eli5 Why is the empty set a subset of any set?

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eli5 Why is the empty set a subset of any set?

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It is a vacuous truth. Any element that is in the empty set also appears inside any other set.
But there are no elements in the empty set.

It’s like saying: “All of these people in my room have purple wings”, when my room is empty. Technically the truth, or in mathy terms, vacuous truth

When you remove all elements from a set, you are left with the empty set. Since you can remove all elements from any set, the empty set must be common to all sets.

A is a subset of B if every element in A is also an element in B. It’s vacuously true that every element in the empty set is also an element in every set.

Put another way, A is **not** a subset of B if there’s an element in A which is not in B. Given some arbitrary set, can you show me at least one element that’s in the empty set but not in this set?

It’ll be easier to explain it with the notion of **superset**.

A set *bigbag* is a superset of *smallbag* if you can obtain *bigbag* by adding objects to *smallbag.* In particular, *bigbag* is always a superset of the empty set, because if you have the empty set, you can just add all the objects of *bigbag* to obtain *bigbag*

The other way around, *smallbag* is a subset of *bigbag* if you can obtain *smallbag* by removing objects from *bigbag*. In particular, the empty set is always a subset of the bigbag, because you can start from *bigbag* and remove all the objects.

If that doesn’t feel intuitive too you, it’s probably you don’t fully grasp that the empty set is a set. It is kind of weird said like that, but using bags: the empty set is a bag with nothing in it, but it’s still a bag.

If you take a set, to make a subset of it, you simply have to remove some number of elements from the set. This includes removing everything from the set, leaving you with the empty set.