DFT have independent variable in between 0 to $\frac{1}{2}$ no of samples
This statement is only true if your time-domain signal $x[n]$ is real. A real signal has a Fourier transform that is conjugate-symmetric. Stated differently the "negative frequencies" are just the conjugate of the corresponding "positive frequencies." Therefore, the negative frequencies add no new information, so they can be discarded from the DFT of a real signal.