24 - LECTURE 24 CARDINALITY(10.1-10.2 We have too many high...

LECTURE 24: CARDINALITY ( § 10.1-10.2)
We have too many high sounding words, and too few actions that correspond with them. Abigail
Adams
1. Cardinality Revisited
We have used | A | to denote the number of elements in A . But for infinite sets this is not precise.
Instead, we will measure the size of sets in a more precise way.
Notice that A = { a, b, c, d } and B = { 1 , 2 , 3 , 4 } have the same number of elements. One way to see
this is to create a bijection between them. We can do a similar thing with infinite sets.
Definition 1. Two sets A and B are said to have the same cardinality if there is a bijection f : A → B .
Note that f - 1 : B → A is also a bijection, so this relation is symmetric. We write | A | = | B | .
If A and B have no bijections between them, we write | A | 6 = | B | .
Theorem 2. The relation of having the same cardinality is an equivalence relation.
Proof. First we show this relation is reflexive. Let A be a set. We want a bijection from A to itself.