Either give an example or state that no example exists You d

Either give an example or state that no example exists. You don\'t need to give any proofs or justifications, but define any sets and functions very clearly and explicitly. Notation: f(A) = {f(x): x Element A}. A countably infinite set A with a countably infinite subset B such that ? notequalto B. Sets A and B and a function f: A rightarrow B such that A is uncountable. B is finite, and f(A) = B. Sets A and B and a function f: A rightarrow B such that A is uncountable, B is countably infinite, and f(A) = B. Sets A and B and a function f: A rightarrow B such that A is countably infinite. B is uncountable, and f(A) = B.

Solution

a). Take Set A to be set of Integers and set B to be set of natural numbers. Then both are countably infinite and B is a subset of A.

b). Take set A to be set of real numbers which is uncountable and take set B to be a singleton set say B = {0}. Then define function f as f(x) = 0 for every x belonging to the set A.

C). Let set A to be set of real numbers which is uncountable and set B to be set of integers which is countably infinite and define the mapping f as f(x) = [x] , where [x] denotes greatest integer function. So, the domain of this function is the set of real numbers and range is the set of integers.

d). No example exists.