Problem 2(i) please! Math 310 - HW 3 (Due Wednesday 09/22)1. Let X, Y be sets with a function f: X→Y. Prove that the following are equivalent:(a) f is 1– 1.(b) f(A - B) = f(A) f(B) for all subsets A and B of X.(c) f-(f(E)) = E for all subsets E of X.(d) f(An B) = f(A) n f(B) for all subsets A and B of X.2. (i) Show that given finitely many countable sets A1, A2, , An, the set A1 x A2 x .. x An is alsocountable.(ii) Is it true that given countably many countable sets A1, A2,.., the set A1 x A2 x ... is also countable? Justify your answer.3. (i) Let F be the collection of all functions f : {0,1} → N. Is F countable ?(ii) Let F be the collection of all functions f : N {0, 1}. Is F countable ?(iii) Let F be the collection of all functions f: N → N. Is F countable ? What about f :R →R ?4. (i) Let (0, 1) be the open interval from 0 to 1. Show that |(0, 1)| = |R+| by finding an explicit bijection.(ii) [Extra-Credit] Show that |(0, 1)| = |R|, by finding an explicit bijection.(iii) Show that |P(N)| = R+| (Hint: Use (i) and binary representation of numbers in (0, 1) )5. A real number x is said to be algebraic (over Q) if it satisfies some polynomial equation ana"+an-1r"-1+...+ a1x + ao = 0, where each a; E Q, an + 0. If x is not algebraic, it is called transcendental.(i) Show that the set of all polynomials over Q (as mentioned above) is countable.(ii) Prove that there are countably many algebraic numbers. (Hint: Use (i) and here you may use theFundamental theorem of Algebra, that any polynomial has finitely many roots.)(iii) Prove that there are uncountably many transcendental numbers. (Remark: e and T are two such num-bers.)Not for Credit6. Prove that the set of points on the unit circle in R2, that is {(x, y) : x² + y? = 1} is uncountable.7. Is the subset of rational numbers {m : m, n e Z,1

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Problem 2(i) please! Math 310 - HW 3 (Due Wednesday 09/22)1. Let X, Y be sets with a function f: X→Y. Prove that the following are equivalent:(a) f is 1– 1.(b) f(A - B) = f(A) f(B) for all subsets A and B of X.(c) f-(f(E)) = E for all subsets E of X.(d) f(An B) = f(A) n f(B) for all subsets A and B of X.2. (i) Show that given finitely many countable sets A1, A2, , An, the set A1 x A2 x .. x An is alsocountable.(ii) Is it true that given countably many countable sets A1, A2,.., the set A1 x A2 x ... is also countable? Justify your answer.3. (i) Let F be the collection of all functions f : {0,1} → N. Is F countable ?(ii) Let F be the collection of all functions f : N {0, 1}. Is F countable ?(iii) Let F be the collection of all functions f: N → N. Is F countable ? What about f :R →R ?4. (i) Let (0, 1) be the open interval from 0 to 1. Show that |(0, 1)| = |R+| by finding an explicit bijection.(ii) [Extra-Credit] Show that |(0, 1)| = |R|, by finding an explicit bijection.(iii) Show that |P(N)| = R+| (Hint: Use (i) and binary representation of numbers in (0, 1) )5. A real number x is said to be algebraic (over Q) if it satisfies some polynomial equation ana&#34;+an-1r&#34;-1+...+ a1x + ao = 0, where each a; E Q, an + 0. If x is not algebraic, it is called transcendental.(i) Show that the set of all polynomials over Q (as mentioned above) is countable.(ii) Prove that there are countably many algebraic numbers. (Hint: Use (i) and here you may use theFundamental theorem of Algebra, that any polynomial has finitely many roots.)(iii) Prove that there are uncountably many transcendental numbers. (Remark: e and T are two such num-bers.)Not for Credit6. Prove that the set of points on the unit circle in R2, that is {(x, y) : x² + y? = 1} is uncountable.7. Is the subset of rational numbers {m : m, n e Z,1 < n < 100} is dense in R ?8. Prove that is N x N is countably infinite by showing that the function f : N x N → N defined byf(m, n) = 2&#34;-1(2m – 1) is a bijection.

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A set A is said to be countable if its cardinality is same as the set of natural numbers ℕ. If A and…

2. (i) Show that given finitely many countable sets A1, A2,, An, the set A1 x A2 x .. x An is also countable.