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Determine whether\({\bf{f}}\)is a function from the set of all bit strings to the set of integers if

a)\({\bf{f}}\left( {\bf{S}} \right)\)is the position of a\({\bf{0}}\)bit in\({\bf{S}}\).

b)\({\bf{f}}\left( {\bf{S}} \right)\)is the number of\({\bf{1}}\)bits in\({\bf{S}}\).

c)\({\bf{f}}\left( {\bf{S}} \right)\)is the smallest integer\({\bf{i}}\)such that the\({\bf{i}}\)th bit of\({\bf{S}}\)is\({\bf{1}}\)and\({\bf{f}}\left( {\bf{S}} \right) = {\bf{0}}\)when\({\bf{S}}\)is the empty string, the string with no bits.

Short Answer

Expert verified

(a)\(f\)is not described as a function as a string can have more than one number of value.

(b)\(f\)is described as a function.

(c) \(f\)is not described as a function as the function does not assign any type of integer in every string.

Step by step solution

01

Significance of the function

The function is described as the relation amongst the inputs and the outputs. Basically, it shows the relationship between the inputs and the outputs of a variable.

02

Determination of the function \({\bf{f}}\left( {\bf{S}} \right)\) in the first case

The given function \(f\left( S \right)\) is described as the position of the \(0\) bit in the sample \(S\). Here, \(f\) is an undefined function, as a string can contain more than one number of value. Taking an example, if \(S = 001\) then the function \(f\left( S \right)\)will be \(f\left( S \right) = 2\)as the string contains zero in the second and in the first position.

There are other strings which are not get assigned to a particular integer. Taking an instance, the function\(f\)does not assigns the string\(S = 111111\)as the string does not have any zeros.

Thus, \(f\)is not described as a function as a string can have more than one number of value.

03

Determination of the function \({\bf{f}}\left( {\bf{S}} \right)\) in the second case

The given function \(f\left( S \right)\) is described as the position of the \(1\) bit in the sample\(S\). Here, \(f\)is mainly defined for all of the strings and also the function \(f\)mainly maps every element of \(S\) to the element of \(Z\).

Thus, \(f\)is described as a function.

04

Determination of the function \({\bf{f}}\left( {\bf{S}} \right)\) in the third case

Here, \(f\left( S \right)\)is described as the smallest integer \(i\) if the \(i\)th bit of the string \(S\) contains the first number \(1\) in the particular string. \(f\left( S \right)\)is described as the number \(0\) if the string is empty. Here, \(f\)is not described as a function as the string no integers has been assigned to the string. For an instance, if the string is \(S = 00111\) then the function \(f\left( S \right)\)will be \(f\left( S \right) = 3\). If the string is \(S = 000000\) then the function \(f\left( S \right)\)will be undefined.

Thus, \(f\)is not described as a function as the function does not assign any type of integer in every string.

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