2.4 Outline

A.  Infinite sets
      1. finite
      2. infinite
      3. one-to-one correspondence
      4. countable/uncountable
          a. rational numbers are countable
          b. real numbers are uncountable
B. Cartesian product of sets
      1. definition
      2. cardinality of a Cartesian product
      3. fundamental counting principle

2.4 Essential Ideas 

One-to-one Correspondence

Two sets A and B are said to be in a one-to-one correspondence if we can find a pairing so that:
(1) Each element of A is paired with precisely one element of B; and
(2) Each element of B is paired with precisely one element of A.

Finite and Infinite

A set is infinite if it can be placed in a one-to-one correspondence with a proper subset of itself. A set is finite if it is not infinite.

Fundamental Counting Principle

If task A can be performed in m ways, and after task A is performed, a second task B can be performed in n different ways, then the fundamental counting principle is that task A followed by task B can be performed in mn different ways.