Graphing set theory: graph {(x, x^2/y) : x in reals, y in naturals}

[hopelesscase]

Hi, I'm just beginning to learn set theory and am really struggling. Going through questions from the Book of Proofs I've gotten stuck on graphing this one.

. . .\(\displaystyle \left\{\left(x,\, \frac{x^2}{y}\right)\, :\, x\, \in\, \mathbb{R},\, y\, \in\, \mathbb{N}\right\}\)

I couldn't manage to answer the one before it either and looked at the answer and can only get a vague idea of what is happening. However this one has no answer and I'm not sure how to go out figuring out the answer without just making up points and substituting them in which seems tedious.

I also tried letting y= x^2/2 and substituting for y meaning that the new point is (x,y) but after starting to substitute some points in it seems like this can't be right either. Basically I'm very lost and help would be appreciated in what direction to go in.

 

[mmm4444bot]

… I also tried letting y = x^2/2 and substituting for y meaning that the new point is (x,y) but after starting to substitute some points in it seems like this can't be right either.

That 2 is a typo, yes?

I agree with you, that y=x^2/y means y^2=x^2, but that doesn't lead to just y=x.

What kind of values did you substitute? We're told that y is a Natural number. Write out a few lines (not so tedious). :wink:

1 = x^2/1

2 = x^2/2

3 = x^2/3

4 = x^2/4

5 = x^2/5

Does this give you an idea about plotting (x, x^2/y) points?

 

[hopelesscase]

Ah so if you make y^2-x^2=0 then it can become -(x-y)(x+y)=0 which is similar to the question after ( (y-x)(y+x) ) which is two intersecting lines. I think I'm beginning to understand better, guess I need to work on my function manipulation and understanding to have better chances in set theory. Thank you so much!


P.S yes that 2 was a mistake sorry about that!

 

[mmm4444bot]

… if you make y^2-x^2=0 then it can become -(x-y)(x+y)=0 …

Yes. This is one of the special factoring patterns; it's called a "Difference of Squares".

y^2 - x^2 = (y + x)(y - x)

Your graph ought to have the same shape as the graph of y=|x|, except that you're plotting discrete points (instead of lines) at Integer values of x. :cool: