Is `f:N rarr N` given by `f(x)=x^(2)`, one-one? Give reason.

Answer

To determine if the function \( f: \mathbb{N} \to \mathbb{N} \) defined by \( f(x) = x^2 \) is one-one (injective), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding One-One Function**:
A function \( f \) is said to be one-one (or injective) if for every pair of distinct elements \( a \) and \( b \) in the domain, \( f(a) \neq f(b) \). This can also be expressed as: if \( f(a) = f(b) \), then \( a = b \).

2. **Assume \( f(a) = f(b) \)**:
Let's assume that \( f(a) = f(b) \) for some \( a, b \in \mathbb{N} \). According to our function, this means:
\[ a^2 = b^2 \]

3. **Solving the Equation**:
From the equation \( a^2 = b^2 \), we can take the square root of both sides:
\[ a = b \quad \text{or} \quad a = -b \]
However, since \( a \) and \( b \) are both natural numbers (i.e., \( a, b \in \mathbb{N} \)), the case \( a = -b \) is not possible because natural numbers are positive integers (1, 2, 3, ...).

4. **Conclusion**:
Since the only possibility is \( a = b \), we conclude that if \( f(a) = f(b) \), then \( a \) must equal \( b \). Therefore, the function \( f(x) = x^2 \) is one-one.

### Final Statement:
Thus, the function \( f: \mathbb{N} \to \mathbb{N} \) defined by \( f(x) = x^2 \) is indeed a one-one function.

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