Doubt about Empty Set's definition as a Set.

Question

A set is a well-defined - collection of distinct objects. The objects that make up a set (also known as the set's elements or members) can be anything: numbers, people, letters of the alphabet, other sets, etc.

My question is that Empty Set (or Null Set), though very well defined as the set which has no elements, why is it considered a set if it doesn't contain any distinct element?

The question might feel little dumb but it's disturbing me for almost an year now, and my teacher said that this is the way we defined Null Set so it is ambiguous for now but in higher studies the ambiguity fades away. But I don't want to settle at that answer.

Edit: My actual concern was not on distinctness of the elements but rather on the number of elements in the set. But nonetheless, thank you all for the help.

Answer 1

Saying that all the elements of a set are distinct does not imply that there are any elements at all. It just says there are not two of the same element.

Answer 2

That definition of a set is the customary one in beginning courses. The objects are mathematical objects.

What "well defined" means in that definition is that you know what set you have precisely when you know what things it contains, independent of how it is described. So, for example, the set of solutions to the equation $3x = 6$ and the set $\{2\}$ are the same set.

Often we want to talk about a set even if we don't know how to specify its elements. For example, we might want to consider the set $S$ of all odd perfect numbers. That's a perfectly good definition of a mathematical set, even though no one knows whether there are any odd perfect numbers. In other words, $S$ might be an empty set.

In fact, $S$ might be the empty set, since there is just one empty set. That's because a set is known when you know what is in it, and any two empty sets contain exactly the same things, namely, none.

The "distinct" in the definition you quote does not mean "particular" it means (essentially) "different". So the sets $\{2\}$ and $\{2,2\}$ are the same set.

Answer 3

Very broadly speaking, given a set $X$ and a property $P$ depending on elements $x \in X$, you can define the set $$Y:=\{ x \in X \text{ such that }x \text{ satisfies the property } P \}$$

For instance, if you have already defined the set of all integers, you can define $\{ x \in \mathbb N : x>2\}$ (the set of all integers larger than $2$).

But if you chose a property that no element of the set $X$ satisfies, then you get the empty set.

For instance, you could define the empty set as $\{ x \in \mathbb N: x \text{ is both even and odd}\}$. This is a set, and it contains no elements.

If you want a more fundamental explanation of why this is an authorized defintition, you have to go back to ZF axioms. Modern mathematics is built on a finite set of axioms from which everything can be deduced; and the fact that the definition of sets I mentionned is well-defined is exactly one of the axioms, see https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory (axiom schema of specification).

Answer 4

It would be perfectly possible to define a set to be a non-empty collection, and then we would get around your objection. However, this would break a couple of different "nice properties" that we really want sets to have: for example, we would stop being able to take intersections (what's the intersection of $\{1,2\}$ and $\{3,4\}$?), or more generally we would stop being able to select subsets (what's the subset of $\{1,3,5\}$ which contains exactly those members which are even?).

It turns out just to be much more convenient if $\emptyset$ is a set.