Yet again, you omitted out the part that proves you wrong.
It says: "Two sets A and B are disjoint if and only if their intersection is the empty set. It follows from this definition that every set is disjoint from the empty set, and that the empty set is the only set that is disjoint from itself.[5]
You have been saying that the empty set is a subset of every set. That is obviously contradicted here.
Further, if the empty set is disjointed with itself. That means {} =/= {} by defintion. Two sets are equal if and only if each set is a subset of the other. And that proves that {} cannot be evaluated.
No. You have claimed that an empty set is a subset of all sets. What you posted above is the OPPOSITE of that. You said that it was so simple and easy to prove. You said it was taught to young children, and is included on the first 1 or 2 pages of a child's math book. So, go ahead. Prove it using the definition that we have agreed on. It should be simple, right? Just remember { 1,2,3 } =/= { 1,2,3,{} }.
There is no contradiction. Do you really believe that millions of mathematicians, logicians, including the greatest minds of the last two centuries, educators, students, etc. could have missed such a huge and obvious contradiction at the root of the fundaments of their discipline? I am sure you must have asked yourself how it can be that all articles, courses, books, etc. about the argument, agree with me, and none whatsoever agrees with you.
The source of your confusion is obvious. You rely too much, and erroneously, on the geometric visualization of sets as bubbles, as you can find in Venn and Euler diagrams. You must find it impossible to visualize a set that is contained in another set and, at the same time, is disjoint, external, to it. After all, a bubble cannot be inside and outside another bubble at the same time, right?
Well, in case of the empty set, this is the case. And it is the bubble analogy that fails, not the properties of the empty set. And it fails because the empty set should not be represented as a bubble. After all, if I have two non-empty sets which are disjoint, and therefore have the empty set as intersection, there is no third ball representing that intersection in any Venn diagram known to woman.
Therefore, best to not think of the empty set as a bubble. It will just mislead you.
You should simply rely on the definitions, like a robot. That is really the safest for you, and will put you back in line with the rest of the educated world.
So, let's take intersection and union as examples, and think only according to the definitions:
Q: What is the intersection of sets A and B?
A: A set
Q: What does that set contain?
A: All the elements, if any, that are common in A and B
Q: Can the result be an empty set?
A: Sure. if the two sets have no elements in common
Q: What if A is the empty set, and B is arbitrary?
A: Nothing. The definition still applies
Q: Will, in that case, the resulting set have elements?
A: Nope. Since if it had even one, it would be common with A, which is empty. And that would be absurd
Q: So, the result is an empty set, too. As in the case of not empty sets with no elements in common?
A: Of course
Ergo: --> {} intersection B = {} for all sets B
Q: What is the union of sets A and B?
A: A set
Q: What does that set contain?
A: All elements, if any, that are either in A. or in B. Or in both
Q: Can A be empty and B arbitrary?
A: Of course. The definition still applies
Q: Will the resulting set contain element of A?
A: Of course not, if A is empty and has, therefore, no elements
Q: So, it will contain only the elements of B?
A: Yes
Q: So, it will be equal to B?
A: Of course
Ergo: --> {} union B = B for all sets B
Any of those two obvious results entail immediately, according to the definition of subset you posted, that {} is a subset of B, for any B.
Now, this is basic stuff, as it is taught to kids as young as 10. You can, in fact, find the same results in
Empty set - Academic Kids
- For any set A, the empty set is a subset of A:∀A: {} ⊆ A
- For any set A, the union of A with the empty set is A:∀A: A ∪ {} = A
- For any set A, the intersection of A with the empty set is the empty set:∀A: A ∩ {} = {}
So, it is your call. Either you admit that you are incapable to understand what little kids are assumed to be able to grasp, or you ponder about it a little longer.
Possibly making a bit of research on your old school books, assuming you have ever been educated on this.
Ciao
- viole
