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

Let S be the collection of lines in the Cartesian plane R2 satisfying:

(i) The line is not vertical.

(ii) The slope of the line is a natural number.

(iii) The line passes through a point of the form (0, k), where k is a natural number.

Determine the cardinality |S| of S. (Hint: 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.)

Equation of the line is: y=mx+c where m is a natural number

line pass through a point: (0,k) where k is a natural number. Substituting gives:

k=c

Hence equation of line becomes: y=mx+k

where m and k can be independently varied.

Let cardinality of natural numbers be N

Hence cardinality of S, |S|=N^2=N