Thread: student in concepts of math here.

1. student in concepts of math here.

so I am an architecture student trying to get into the programming game so I'm taking a concepts of math corse at Carnegie Mellon. I'm horrible at writing proofs. I have no idea where to start even. but I get the concepts part. so I need to write a proof for why The Cartesian product of set R x N is uncountable. I think I know that it's because R is uncountable so that alone means that there is an uncountable amount elements in the set containing the solution. but how do I write the proof formally? am I even correct so far!? please help!

2. Re: student in concepts of math here.

Originally Posted by ifoundfluffy
so I need to write a proof for why The Cartesian product of set R x N is uncountable. I think I know that it's because R is uncountable so that alone means that there is an uncountable amount elements in the set containing the solution.
That's the right idea. To write a formal proof, you need to know definitions and auxiliary facts. When exactly is a set called countable? What theorems do you know about countability and cardinalities?

3. Re: student in concepts of math here.

I learned all about countable sets being plugged in to a function that outputs some N and is bijective. so I know how to do the proof if it were just R. I also know about cardinality and how for something to be bijective it must have the same cardinality.

4. Re: student in concepts of math here.

Originally Posted by ifoundfluffy
so I am an architecture student trying to get into the programming game so I'm taking a concepts of math corse at Carnegie Mellon. I'm horrible at writing proofs. I have no idea where to start even. but I get the concepts part. so I need to write a proof for why The Cartesian product of set R x N is uncountable. I think I know that it's because R is uncountable so that alone means that there is an uncountable amount elements in the set containing the solution. but how do I write the proof formally? am I even correct so far!? please help!
If a set contains an uncountable subset than the set is uncountable.
It is clear that $\{(x,1):x\in\mathbb{R}\}\subset~\mathbb{R}\times \mathbb{N}$.
Map $\{(x,1):x\in\mathbb{R}\}\to\mathbb{R}$ by $\Phi: (x,1) \mapsto x$.
It follows that $\Phi$ is a bijection so $\{(x,1):x\in\mathbb{R}\}$ is uncountable, thus so is $\mathbb{R}\times \mathbb{N}.$

5. Re: student in concepts of math here.

Originally Posted by ifoundfluffy
I learned all about countable sets being plugged in to a function that outputs some N and is bijective.
Did your textbook or lecture notes give the definition using this language?

I suggest you rigorously prove the fact that Plato used: If a set contains an uncountable subset than the set is uncountable.