Let `f: N to N` be defined by
`f(x) = x-(1)^(x) AA x in N`, Then f is

Options

A. one-to-one but not onto
B. one-to-one and onto
C. onto but not one-to-one
D. neither onto nor one-to-one

Answer

To analyze the function \( f: \mathbb{N} \to \mathbb{N} \) defined by \( f(x) = x - 1^x \), we will determine whether it is one-to-one (injective) and onto (surjective).

### Step 1: Simplify the function
The function is given as:
\[ f(x) = x - 1^x \]
Since \( 1^x = 1 \) for any natural number \( x \), we can simplify the function:
\[ f(x) = x - 1 \]

### Step 2: Determine the domain and range
The domain of \( f \) is \( \mathbb{N} \) (natural numbers), which typically starts from 1. Thus, the values of \( x \) can be \( 1, 2, 3, \ldots \).

Now, we can calculate the values of \( f(x) \):
- For \( x = 1 \): \( f(1) = 1 - 1 = 0 \)
- For \( x = 2 \): \( f(2) = 2 - 1 = 1 \)
- For \( x = 3 \): \( f(3) = 3 - 1 = 2 \)
- For \( x = 4 \): \( f(4) = 4 - 1 = 3 \)
- And so on...

The range of \( f(x) \) is \( \{0, 1, 2, 3, \ldots\} \).

### Step 3: Check if the function is one-to-one (injective)
A function is one-to-one if different inputs produce different outputs. We can check this by assuming \( f(a) = f(b) \) for \( a, b \in \mathbb{N} \):
\[ a - 1 = b - 1 \]
This simplifies to:
\[ a = b \]
Since \( a \) must equal \( b \) for \( f(a) = f(b) \), the function is one-to-one.

### Step 4: Check if the function is onto (surjective)
A function is onto if every element in the codomain has a pre-image in the domain. The codomain of \( f \) is \( \mathbb{N} \), which includes all natural numbers \( 1, 2, 3, \ldots \).

From our earlier calculations, the range of \( f \) is \( \{0, 1, 2, 3, \ldots\} \). Notice that \( 0 \) is included in the range, but it is not a natural number according to the standard definition (which typically starts from 1). Therefore, there are no natural numbers that map to \( 0 \), meaning not every natural number has a pre-image.

### Conclusion
Since the function is one-to-one but not onto, we conclude that:
\[ \text{The function } f \text{ is one-to-one but not onto.} \]