Are Disjoint Sets Truly Without Common Elements?

[Townsend]

I have been through elementary set theory but there are still a few things that I never really understood in that class. One that keeps coming up and I still cannot resolve is this.

If two sets A and B are disjoint sets then neither is a subset of the other unless one is the empty set.

Looking at the definition of disjoint we have:

Two sets A and B are disjoint if they have no elements in common.

And we also have that the empty set is the subset of every set.

So what if we let A be the empty set and B be the empty set?

Are they disjoint then?

Or consider A={1,2,3,{}} and B={}

Now A and B are disjoint and B is a subset of A. What does that mean? They both contain the empty set, and that is a common element for every set, right. I mean in order for the empty set to be a subset of every set then every set must contain the empty set.

I think I don't really understand the concept of what an empty set is. I realize that the set with no element is the empty set but the set itself is an element, no? If I give you three sets with definitions for all the elements in them then the three sets can also be considered elements of another set right? So if one of those sets is defined to be the null set then it is still an element of another set.

So now we look at sets A and B above and we see they both contain the empty set. So they are not really disjoint and no two sets can be disjoint if they all have a common element namely the element that is the empty set.

I know that I have this all wrong but I really need to understand this concept. If someone from PF can make this as simple as possible I would really appreciate it a lot.

Thanks

 

[matt grime]

So what if we let A be the empty set and B be the empty set?

Are they disjoint then?

yes, AnB is empty.

Or consider A={1,2,3,{}} and B={}

Now A and B are disjoint and B is a subset of A. What does that mean? They both contain the empty set, and that is a common element for every set, right. I mean in order for the empty set to be a subset of every set then every set must contain the empty set.

B does not contain the empty set, it is the empty set. B contains no elements. The empty set is a subset of every set, not an element of every set. You are confusing the two things.

I think I don't really understand the concept of what an empty set is. I realize that the set with no element is the empty set but the set itself is an element, no?

No, the empty set is not an element of itself.

 

[tacman]

First of all, it's important to understand that the concept of sets and subsets is based on the idea of containment. A set A is a subset of another set B if all the elements of A are also elements of B. In other words, A is contained within B.

Now, let's address your question about disjoint sets. If two sets A and B are disjoint, it means that they have no elements in common. In your example, A={1,2,3,{}} and B={} are disjoint because they don't have any elements in common. The fact that they both contain the empty set does not change this, because the empty set is not considered an element of itself. It's simply a placeholder for a set with no elements.

Now, let's consider the case where A and B are not disjoint. For example, let A={1,2,3} and B={3,4,5}. In this case, A is not a subset of B, because A contains elements that are not in B (1, 2), and B is not a subset of A because B contains elements that are not in A (4, 5). The only way for A to be a subset of B is if A and B are disjoint, meaning they have no elements in common.

So, to answer your question, if A and B are both the empty set, they are still considered disjoint because they have no elements in common. The fact that they both contain the empty set does not change this. And just to clarify, A and B are not considered to be elements of each other, they are sets themselves.

I hope this helps clarify the concept of disjoint sets for you. Remember, the key is to understand the idea of containment and how subsets are defined based on that. Keep practicing and asking questions, and you'll continue to improve your understanding of sets.