Problem 3 - Calculus 1 Course in Mathematics for the IIT JEE and Other Engineering Exams

Question

A function \(f\) from the set of natural numbers to integer defined by \(f(n)=\left\\{\begin{array}{cc}\frac{n-1}{2}, & \text { when } n \text { is odd } \\ -n / 2 & \text { when } n \text { is even }\end{array}\right.\) (a) neither one-one nor onto (b) one-one but not onto (c) onto but not one-one (d) one-one and onto both

Step-by-Step Solution

Answer

The function is one-one but not onto (option b).

Step 1: Determine Function Behavior for Odd n

To understand how the function behaves when \(n\) is odd, substitute odd values such as \(n=1,3,5\). We know odd numbers are of the form \(2k+1\). Thus, \(f(n) = \frac{n-1}{2} = \frac{2k+1-1}{2} = \frac{2k}{2} = k\). So, each odd number \(n\) maps to a distinct integer \(k\).

Step 2: Determine Function Behavior for Even n

For even \(n\), substitute even values such as \(n=2,4,6\). Even numbers are of the form \(2k\). Thus, \(f(n) = -\frac{n}{2} = -k\). So, each even number \(n\) maps to a distinct negative integer \(-k\).

Step 3: Analyze One-One Property

A function is one-one if distinct inputs map to distinct outputs. For odd \(n\), \(f(n) = k\) is always an integer based on \(k\). For even \(n\), \(f(n) = -k\) is also distinct. Thus, each distinct \(n\) gives a unique \(f(n)\), confirming the function is one-one.

Step 4: Analyze Onto Property

A function is onto if every integer has a pre-image in the natural numbers. However, consider an integer like \(-1\), which cannot be expressed as either \(k\) (for odd \(n\)) or \(-k\) (for even \(n\)). Therefore, not all integers are achievable, making the function not onto.

Step 5: Conclusion

Based on the analysis, the function \(f\) is one-one (injective) but not onto (surjective) because it cannot map to all integers.

Key Concepts

Understanding Natural Numbers

Natural numbers are the backbone of counting. They are the numbers we use daily for enumeration. Think of the sequence 1, 2, 3, and so on. These numbers are called natural because they're the most basic form of number we naturally use to count objects.

• They start from 1 and go on infinitely.
• No fractions or decimals, just whole numbers.
• Innately positive and non-negative.

In the exercise, a function is defined from the set of natural numbers, hence every input for the function is a natural number. This is crucial to determining how the function behaves since they form the domain of function in this context.

Decoding Integer Function

An integer function is a function that returns integer values, which can be both positive and negative, including zero.

• Integers are a broader set including negative numbers, zero, and positive numbers.
• When defining functions between sets, it's crucial to understand the range or set of possible outputs.

In this function analysis, the function maps natural numbers to integers, meaning every natural number produces an integer output. This includes scenarios where a natural number can be transformed into a negative integer, which happens in this function when dealing with even natural numbers.

Exploring One-One Mapping

A one-one mapping, or injective function, is when each input maps to a unique output. There's no repetition among the outputs for distinct inputs.
For the given function:

• When \( n \) is odd, it maps distinctly to \( \frac{n-1}{2} \). No two odd numbers share an output.
• When \( n \) is even, it maps distinctly to \( -\frac{n}{2} \). Similarly, no two even numbers share an output.

Together, these mappings ensure one-one behavior, and that each natural number points to a unique integer outcome.

Analyzing Function Behavior

To fully understand a function, analyzing its behavior is essential. Here's how it works for the given function:
1. **Evaluate for Odd and Even Natural Numbers:**
- For odd \( n \), the function resulted in integers such as 0, 1, 2, etc.- For even \( n \), it resulted in negative integers like -1, -2, etc.
2. **Determine If the Function is Onto:**
- An onto function covers every possible integer, but in this case, integers like -1 cannot be formed by any natural \( n \).- So, not all integers are used as outputs.
Through this behavior analysis, we discern the function is injective but not surjective, it covers unique mappings but doesn't reach every possible integer.