Introduction:
The given problem involves a continuous-time, real-valued signal x(t) which is band-limited to F Hz. We are asked to find the Nyquist sampling rate in Hz for two different cases: y(t) = x(0.5t) and z(t) = x(t) - x(2t).

Explanation:

1. Band-limited signal:
A signal is said to be band-limited to F Hz if its frequency content is limited to the range [-F, F]. This means that the signal does not contain any frequency components beyond F Hz.

2. Nyquist-Shannon sampling theorem:
According to the Nyquist-Shannon sampling theorem, in order to accurately reconstruct a continuous-time signal, it must be sampled at a rate higher than twice its highest frequency component. This is known as the Nyquist sampling rate.

3. Sampling rate for y(t) = x(0.5t):
In the given case, the signal y(t) is obtained by compressing the time axis of x(t) by a factor of 0.5. This compression results in a stretching of the frequency axis by a factor of 2. Therefore, the band-limited signal x(t) is now band-limited to F/2 Hz.

According to the Nyquist-Shannon sampling theorem, the sampling rate for y(t) should be higher than twice the highest frequency component of y(t), which is F/2 Hz. Therefore, the Nyquist sampling rate for y(t) is 2*(F/2) = F Hz.

4. Sampling rate for z(t) = x(t) - x(2t):
In the given case, the signal z(t) is obtained by subtracting the signal x(2t) from x(t). The signal x(2t) is obtained by compressing the time axis of x(t) by a factor of 2. This compression results in a stretching of the frequency axis by a factor of 0.5. Therefore, the band-limited signal x(2t) is now band-limited to 2F Hz.

The resulting signal z(t) is the difference between two band-limited signals: x(t) (band-limited to F Hz) and x(2t) (band-limited to 2F Hz). The difference of two band-limited signals is also band-limited to the maximum of their individual bandwidths, which is 2F Hz in this case.

According to the Nyquist-Shannon sampling theorem, the sampling rate for z(t) should be higher than twice the highest frequency component of z(t), which is 2F Hz. Therefore, the Nyquist sampling rate for z(t) is 2*(2F) = 4F Hz.

Conclusion:
The Nyquist sampling rate for the signal y(t) = x(0.5t) is F Hz, and the Nyquist sampling rate for the signal z(t) = x(t) - x(2t) is 4F Hz.