Question

a) Explain why a function\(f\)from the set of positive integers to the set of real numbers is well-defined if it is defined recursively by specifying\(f\left( 1 \right)\)and a rule for finding\(f\left( n \right)\)from\(f\left( {n - 1} \right)\).

b) Provide a recursive definition of the function\(f\left( n \right) = \left( {n + 1} \right)!\).

Short answer

a) Thus, a function\(f\)from the set of positive integers to the set of real numbers is well-defined.

b)Recursive definition\(f\left( n \right)\)is,

\(\begin{array}{c}f\left( 0 \right) = 1\\f\left( n \right) = \left( {n + 1} \right)f\left( {n - 1} \right)\end{array}\)

Step-by-step solution

Function

A function is well defined if it will give the same value when the main of the input is changed without changing the value of the input.

Recursive definition.

A recursive function is a function whose value at any point can be calculated from the values of the function at some previous points.

Step 3: \(\left( a \right)\)Specifying \(f\left( 1 \right)\)and a rule for finding \(f\left( n \right)\) from\(f\left( {n - 1} \right)\).

1) Solution.

\(f\)is well defined, because when\(n = 1\).

Basis step for\(f\left( 1 \right)\)and when\(n > 1\).

Then use the recursive definition to determine,\(f\left( n \right)\)from\(f\left( {n - 1} \right)\).

If \(f\) is not well defined the rule for finding\(f\left( 0 \right)\)from \(f\left( {n - 1} \right)\)states holds for \(n \ge 1\) is defined and \(f\left( 0 \right)\) is undefined.

Prove recursive definition\(f\left( n \right) = f\left( {n + 1} \right)!\).

1) explanation

Given in the question\(f\left( n \right) = f\left( {n + 1} \right)!\_\_\_\_\_\_\_(i)\)

If put\(n = 0\)in equation (i)

\(\begin{array}{c}f\left( 0 \right) = \left( {0 + 1} \right)!\\ = 1!\\ = 1\end{array}\)

2) Solution.

Let’s try to write\(f\left( n \right)\)in terms of\(f\left( {n - 1} \right)\).

By equation (i)

\(\begin{array}{c}f\left( n \right) = f\left( {n + 1} \right)!\\ = \left( {n + 1} \right)\left( {n!} \right)\\ = f\left( {n + 1} \right)\left( {\left( {n - 1} \right) + 1} \right)\\ = \left( {n + 1} \right)f\left( {n - 1} \right)\end{array}\)

Hence, a recursive definition\(f\left( n \right)\)is,

\(\begin{array}{c}f\left( 0 \right) = 1\\f\left( n \right) = \left( {n + 1} \right)f\left( {n - 1} \right)\end{array}\)