`The Schroeder-Bernstein theorem states the following: If A and B are any sets with the property that there is a one-to-one function from B to A, then A and B have the same cardinality. Use this theorem to prove that there are as many functions from
To prove:
There are as many functions from
Given information:
Concept used:
A function is said to be one-to-one function if the distinct elements in domain must be mapped with distinct elements in co-domain.
A function is onto function if each element in co-domain is mapped with atleast one element in domain.
Proof:
Consider the Schroeder-Bernstein theorem states that:
Let
The object is use the Schroeder-Bernstein theorem to prove that there are many functions from
Consider
Consider
Let us show that the cardinality of
Let
Let us write a number
For example, if the function
Thus, for any