I am trying to prove whether or not the set of all straightlines in the Cartesia
ID: 2940040 • Letter: I
Question
I am trying to prove whether or not the set of all straightlines in the Cartesian plane, each of which passes through theorigin, is a countable set. I'm sure that it isuncountable. I have that any line would be of the for L = mx, where m isthe slope. Furthermore, it would pass through a point (a,b)in the plane, where a and b are real numbers. So every line could be defined as the unique coordinate (a, b,m), where a,b,m are all real numbers. So (a,b,m) is in RXRXR (three-space). But I'm not surehow to prove that this is uncountable. Any help? I am trying to prove whether or not the set of all straightlines in the Cartesian plane, each of which passes through theorigin, is a countable set. I'm sure that it isuncountable. I have that any line would be of the for L = mx, where m isthe slope. Furthermore, it would pass through a point (a,b)in the plane, where a and b are real numbers. So every line could be defined as the unique coordinate (a, b,m), where a,b,m are all real numbers. So (a,b,m) is in RXRXR (three-space). But I'm not surehow to prove that this is uncountable. Any help?Explanation / Answer
I don't think you need all of the a,b stuff. Basically, this set is defined by the slope, which can be any realnumber, although I guess you also have to take into account thevertical line passing through the origin. Basically, (if you don't count the vertical line through theorigin), you have a one-to-one correspondence between this set andthe set of all real numbers which are uncountable. Adding inanother element (the vertical line), still means it isuncountable.