A function f from the set of natural numbers | vedclass

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(C) Given the function $f: \mathbb{N} 	o \mathbb{Z}$ defined as:$f(n) = \begin{cases} \frac{n-1}{2}, & 	ext{if } n 	ext{ is odd} \ -\frac{n}{2}, &

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One-one and onto both

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(C) Given the function $f: \mathbb{N} 	o \mathbb{Z}$ defined as:$f(n) = \begin{cases} \frac{n-1}{2}, & 	ext{if } n 	ext{ is odd} \ -\frac{n}{2}, & 	ext{if } n 	ext{ is even} \end{cases}$Let us calculate the images of the first few natural numbers:$f(1) = \frac{1-1}{2} = 0$$f(2) = -\frac{2}{2} = -1$$f(3) = \frac{3-1}{2} = 1$$f(4) = -\frac{4}{2} = -2$$f(5) = \frac{5-1}{2} = 2$$f(6) = -\frac{6}{2} = -3$$1$. One-one check: For every distinct $n_1, n_2 \in \mathbb{N}$,we get distinct images in $\mathbb{Z}$. Since each input maps to a unique output,the function is one-one.$2$. Onto check: The range of the function is $\{0, -1, 1, -2, 2, -3, 3, \dots\}$,which is the set of all integers $\mathbb{Z}$. Since the range equals the codomain,the function is onto.Therefore,the function is both one-one and onto (bijective).

$A$ function $f$ from the set of natural numbers $\mathbb{N}$ to the set of integers $\mathbb{Z}$ is defined by $f(n) = \begin{cases} \frac{n-1}{2}, & \text{if } n \text{ is odd} \\ -\frac{n}{2}, & \text{if } n \text{ is even} \end{cases}$. The function $f$ is:

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