Question

Can be proved that \(\emptyset \) is a subset of every set?

Answer 1

Since the empty set contains no elements, the fact that:\[(\forall A), \emptyset\subseteq A\]is whats called vacuously true. In other words, its true by default, because there is nothing to check.

It would be like going into a room of people where there are no cell phones, and saying "there are no cell phones on." That is vacuously true, because there arent any cell phones to check.