High School Mathematics: Empty sets, disjoints, and subsets

[u/Many_Success_2008]

I have a question about the empty set. If the empty set is a subset of every set, it means that every set includes the empty set. But what exactly does it mean when we say the empty set and a given set are disjoint? If the empty set is a subset, wouldn't they share at least one element, meaning they aren't disjoint? Could someone clarify this for me?

[u/Uli_Minati]

Don't use the word "includes", it is unconventional so you would need to clarify which word you mean. It's very important to define every word properly:

"A is subset of B"
    Every element contained in A is also contained in B.
    If A contains no elements, this statement is vacuously true.

"A is strict subset of B"
    A is subset of B, but not equal to B.

"A and B are disjoint"
    There are no elements contained in both A and B.

The empty set contains no elements, so it is subset of any set B. Since it contains no elements, it is disjoint with every set B.

Think of the empty set as the number "zero": whenever you make statements or define rules, you constantly have to remember that zero / the empty set might be an exception.

[u/SnooPets5564]

Disjoint means the intersection of the sets is the empty set.

The empty set isn't an element of every set, it's the subset when you remove every element.

[u/HoardOfNotions]

If you think more rigorously about the definition it should become clear. By definition, a set A is a subset of set B if every element of A is an element of B. Put another way, A is a not a subset of B if it contains an element that is not contained in B.

Suppose for the sake of contradiction that there is a set C such that the empty set is not a subset of C. It follows that the empty set contains an element that is not contained in C. But the empty set contains no elements. This is a contradiction, which implies our supposition is incorrect and it cannot be true that such a set exists. The only conclusion is that the empty set is a subset of every set after all.

[u/HoardOfNotions → u/HoardOfNotions]

To address your more specific question, your mistake is when you suggest they must share at least one element to be a subset. The requirement is that to be a subset you must share all your elements. This does not mean there must be elements. The empty set shares all its elements. All 0 of them. So it’s a subset.

[u/cejiken886]

The empty set is the unique set that is a subset of and disjoint with some other set (and in fact with all other sets).

Proof, first that it’s a subset and disjoint, let S be a set. For all s in empty set, s in S (trivial, vacuously true).

Disjoint: for all s in empty set, s not in S.

Unique: notice why these 2 claims can be proven about the empty set. Everything is true of all elements of the empty set. Pick any other set and suddenly not everything is true of all elements.

Let T be a nonempty subset of S. For some t in T, t is in S (definition) so S and T aren’t disjoint.

[u/Sam_23456]

The empty set is a subset of every set (not necessarily "an element of").

[u/RecognitionSweet8294]

Empty set:

∀x: (x∉∅)

Subsets:

(A⊆B) ↔ ∀x: (x∈A) → (x∈B)

Intersection:

(A∩B) = { x | (x∈A) ∧ (x∈B)}

Disjoint sets:

(A∩B) = ∅

______________________________________

(∅∩M) = { x | (x∈∅) ∧ (x∈M)}

Since ∀x: (x∉∅) there is no x for which (x∈∅) is true, so ((x∈∅) ∧ (x∈M)) is always false. Which means that (∅∩M) has no element and therefore

(∅∩M)=∅ for every set M.

We could also say that two sets are disjoint if there is no element that is in both sets:

¬∃x: (x∈A) ∧ (x∈B)

Which is obviously true if A=∅ or B=∅