Recent content by milena24

Well, I agree, I should be able to do a bijection as visually and logically we would expect the numbers between 9 and 10 to have the same cardinality as between 0 and 1, but I am not sure on how I would prove this... in class we taught the diagonaliation proof for reals between 0 and 1, and the...

Homework Statement The set of irrational numbers between 9 and 10 is countable. Homework Equations The Attempt at a Solution My belief is that I can prove by contradiction. first, i must prove by contradiction using diagonalization that the real numbers between 9 and 10 are...

Yeah, i wold use words, but this professor likes the function notation and writing out each step as so. so I'm going to have to do like f: (ZxZ)^2 -> N, and prove it so on... Would that be the correct start? Then i would need to proceed to defining a restriction where (ZxZ)_1 doesn't equal...

ah, i get it now! so what we're saying is that since (ZxZ)^2 is countable, and all of its subsets are countable, the subsets corresponding to the rule that (ZxZ)_{1} \neq (ZxZ)_{2} are surjective onto the set of all lines, hence the et of lines is also countable. Is that right? Those...

hmm... i see what you mean... but given the specific scenario... wouldn't it be (QxQ)x(ZxZ)? i might be missing the point of the (ZxZ)^2... are we saying the set of lines conforming to (ZxZ)^2 is a surjection on N, thus making it uncountable? I am so confused now...

Yeah, I don't want to, the problem is that I have to use notation for the class, so I am trying to write it out in a non-paragraph form, so that's why I'm a bit mized up in how to write the opening statement... the construction step

Ok i see why that line wouldn't conform... because it would take a irrational x to get an integer y... so basically, that would mean I need the slope to be a rational, also, B would have to be a rational for it to work... so basically the ordered pair must both be of the same type... is m is...

Homework Statement I have to prove the countability of the set of all lines on the Euclidean plane passing through at least two points whose coordinates are both integers.Homework Equations Proofs don't have particular equations (at least that's what my book says)The Attempt at a Solution First...