da Cannone Speciale » 13/01/2024, 19:39
Riporto una parte del libro di topologia di James Munkres che secondo me chiarifica l'utilizzo dell'insieme vuoto:
when the sets A and B happen to have no elements in common. What meaning does the symbol A ∩ B have
in such a case? To take care of this eventuality, we make a special convention. We introduce a special set that we call the empty set, denoted by ∅, which we think of as “the set having no elements.” Using this convention, we express the statement that A and B have no elements in common by the equation A ∩ B = ∅. We also express this fact by saying that A and B are disjoint. Now some students are bothered by the notion of an “empty set.” “How,” they say, “can you have a set with nothing in it?” The problem is similar to that which arose many years ago when the number 0 was first introduced. The empty set is only a convention, and mathematics could very well get along without it. But it is a very convenient convention, for it saves us a good deal of awkwardness in stating theorems and in proving them. Without this convention, for instance, one would have to prove that the two sets A and B do have elements in common before one could use the notation A ∩ B. Similarly, the notation C = {x | x ∈ A and x has a certain property} could not be used if it happened that no element x of A had the given property. It is much more convenient to agree that A ∩ B and C equal the empty set in such cases. Since the empty set ∅ is merely a convention, we must make conventions relating it to the concepts already introduced. Because ∅ is thought of as “the set with no elements,” it is clear we should make the convention that for each object x, the relation x ∈ ∅ does not hold. Similarly, the definitions of union and intersection show that for every set A we should have the equations A ∪ ∅ = A and A ∩ ∅ = ∅. The inclusion relation is a bit more tricky. Given a set A, should we agree that ∅ ⊂ A? Once more, we must be careful about the way mathematicians use the English language. The expression ∅ ⊂ A is a shorthand way of writing the sentence, “Every element that belongs to the empty set also belongs to the set A.” Or to put it more formally, “For every object x, if x belongs to the empty set, then x also belongs to the set A.” Is this statement true or not? Some might say “yes” and others say “no.” You will never settle the question by argument, only by agreement. This is a statement of the form “If P, then Q,” and in everyday English the meaning of the “if ... then” construction is ambiguous.
[...]
Again, mathematics cannot tolerate ambiguity, so a choice of meanings must be made. Mathematicians have agreed always to use “if ... then” in the first sense, so that a statement of the form “If P, then Q” means that if P is true, Q is true also, but if P is false, Q may be either true or false. As an example, consider the following statement about real numbers: If x > 0, then $x^3 != 0$. It is a statement of the form, “If P, then Q,” where P is the phrase “x > 0” (called the hypothesis of the statement) and Q is the phrase “$x^3 != 0$” (called the conclusion of the statement). This is a true statement, for in every case for which the hypothesis x > 0 holds, the conclusion $x^3 != 0$ holds as well. Another true statement about real numbers is the following: If $x^2 < 0 $, then x = 23; in every case for which the hypothesis holds, the conclusion holds as well. Of course, it happens in this example that there are no cases for which the hypothesis holds. A statement of this sort is sometimes said to be vacuously true. To return now to the empty set and inclusion, we see that the inclusion $ O/ sub A $ does hold for every set A. Writing ∅ ⊂ A is the same as saying, “If x ∈ ∅, then x ∈ A,” and this statement is vacuously true.
Ad Maiora!
