Question

Prove that an empty set is a subset of every set ?

Answer 1

I like to prove this statement using contradiction as I think it best illustrates the concept. See below.

Explanation:

Given two sets `A` and `B`, let `A = emptyset`.

By definition, `A` is a subset of `B` if and only if every element in `A` is also in `B`.

This means that `A` would not be a subset of `B` if there exists an element in `A` that is not in `B`.

However, there are no elements in `A`. This means there cannot exist an element in `A` that is not in `B`.

Thus, `A` is a subset of `B`.

Since `A=emptyset` and `B` is an arbitrary set, the `emptyset` must be a subset of all sets.