Vacuously True

Summary

Can an empty set be a subset of any set containing elements? That is the question I mull over in this short post.

It's the weekend. While I sit at my desk convincing myself not to flake on an outing I agreed to weeks ago, a discrete mathematics video¹ on YT catches my eyes. I was presented with an interesting set edge case.

Given two sets A and B where:

$A = \{ x | 2 < x < 10 \}$

i.e. A is a set of variables x, such that x is greater than 2 and less than 10.

And $B = \{5,6\}$

i.e. B is a set containing the elements 5 and 6.

We know that $B \subseteq A$

i.e. B is a subset of A. Because, a set is a subset of another if all of its elements exist in the other set.

If we have another set C where:

$C = \emptyset$

i.e. C is an empty set. An empty set described as $\emptyset$ or $\{ \}$ contain no elements.

The question now is: Is C a subset of A? Can an empty set be a subset of another set that contains elements?

The answer

To answer this, we need to look at the requirement of a subset: all elements of a subset must be found in the superset.

Is there an element we can retrieve from set C that is not in set A? No, there is no such element. Hence, saying C is NOT a subset of A is False, meaning it is True. C IS indeed a subset of A. This type of statement is called a "Vacuous Truth". There is no way to verify it because C contains no element for us to check.

In mathematics and logic, if the predicate of a conditional or universal statement cannot be satisfied, then the statement is vacuously true.²

I made a new friend at the event today is vacuously true because I didn't go to the event today :(