In this note we’ll introduce the notion of a “vacuous statement”—a statement that is true, but completely devoid of meaning. In particular, we’ll learn what it means for some statement to be “vacuously true”. Statements that are vacuously true come up from time to time in mathematics, and their existence (and truth) often are crucial for certain mathematical constructions to be logically sound. For example, we made use of a vacuous truth in our lesson on subsets, where we noticed that the empty set is a subset of every set. This is a great example of a vacuous statement, so let’s explore it in detail again.
Let us first remind ourselves briefly about subsets. Recall that if we have some set A, then a subset of A is some set whose every element is also in A. If the enemy hands me some set B and asks me if it’s a subset of A, all I need to do is consider every single element in B and ask if it’s also an element of A. If the answer is yes for every element in B, then the answer to the enemy’s question is yes. Suppose, however, that the enemy handed me the empty set and asked if it was a subset of A. What would be my answer? Well, what I would have to do is look at every element of the empty set and ask if it is also an element of A. But there are no elements in the empty set. Thus the statement “every element in the empty set is also in A” is true simply because there are no elements in the empty set to even consider!
Let’s take another example, which will hopefully make this notion clearer. This example won’t be as mathematically precise, but it will hopefully bring out the essential features of a vacuous statement, thus making the mathematically precise ones easier to handle. Consider the following statement: “Whenever there are cows on the moon, I can fly”. (I know this sounds totally absurd, but it’s actually mathematically relevant!) Now, I know that I can’t fly (despite my wishes). You know I can’t fly. Thus the statement “I can fly” is certainly false. However, it happens to be true that I can fly, provided that there are cows on the moon. Namely, every time in the history of the universe that there has been a cow on the moon, I have had the ability to fly. Of course (presumably), there has never been a cow on the moon. Also of course (at least I’m quite sure), I have never been able to fly. Thus it is indeed the case that the presence of cows on the moon has coincided perfectly with the instances of me being able to fly. Accordingly, the statement “Whenever there are cows on the moon, I can fly” is indeed true!
Now before you go crazy proving statements like “As long as pigs can speak French, Lebron is better than Kobe” (which would be true only in this circumstance), I should warn you that there is a reason why we call these statements “vacuous”. Namely, they are devoid of any content (like a vacuum). Thus, even though they’re true, they don’t tell us anything new. No one will be handed a Fields Medal for proving a vacuous statement, simply because a) they’re automatically true, and b) they don’t give us any new information. Just because I can prove that Lebron is better than Kobe whenever a pig can speak French, this does not mean that I can prove it in any scenario that might actually be relevant to the real world. The truth of the statement just kind of sits “out there” lacking any real substance. Nonetheless, the validity of some vacuous statements is important for certain mathematical results, and certain logical consistency. In particular, we used the fact that the empty set is a subset of every set (even itself) in order to establish the general pattern first described in lesson 3, and proved in lesson 15. These manipulations of the empty set will also come into play when we define topologies, as well as in category theory. Thus, despite the fact that the statement “the empty set is a subset of every set” is only vacuously true, its truth is necessary for much of the logical consistency of mathematics.
Now go have fun owning your friends in arguments by inserting various vacuously true statements in such a way that your argument is untouchable. Perfect this craft and you could seriously consider a career in politics!
This is the most clear explaination I’ve ever seen
Thanks! Glad you found it useful 🙂
So, we say that “there is no element in the empty set that is not also an element in A” and conclude that every element in the empty set is also an element in A which, of course, implies that the empty set is a subset of A.
What about the opposite? What if we say that “there is no element in the empty set that is also in A” (simply because there is no element in the empty set) and conclude that every element of the empty set is not to be found in A, which would imply that the empty set is not a subset of A.
Wouldn’t the second statement too be true (vacuously?)? If yes, why did mathematicians choose the first?
This leads me to another question I have : What happens, to set theory, mathematics etc, if the following statements are not considered true : 1. Any set is a subset of itself, 2. The empty set is a subset of every set?
That’s a fantastic question! This logic is indeed very subtle, so let’s look at it more carefully. It is indeed true that we can say “There is no element in the empty set that is also in A.” However, this does not allow us to imply that the empty set is NOT a subset of A. The reason is as follows. In order for some set B to NOT be a subset of another set A, it MUST be the case that there IS an element in B that is NOT in A. We then immediately see that this isn’t the case for the empty set. Namely, it isn’t enough to simply say that no element in the empty set is in A. We must ALSO be able to say that there IS some element in the empty set that is NOT in the set A, and we clearly cannot say that. Does that make sense?
As for your more general question: not much would change. Namely, we would just have to make slightly more detailed definitions and some of our results would be less “pretty,” but the content of math itself wouldn’t change. For example, we’d have to define a subset B of a set A to be a set that is not equal to A, not empty, and such that every element in B is in A. This is clearly more cumbersome than the definition that B is a subset of A if every element in B is in A. Then, when we count subsets of a set with N elements in it, we’d have to say that the empty set has no subsets, that any 1-element set has no subsets, and that any set with N=2 or more elements has (2^N)-2 subsets. Clearly, this is less “pretty” than the simple result that any set with N elements has 2^N subsets. However, I also hope that it’s as clear that the real content of these statements is unaltered. The next place that these other possible definitions would make an appearance will be in topology, and we’ll eventually (not for a while, though) see that those results are effectively unaltered as well.
Does that all make sense? Great questions!
1). When is set B a subset of A?
Ans : When *all* of its elements are also in A. In other words, when it has no elements that are not in A.
The previous two sentences say the same thing. However, the first one cannot be directly applied to the empty set, simply because there are no elements in it. Is that why the second one is applied to the empty set to derive the first one and conclude that the empty set is a subset of A?
2). When is set B *not* a subset of A?
Ans : When set B has even one element that is not in A. But this cannot be translated to my earlier statement “when there is no element in B that is also in A”. They are not equivalent. In fact, the second statement implies that *all* elements of B must not be in A for B to be a subset of A. This is not needed. Just one extra element in B is enough.
So the second sentence cannot be used on a normal set B to answer this question. Only the first sentence can be used. Is the same rule extended to the empty set also, even though the second sentence can be used on the empty set?
Hi Karthik, I think all the confusion comes from thinking that the phrase “set B is a subset of set A when all of the elements in B are in A” cannot be applied to the empty set. This isn’t the case. Namely, this set can be applied to the empty set all the same. In particular, for ANY set A, it is true that all of the empty set’s elements are in A, simply BECAUSE the empty set has no elements. For example, suppose I have a savings account and suppose I have absolutely no money to my name. I literally have zero dollars and zero sense. I can then truthfully say that ALL of my money is saved in my savings account, because I have no money at all! That’s precisely why the empty set is subset of every set 🙂 Does that make sense?