Union & Intersection for Propositional Logic How is it that the union of two sets of sentences (that, individually, logically entail a sentence) logically entails a sentence while the intersection of the two sets does not logically entail a sentence? !visual of the question

1. If it is raining, then my car is wet.

2. It is raining.

3. If I am in the carwash, then my car is wet.

4. I am in the carwash.




Set $S$ to be the first two propositions, $T$ to be the last two. Each of $S,T$ imply that my car is wet. Their union, i.e. all four propositions above, also implies that my car is wet. However $S\cap T=\emptyset$, which does not imply that my car is wet.