Mathematics Problem Of the Week

Fall 2006

POW #11

Circle Map

Let the set S ={(x,y)| x²+(y-1)² =1} - {(0,2)} be the set of all points on a circle of radius one, centered at (0,1), except for the top point. (as in the diagram)

Let the set R = {(x,y)| y=0) be the x-axis.

Define a function f: S→R by f(a,b)=c as in the diagram.

(1) Find c in terms of a and b.

(2) For this function to be onto, every element in R must have an element in S mapped to it. Is this function onto? Justify.

(3) For this function to be one to one, no two elements of S can map to the same element in R. Is the function 1-1? Justify.

Due Friday, December 8^th at Noon.

Solutions should be submitted to Dr. R. Lock's mailbox in the Math office or sent via e-mail to rlock@stlawu.edu.