I am currently taking analysis for math major student, I have some problems with proves related to Cardinality of Sets, would you please help me with it? thank you

3. A real number x E R is called algebraic if there exist integers a0, a1, . . . , an E Z, not all zero, such that and\" + an_1m\"_1 + ' ' ' + ala: + a0 = 0. Said another way, a real number is algebraic if it is the root of a polynomial with integer coeicients. Real numbers that are not algebraic are called transcendental numbers. (a) Show that x/Z W, and \\/ + x/g are algebraic. (b) Prove that the set of all algebraic numbers is countable. What may we conclude from this regarding the set of all transcendental numbers? Hint: First show that the set of all polynomials with integer coeicients of degree n is countable. 4. (a) Let C Q [0, 1] be uncountable. Show that there exists a E (0, 1) Such that CF] [a, 1] is uncountable. (b) Now let A be the set of all a E (0, 1) such that 0 n [a, 1] is uncountable, and set as = sup A. Is C H [(3, 1] uncountable? (c) Does the statement in (a) remain true if \"uncountable\" is replaced with \"innite\"? 5. (a) Let A be a given set and P(A) denote the power set of A, namely the collection of all subsets of A. Prove that there does not exist a function f : A > P(A) that is onto. Hint: Assume that such a function does exist and arrive at a contradiction by considering the set B={aEA:af(a)}. (b) Prove that the set of all innite subsets of N is uncountable. Hint: Show directly that the set of all finite subsets of N is countable