Finite Sets

We have mentioned finite sets on a few occansions, but have yet to give a formal definition. This section will define what it means for a set to be finite, and properties of finite sets.

Definition

There a few definitions of finite sets, we will use the definition found in Enderton [1977].

Definition

A set A is finite if it is equinumerous to some natural number.

This definition is simple, and only relies on the concept of equinumerosity which we just discussed, and the natural numbers.

Example

Let A = \{x,y\}. The bijective function

f = \{(x,0),(y,1)\}

shows that A \approx 2, and since 2 \in \mathbb{N} we have that A is finite.

Properties

We will begin with a handful of simple theorems expanding our knowledge on finite sets.

Theorem

The empty set is finite

Theorem

Let A be a set. Then the singleton \{A\} is finite.

Theorem

Let A be a set. If A is finite and B \subseteq A, then B is finite.