Solve the given simultaneous equation

[malawi_glenn]

Since both sets are aleph one, it is possible to construct a bijection between them.

I was thinking about that too. I recalled a problem in an real analysis book of mine which I did not read that long time ago which asked how many algebraic vs. trancendetal numbers (over the rationals) there are. If I remember correctly, there should be countable many ($\aleph_0$) algebraic numbers and uncountable ($\aleph_1$) trancendental numbers... I should revisit my notes at some points

 

[WWGD]

Hmm, can we say that the probability a random polynomial to have only real roots is 50%? Hard though to imagine how we define this probability as if we try to define it as a fraction, both the numerator and the denominator are uncountable many...

I guess we'd have to choose a distribution to determine probabilities.

 

[neilparker62]

Homework Statement:: Solve for $x$ and $y$ given;
$\sqrt x+ y=7$
$x+\sqrt y=11$
Relevant Equations:: Simultaneous equations

Presuming this is a Diophantine (integer only) equation , we don't have too many choices:

1 + 6
2 + 5
3 + 4

One number has to be a perfect square. Exclude 2 + 5. If x=1, y=6 and that doesn't work either.