^Introduction
In this question, we are given a continuous-time, real-valued signal x(t) that is band-limited to F Hz. We need to find the Nyquist sampling rate for the two given signals y(t) = x(0.5t) and z(t) = x(t) - x(2t).

^Explanation
To understand the Nyquist sampling rate, let's start with the Nyquist-Shannon sampling theorem. According to this theorem, in order to accurately represent a continuous-time signal in the digital domain without any loss of information, the sampling rate should be at least twice the bandwidth of the signal.

^{Finding the bandwidth}
To find the Nyquist sampling rate, we first need to determine the bandwidth of the signal x(t). The given signal x(t) is band-limited to F Hz, which means it contains frequency components up to F Hz and no frequencies higher than F Hz.

^{Sampling rate for y(t)}
The signal y(t) is obtained by compressing the time axis of x(t) by a factor of 0.5. This means that the original bandwidth F is compressed by a factor of 0.5, resulting in a new bandwidth of F/0.5 = 2F. Therefore, the Nyquist sampling rate for y(t) should be at least 2 times the new bandwidth, which is 2 * 2F = 4F Hz.

^{Sampling rate for z(t)}
The signal z(t) is obtained by subtracting x(2t) from x(t). The highest frequency component in x(t) is F Hz, and when we multiply t by 2, the highest frequency component becomes 2F Hz. Therefore, the bandwidth of x(2t) is 2F Hz. Subtracting x(2t) from x(t) does not introduce any new frequencies, so the bandwidth of z(t) remains F Hz. Hence, the Nyquist sampling rate for z(t) should be at least 2 times the bandwidth, which is 2F Hz.

^Conclusion
The Nyquist sampling rate for y(t) = x(0.5t) is 4F Hz, and the Nyquist sampling rate for z(t) = x(t) - x(2t) is also 4F Hz. Therefore, the correct answer is option 'C' (4F).