncountable set.This A-B is uncountable set.That is C is uncountable set.

Part (b), it asks if C is uncountable, then what can be the possibilities on A and B. I will answer this part through a table, hopefully it will make better sense to you.

Therefore, we have two \"Possibilities\" for A and B.

Possibility1: Both A and B are uncountable

Possibility2: A is contable and B is uncountable.

Part (c):

It asks if C is contable, then what can be the possibilities on A and B. Let us see a table for this one as well:

Therefore, again we have two possibilities.

Possibility1: Set B is countable and Set A is also countable.

Possibility2: Set B is uncountable and set A is also uncountable.