Problem (a) Please count how many functions can be defined if the domain D is a finite set with the cardinality IDI = n. (b) Can you find a bijection between the set of all such functions and the power set P(D)? Solution Cardinality of D =n f is a function defined from D to [0,1] Then any element in D can be mapped onto either 0 or 1 If all elements are mapped on to 0 then we have one funciton All onto 1 another if n-1 to 0 then we have n-1 On the whole no of functions = nC0+nC1+nC2+...+nCn = 2n yes we can find a power set as no of elements i.e. n p(D) = 2n and no of functions 2n As cardinality is the same for both we can find a bijection..

1. Problem (a) Please count how many functions can be defined if the domain D is a finite set with
the cardinality IDI = n. (b) Can you find a bijection between the set of all such functions and the
power set P(D)?
Solution
Cardinality of D =n
f is a function defined from D to [0,1]
Then any element in D can be mapped onto either 0 or 1
If all elements are mapped on to 0 then we have one funciton
All onto 1 another
if n-1 to 0 then we have n-1
On the whole no of functions = nC0+nC1+nC2+...+nCn = 2n
yes we can find a power set as no of elements
i.e. n p(D) = 2n and no of functions 2n
As cardinality is the same for both we can find a bijection.