Determining the Nyquist rate of signals added and multiplied together
Let $Y(\omega)=X(\omega)+X(\omega)e^{-j\omega}$ with $X(\omega)$satisfying $X(\omega)=0$ for $|\omega|&gt;\omega_m$. Then, by its definition, $Y(\omega)$ also satisfies $Y(\omega)=0$ for $|\omega|&gt;\omega_m$, i.e., the maximum frequency remains unchanged, and, consequently, also the Nyquist frequency remains unchanged.

Note that if $x(t)$ is the signal corresponding to the spectrum $X(\omega)$, then $X(\omega)e^{-j\omega}$ corresponds to the signal $x(t-1)$, which clearly has the same maximum frequency as $x(t)$.