Prove or disprove For any nonempty set A there is no onto fu

Prove or disprove: For any non-empty set A there is no onto function from A to (A) - the power set of A.

Solution

Solution :

Let there exist a nonempty set A.
Define the identity function i such that i(a)=a for every a in A.
By construction i is an injective function from A onto A

It occurs to me now that P(A) might represent the power set of A, which is the set of all subsets of A. In This case:

Let there exist a nonempty set A.
Since A is nonempty, it follows that the power set of A, P(A) contains at least 1 element.
In particular, for every element a in A, P(A) contains the set {a}.
Define the function f:
f: A ---> P(A)
a------>{a}.
By construction the function f is an injective function from A into P(A), which uniquely maps an element a in A to the corresponding set {a} in P(A).