provide a full handwritten solution to all parts

1. Recursively defining bijections. 8) b) C) d) f) Given a bijection f : A x A l- A, use it to define another bijection g : A x A x A > A. Make sure to prove y is a bijection given that f is a bijection. Given a bijection f : A x A > A (as in part a), present a short recursive argument to prove for each ii... there is a bijection between ATl [the Cartesian product of 11 copies of the set A}, and the set A itself. Apply a problem from PS3 to conclude that any finite Cartesian product of copies of N must be of the same cardinality [see Definition 10.1.7] with N. Let AH be sets of objects that are of the same cardinality as N: that is. for each ii, there is a bijection f" : N > A". Suppose additionally that Ans are disjoint. Let A be the union of all Ans. that is. .4 = U A". Construct a bijection F : N x N > A and prove that it is a bijection. (Hint: .L' E .4 iff there is some 7:. E N such that :r: E A". ] Given an infinite countable set A. use parts c) and d} above to prove the set of all finite sequences of elements of A must also be countable. (Hint: a finite sequence of length 'l'l- would be a member of a Cartesian product of ri. copies of the set A. Also note that the textbook presents an alternative proof of this fact that is far shorter and more elegant. But we will appreciate the simplicity and elegance of that proof when we go through this standard argument.) Let P be the set of polynomials with integer coefficients. Identify each such polynomial with a finite sequence and use part e to prove that 'F' is countable