# Is the empty set always a member of the universal set?

Consider the following universal sets:

$$A = \{\emptyset,1, 2, 3, \dots \}$$

$$B = \{1, 2, 3, \dots \}$$

According to my understanding $\emptyset \subseteq A$, $\emptyset \subseteq B$, $\emptyset \in A$ and $\emptyset \notin B$

So, is this site wrong when it says "Note that the empty set is a member of the universal set"?

http://kias.dyndns.org/comath/26.html

• I'd take anything someone says about set theory if they talk about a universal set. Not that it is impossible to have a universal set, but usually one has to actually know what one is talking about. And it requires a very clear context as to what is that universal set. – Asaf Karagila Mar 8 '17 at 15:21
• The box is the universal set, it has three subsets A, B and C, that seems simple enough. But where is the empty set ? And I have to confess that when I start talking about the empty set, I always end up sounding like Abbott and Costello talking about who is on first. – Ivan Hieno Mar 8 '17 at 15:35
• I don't know what is the box or what's inside of it. That's why you need context for the term "universal set". The fact you even had to ask this question is an ironclad evidence of the inherent problem with the term. – Asaf Karagila Mar 8 '17 at 15:39
• Okay, the box is the set of all socks, A is the set of all socks with holes in them, B is the set of all socks that are black, and C is the set of all socks that are dirty. Or are you looking for some other "context"? – Ivan Hieno Mar 8 '17 at 15:49
• I want to understand what is a universal set in this context. For you, it's the set of all "atomic objects", but in other contexts it might also mean all the mathematical objects of interest, including subsets and the empty set specifically. Context is always key. – Asaf Karagila Mar 8 '17 at 15:55

I wouldn't learn set theory from that website... Here's all they set about universal sets:

There are two special sets: ... and the "universal set". ... The universal set is denoted by the capital letter $\mathbf{U}$.

And that's it??!! This doesn't say absolutely anything about what a universal set is, so any discussion of its properties based on this website is futile and meaningless.

I think your confusion arose from mixing two different understandings of the term "universal set" with each other.

1. For a formal definition, even Wikipedia is a better resource on the concept of a universal set — see https://en.wikipedia.org/wiki/Universal_set. Short version: understood as the set of all objects, this leads to paradoxes, and therefore this concept is not allowed.

2. However, in textbooks the term "universal set" is often used in a context-dependent manner, meaning the set of all elements under consideration in the current context. When we draw Venn diagrams, the all-enclosing box represents such a universal set. In any specific example, the universal set for that example would have to be provided or understood from the context.

So when they say "Note that the empty set is a member of the universal set", that would be true in the first interpretation… but alas, there's no such thing as "the universal set".

On the other hand, either of your examples could be a universal set for whatever purposes you want to define them as such, in the sense of the second interpretation. But then there's no contradiction with the stated "property" simply because there's no such property.

• Whether there's a universal set in the sense of (1) is dependent on the other axioms involved. As the Wikipedia article itself mentions, there are theories which accommodate a universal set just fine. That said, I agree that the problem is that the linked material conflates the two meanings... – Malice Vidrine Mar 9 '17 at 3:39