Mathematical induction is a tool or technique which is used to prove a proposition about all

Options

A. (a) Number
B. (b) Integers
C. (c) Whole number
D. (d) None of these

Answer

To solve the question regarding the use of mathematical induction, we will analyze the options provided and determine the correct answer step by step.

### Step 1: Understanding Mathematical Induction
Mathematical induction is a method used to prove statements or propositions that are asserted to be true for all natural numbers. The process involves two main steps:
1. **Base Case**: Prove that the statement holds for the initial value (usually n = 1).
2. **Inductive Step**: Assume the statement is true for n = m (inductive hypothesis), and then prove it for n = m + 1.

### Step 2: Analyzing the Options
We have the following options:
- Option A: Number
- Option B: Integers
- Option C: Whole Numbers
- Option D: None of these

### Step 3: Evaluating Each Option
1. **Option A: Number** - The term "number" is too broad as it can refer to any type of number (natural, whole, integers, rational, etc.). Thus, this option is not specific enough.

2. **Option B: Integers** - Integers include negative numbers, zero, and positive numbers. However, mathematical induction typically starts from natural numbers (1, 2, 3, ...), not including negative integers. Therefore, this option is incorrect.

3. **Option C: Whole Numbers** - Whole numbers include zero and positive integers (0, 1, 2, 3, ...). Since mathematical induction usually starts from 1, this option is also not correct.

4. **Option D: None of these** - Since none of the previous options accurately describe the scope of mathematical induction, this option appears to be the most appropriate.

### Step 4: Conclusion
Based on the analysis, the correct answer is:
**Option D: None of these**