# Vacuously True

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### 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?