Express each of the following using LTL, assuming the atoms rain, volcano, and ashcloud and a time granularity of 1 day (i.e., next means tomorrow). You only need to do 1e if you are confident its correct (and dont want partial credit), as the other parts are optional and given only in case you need intermediate steps.

(a) If a volcano erupts, there is an ash cloud (starting the day of the eruption). There is also no more rain after the eruption until the ash cloud disperses (and there is no guarantee that the cloud will actually disperse).

(b) It takes at least 3 days for an ash cloud to disperse after a volcano.

(c) Today (day 0) is rainy and weather patterns are such that rainy and sunny days alternate starting on day 3. But if a volcano erupts (which doesnt occur until at least day 3 but will eventually happen), there will be a [possibly permanent] ash cloud starting that day and the above alternation will cease with no rain during this ash cloud (but the alternation will resume after that). Moreover, it takes at least 3 days for an ash cloud to disperse after a volcano.