Let $X$ be a set with exactly $5$ elements and $Y$ be a set with exactly $7$ elements. If $\alpha$ is the number of one-one functions from $X$ to $Y$ and $\beta$ is the number of onto functions from $Y$ to $X$,then the value of $\frac{1}{5!}(\beta-\alpha)$ is.

Given below are two statements:
Statement $I$: The function $f:R \rightarrow R$ defined by $f(x) = \frac{x}{1+|x|}$ is one-one.
Statement $II$: The function $f:R \rightarrow R$ defined by $f(x) = \frac{x^{2}+4x-30}{x^{2}-8x+18}$ is many-one.
In the light of the above statements,choose the correct answer from the options given below:

Let $f, g: N \rightarrow N$ such that $f(n+1)=f(n)+f(1)$ for all $n \in N$ and $g$ be any arbitrary function. Which of the following statements is $NOT$ true?

Let $f: R \rightarrow R$ be defined as $f(x)=x^{4}$. Choose the correct answer.

Let $R = \{ a, b, c, d, e \}$ and $S = \{1, 2, 3, 4\}$. The total number of onto functions $f: R \rightarrow S$ such that $f(a) \neq 1$ is equal to $.............$.

If $f(x) = \sin([\pi^2]x) - \sin([-\pi^2]x)$,where $[x]$ denotes the greatest integer function $\leq x$,then which of the following is not true?