Vacuously True

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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={x2<x<10}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}B = \{5,6\}

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

We know that BAB \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= 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 :(


  1. The Empty Set and Vacuous Truth

  2. Vacuous Truth