The empty set revisited. How does the empty set ∅ interact with subsets?

Consider any set S. Is the empty set a subset of S ? Recall the definition of subset: AB⊆ precisely when every element of A is also an element of B . The empty set doesn't contain any elements, so how does it interact with this claim? If we plug ∅ and the set S into the above, we get:

∅⊆SS if every element of ∅ is an element of .

1 Look at the last part ... "if every element of ∅ is an element of S ." What does this mean?

Recall that there aren't any elements of ∅∅ , because doesn't contain any elements!

Given this, is the above statement true or false ?

2 There are two ways to think about this:

1. Since ∅ contains no elements, the claim "every element of ∅ is an element of S." is false , because we can't find even a single example of an element of ∅ that is contained in S .

2. Since ∅ contains no elements, the claim "every element of ∅ is an element of S." is true , because we can't find even a single example of an element of ∅ that isn't contained in S .

3 So which line of thinking is mathematically correct? It turns out that it is the second approach, and indeed it is true that ∅⊆S.

2. Since ∅ contains no elements, the claim "every element of ∅ is an element of S ." is true , because we can't find even a single example of an element of ∅ that isn't contained in S .

Why is that?

4 We need to introduce the idea of a vacuous truth. Informally, a statement is vacuously true if it's true simply because it doesn't actually assert anything at all.

Example: Consider the statement "if I am a cheeseburger, then the earth is flat." This statement is completely meaningless since the statement "I am a cheeseburger" is false.

As a consequence, the statement "if I am a cheeseburger, then the earth is flat." doesn't actually assert anything because I'm not a cheeseburger.

5 More formally,

The statement "if PQ , then " is vacuously true if P is always false.

6 More generally,

The statement "Every XY has property is vacuously true if there are no X 's.

7 Back to the original question: is ∅ a subset of any set S ? Recall that ∅∅ a subset of set S if every element of is also an element of S .

But the statement "every element of ∅ is an element of S " is vacuously true because there are no elements of ∅ .

We have

For any set SS , ∅⊆ .

8 Note that since, for any set SS , ∅⊆ , we have

∅⊆{ 1,2,3} , ∅⊆{ pig, chicken, cow} , ∅⊆ and even ∅⊆∅

Is the empty set unique?

9 Recall that two sets ST and are equal if and only if they have the same elements. This means S= T ⇔ for all xx , ∈⇔∈ S x T (1)

Theorem: The empty set is unique. Proof: Suppose, BWOC, that ∅≠∅′ are empty sets. We have, by definition, if ∅ is an empty set, then ∅⊆∅′ also, if ∅′′ is an empty set, then ∅ ⊆∅ We have both ∅⊆∅′′ and ∅ ⊆∅ . By (1) above we have ∅=∅′ and thus the empty set is unique.

10 Let AB and be two sets. We say that A is a subset of B , denoted by AB⊆ , if and only if every element of A is also an element of B.

In symbols [ A⊆ B] ⇔[ xA ∈ ⇒∈ xB].

If there exists an element of A which is not in B we write AB⊆ .

11 Note that, for any set A , the proposition x∈∅⇒ xA ∈ is vacuously true since xA∈∅ is always false. So ∅⊆ .

Two sets AB and are said to be equal if and only if BA⊆⊆ and AB . In this case we can write AB= . If two sets are not equal we write AB≠ .

12 End Presentation