Empty Set

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The empty set (∅) has no members. This placeholder is equivalent to the role of “zero” in any number system.

Examples of empty sets include:

• The set of real numbers x such that x² + 5,
• The number of dogs sitting the PSAT.

We can also define it as “the set of all objects not equal to themselves,” because there are no objects not equal to themselves (Hersh, 1997). This may be challenging to wrap your head around, but imagine an object. For this example, let’s say it’s a pink poodle. Are there any pink poodles that aren’t pink poodles? Obviously, no. The thing itself doesn’t exist (a pink poodle that isn’t a pink poodle), but the idea of it does, just like the idea of “zero” exists.

What is the Purpose of the Empty Set?

Although the empty set doesn’t have any “real life” practical purposes, it’s extremely important in number theory because the natural numbers are formed from the empty set. In other words, without it, the natural numbers couldn’t exist mathematically. The building blocks of the natural numbers are:

• The empty set (no members)
• The set containing the empty set (one member). This set has a cardinality of 1.

If you have trouble wrapping your head around this, imagine a gift box with no gift in it. This empty box is analogous to the empty set, because there’s nothing in it. But if you place that box inside another box, you have created a nonempty set with one member—the original empty gift box. You can continue to put boxes inside of boxes to create the natural number system, even though you started with nothing.

Properties of the Empty Set

For any set X:

• Union: X ∪ ∅ = X
• Intersection: X ∩ ∅ ; If two sets X and Y are disjoint, then X ∩ Y = ∅

Also, if X is a subset of Y then X \ Y = ∅.

References

Grinshpan, A. (2020). The Empty Set. Retrieved October 22, 2020 from: https://www.math.drexel.edu/~tolya/emptyset.pdf
Hersh, R. (1997). What is Mathematics, Really? Oxford University Press.