May 17th, 2017, 10:01 PM  #1 
Senior Member Joined: Nov 2015 From: hyderabad Posts: 232 Thanks: 2  Uncountable Subset
Which of the following is an uncountable subset of $R^2$ ? A) $\displaystyle {(x,y) \epsilon R^2 : {x \epsilon Q} { or} {(x+y ) \epsilon Q} } $ B)$\displaystyle {(x,y) \epsilon R^2 : {x \epsilon Q} and {y \epsilon Q} } $ C) $\displaystyle {(x,y) \epsilon R^2 : {x \epsilon Q} { or} {y \epsilon Q} } $ D) $\displaystyle {(x,y) \epsilon R^2 : {x \epsilon Q} { or} {y^2 \epsilon Q} } $ In my view Option D contains all irrational and rational values as y can be an irrational whose square gives us a rational number which belongs to $Q$. Please correct me If I'm wrong. Thanks 
May 18th, 2017, 12:47 AM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 1,930 Thanks: 998 
It will be (C) if $x \in \mathbb{Q}$ then $y$ is only restricted in that it is real. the same holds if $y \in \mathbb{Q}$ for $x$ The fact that $x$ or $y$ can be reals makes the pair uncountable. The reason it is not (D) is that $x \in \mathbb{Q}$ is the rationals in $x$ and $y^2 \in \mathbb{Q}$ is a subset of the algebraic numbers. Both are countable. 
May 18th, 2017, 12:59 AM  #3  
Senior Member Joined: Nov 2015 From: hyderabad Posts: 232 Thanks: 2  Quote:
and either $x$ or $y$ in option C covers the real numbers ??  
May 18th, 2017, 02:10 AM  #4 
Senior Member Joined: Sep 2015 From: USA Posts: 1,930 Thanks: 998  
May 18th, 2017, 02:55 AM  #5 
Senior Member Joined: Nov 2015 From: hyderabad Posts: 232 Thanks: 2  
May 18th, 2017, 04:06 AM  #6 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,159 Thanks: 866 
No. An "algebraic number" is a real number that is a solution to some polynomial equation with integer coefficients. A real number that is not "algebraic" is called "transcendental". All rational numbers are algebraic (since a/b satisfies bx= a). is algebraic since it satisfies . e and are transcendental.

May 19th, 2017, 03:34 AM  #7  
Math Team Joined: Jan 2015 From: Alabama Posts: 3,159 Thanks: 866  Quote:
 
May 19th, 2017, 06:00 AM  #8 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,305 Thanks: 2443 Math Focus: Mainly analysis and algebra 
$\{(x,y) \in \mathbb R^2: x \in \mathbb Q\}$ is uncountable. It contains a "copy" of $\mathbb R$ for every $x \in \mathbb Q\}$. That makes both A, C and D uncountable I think. B is countable because it is exactly $\mathbb Q^2$. 

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