Question

Determine whether f is a function from the set of all bit strings to the set of integers if

a) f(S) is the position of a 0 bit in S
b) f(S) is the number of 1 bits in S
c) f(S) is the smallest integer i such that the i-th bit of S is 1 and f(S) = 0 when S is the empty string, the string with no bits

Answer 1

f is a function from the set of all bit strings to the set of integers for all three conditions.

Step-by-step explanation:

a) f is a function from the set of all bit strings to the set of integers. Since f(S) is the position of a 0 bit in S, each input string S will have a unique output position for the 0 bit. Therefore, f is a function.

b) f is also a function in this case. Since f(S) is the number of 1 bits in S, each input string S will have a unique output count of 1 bits.

c) f is still a function under this condition. Since f(S) is the smallest integer i such that the i-th bit of S is 1, each input string S will have a unique output i. Additionally, when S is the empty string, f(S) is defined as 0, which does not conflict with the definition of a function.