If the function $f: N \rightarrow N$ is defined as $f(n)=\left\{\begin{array}{ll}n-1, & \text { if } n \text { is even } \\ n+1, & \text { if } n \text { is odd }\end{array}\right.$, then

(A) f is injective
(B) f is into
(C) f is surjective
(D) f is invertible

Choose the correct answer from the options given below:

(A), (C) and (D) only

The correct answer is Option (4) →(A), (C) and (D) only

injective test

(if n → odd) ⇒ $f(x_1)=f(x_2)$

$x_1+1=x_2+1⇒x_1=x_2$

(if n → even) ⇒ $f(x_1)=f(x_2)$

$x_1-1=x_2-1⇒x_1=x_2$

surjective test

$y=\left\{\begin{matrix}x+1,&x\,odd\\x-1,&x\,even\end{matrix}\right.$

so $x=\left\{\begin{matrix}y-1,&y\,is\,even\\y+1,&y\,is\,odd\end{matrix}\right.$

for every y there exist atleast some x

f is surjective

⇒ f is bijective ⇒ f is invertible