set countability

title

Countable & Uncountable Set

• countable if it has same number of elements as some subset of ${\mathbb{N}}^{+}$ or ${\mathbb{N}}^{+}$ itself
• include finite and infinite set that is bijective with ${\mathbb{N}}^{+}$
• a set is countable IFF $|S|\le |N^{+}|$
• eg:
  • set $\mathbb{Z}$ is also countable: alternating between non-negative and negative numbers → here
• prove S is countable set:
  • find a bijection between the set S and ${\mathbb{N}}^{+}$
• a set not countable is uncountable
  • Proposition 2: eg. set of (0, 1); $\mathbb{R}$ is uncountable

Theorem 18

• if A and B are countable sets, $A\cup B$ is also countable