please give the reason why every notation for DFT is valid?

$X$ represents sample in frequency domain and $x$ represents samples in time domain.

NOTATION 1

$ X[k] = \sum\limits_{n=0}^{N-1} x[n] \ e^{-j \frac{2\pi}{N} n k} $

$ x[n] = \frac{1}{N} \sum\limits_{k=0}^{N-1} X[k] \ e^{j \frac{2\pi}{N} n k} $

NOTATION 2

$ X[k] = \frac{1}{N} \sum\limits_{n=0}^{N-1} x[n] \ e^{-j \frac{2\pi}{N} n k} $

$ x[n] = \sum\limits_{k=0}^{N-1} X[k] \ e^{j \frac{2\pi}{N} n k} $

NOTATION 3

$ X[k] = \frac{1}{\sqrt{N}} \sum\limits_{n=0}^{N-1} x[n] \ e^{-j \frac{2\pi}{N} n k} $

$ x[n] = \frac{1}{\sqrt{N}} \sum\limits_{k=0}^{N-1} X[k] \ e^{j \frac{2\pi}{N} n k} $

NOTATION 4

$ X[k] = \sum\limits_{n=0}^{N-1} x[n] \ e^{j \frac{2\pi}{N} n k} $

$ x[n] = \frac{1}{N}\sum\limits_{k=0}^{N-1} X[k] \ e^{-j \frac{2\pi}{N} n k} $

Please observe the scaling factor $\frac{1}{N}$ and change of a negative sign $-$ over the exponent term ${e^{\pm j\frac{{2\pi }}{N}nk} }$. Why is every notation of DFT valid?

Answer

It simply does not matter and it is just a matter of convention which version is used. I always use the first definition, which is also the one you will find in (almost) all textbooks on DSP. If you take always the same time domain signal $x[n]$ then it is obvious that $X_2[k]$ (according to notation 2) and $X_3[k]$ (according to notation 3) are simply scaled versions of $X[k]$ (according to notation 1). The one with scaling $1/\sqrt{N}$ is the unitary version of the DFT (i.e. the DFT becomes a unitary transformation). It's just notation number 4 that is a bit weird. In this case, $X_4[k]=X[-k]$, but I have never seen the sign convention like this. In this case, $x[n]$ can be interpreted as the Fourier series coefficients of $X[k]$. But again, all four versions are OK and you're free to use any of them as long as everybody knows which version of the DFT you mean.