Let S be the collection of lines in the Cartesian plane R2 s

Let S be the collection of lines in the Cartesian plane R^2 satisfying: The line is not vertical. The slope of the line is a natural number. The line passes through a point of the form (0, k), where k is a natural number. Determine the cardinality \\S\\ of S. (What can you say about the equation of a line with the indicated properties? Note that in this question you are counting lines - you are not counting points on lines.)

Solution

i)

Consider the line: y=mx+c

for m any real number this line is not vertical. y=mx+c is a vertical line for m=infinity.

Hence, |S|=|R|

where R is set of real numbers

ii)

y=mx+c

Here ,m is a natural but c can be any real number.

Hence again. |S|=|R|

iii)

y=mx+c where m is a natural number.

Substituting: (0,k) gives:

c=k is c is a natural number

So equation of line becomes:

y=mx+k

Hence, |S|=|N|

where N is set of natural numbers.