Question

Use mathematical induction to prove that a function F defined by specifying F (0) and a rule for obtaining F (n + 1) from F (n) is well defined.

Short answer

The function F is well-defined.

Step-by-step solution

Introduction

Mathematical Induction is a mathematical technique which is used to prove a statement, a formula or a theorem is true for every natural number. Step 1(Base step) − It proves that a statement is true for the initial value.

Solution

We define a function F (0) such that is specified.

Using mathematics induction, let us prove that F (n + 1) can be obtain from F (n) is well defined.

We assume that T (n) be the statement that function F well defined at n .

Basis step: n = 0

We know that F = (0) is specified.

Thus, T (0) is true.

From induction hypothesis, we assume that T (n) is true.

We now have to prove that T (n + 1) is also true.

Induction step:

As we know that T (n) is true, therefore, function F is well-defined at n.

Since, F (n + 1) can be obtained from .

Thus, is also well-defined at n + 1 , that is, T (n + 1) is true.

Using mathematical induction, T (n) is true for all n

Thus, we define F as a function on the set of all non-negative integers.

Hence, the function F is well-defined