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.

(a) Not a function, because a string can be assigned to more than one value.

(b) Function

(c) Not a function, because the function does not assign an integer to every string.

Step-by-step explanation:

Definition:

A function f from A to B has the property that has each element of A has been assigned to exactly one element of B.

Solution

            A =Set of all bits strings

              B= Z

(a) Given: f(S) ={position of a 0 bit in S}

     f is not a function,because a string can be assigned to more than one value.

     for example if S = 001, then f(S) =1 and f(S) =2,because S =001

contain a 0 in the first and second position.

Moreover there are also strings that do not get assign to an integer.For example,the string S= 111111 does not get assign by f,because S does not contain any 0's .

b) Given:f(S) ={number of 1 bits in S}

    f is defined for all strings and  the action f maps every element of A to exactly one element in Z,thus f is a function.

c)Given:

f(S) =i if the ith bit of S is the first 1 in the string

f(S)= 0 if S is the empty string

f is not a function because the string does not assign an integer to every string. More precisely,no integer are assigned to all string not containing a 1.

For example:

if S=00111,then f(S) =3.

if S=000000,then f(S) is not defined.

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