Three electromagnetic waves X , Y , and Z are identified by their...

Wave $X$ is produced by the radioactive decay of a nucleus, which identifies it as a gamma ray.

Wave $Y$ is produced by the vibration of atoms and molecules, which identifies it as an infrared ray.

Wave $Z$ is produced by a magnetron valve, which identifies it as a microwave.

The electromagnetic spectrum, in order of increasing wavelength, is: gamma rays $<$ X-rays $<$ ultraviolet $<$ visible $<$ infrared $<$ microwaves $<$ radio waves.

Therefore, the increasing order of their wavelengths is $X < Y < Z$.

Answer: $X < Y < Z$