Cardinality Proof Problems Cardinality Proof Problems (1) Suppose f: S right arrow T, and that f is one to one. Prove that S ~ f(S). (2) Show that R~(a,b) for every a Solution f is a function from S to T f is given to be one to one i.e. each element in f has a unique image in T and also if f(a) = f(b) then a = b Hence if n is the no of elements in S, f(S) will have image for each of the n element. Hence n[f(S)] = n(S) S~f(S) ------------------------------------------------------------------------------------------------------- 2) b) Consider the function f(x) = sinx, where 0<=x<=pi/2 sin 0 =0 and sin pi/2 =1 Also sinx is one to one Hence (0,pi/2)~(0,1) c) Consider the interval (-pi/2, pi/2) as domain for the function f(x) = tan x As from -pi/2 to 0 tan is negative, we see that f is one to one. Hence f(x) = (-tan pi/2, tanpi/2) = (-infinity, infinity) Or (-pi/2, pi/2)~R.