1) (a) Showthat|(0,1)| R: x |-> tan x is a bijection. (d)Use the CantorSchroderBernstein Theorem to conclude that|RN|=|R|=c. 2)Let X be the set of all

1) (a) Showthat|(0,1)|<=|R\N|<=|R|. (b) Construct a bijection f : (0, 1) -> ( -pi/2 , -pi/2 ). (Try a linear function) (c) Show that g:(-pi/2 , -pi/2 ) -> R: x |-> tan x is a bijection. (d)Use the CantorSchroderBernstein Theorem to conclude that|R\N|=|R|=c. 2)Let X be the set of all humans.If x in X,we define the set: Ax = {people who had the same breakfast or lunch as x}. (a) Does the collection {Ax}(x in X) partition X? Explain. (b) Is your answer different if the or in the definition of Ax is changed to and? 3)Define a partition of the sphere S^2 = (x,y,z) :x2+y2+z2 =1 into subsets of the form {(x,y,z),(-x,-y,-z)} . Each subset consists of two points directly opposite each other on the sphere (antipodal points). Let ~ be the equivalence relation whose equivalence classes are the above subsets. (a) f : S^2/~ -> R : [(x, y, z)] -> xyz is not well-defined. Explain why. (b) Prove that f : S^2/~ -> R3 : [(x, y, z)] -> (yz, xz, xy) is a well-defined function