Section 2.4: Finite and Infinite Sets

2.4 Outline

  1. Infinite sets
    1. finite
    2. infinite
    3. one-to-one
    4. correspondence
    5. countable/uncountable
      1. rational numbers are countable
      2. real numbers are uncountable
  2. 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.