Is there an empty set in the complement of an empty set?

Question

Currently taking a logic class and trying to understand this.

You have two set $A$ and $B$.

Both sets are empty sets.

Is set $A$ a subset of the complement of set $B$?

Assume the context is the universal set.

Answer 1

The answer is yes. But there are several comments that need to be made:

1. There is only one empty set. So it is better to say that $A=B=\emptyset$ rather than saying that "both $A$ and $B$ are empty sets", as the latter erroneously suggests that there is more than one. This is because two sets are equal precisely if they have the same elements. So any two empty sets are equal, since they have precisely the same elements (namely, none).

2. I assume by "context" you mean that the complement of $B$ is computed with respect to the "universal set." In the standard system of set theory, there can be no "universal set", as assuming its existence leads to problems (Russell paradox). [Though, yes, it is usual to talk of a "universal set" as a way to delineate what objects we are interested in.]

The reason why $A$ is contained in the complement of $B$ is that $A$ (being the empty set) is a subset of any set. This is because we define "$A$ is contained in $C$" to mean that any element of $A$ is also an element of $C$. Now, since nothing is an element of $A$, this condition is satisfied in this case (one typically says that it is satisfied vacuously.)

Answer 2

The empty set is a subset of any other set.