136. Countability

136.1. Countable and Uncoutable Sets

1. Countable (Discrete)

One-to-one correspondence with positive integers

E.g.,

• Positive integers $\left\{1,2,3,\dots \right\}$
• Integers $\left\{0,-1,1,-2,2,-3,3,\dots \right\}$
• Pairs, triples, quadruples, … of positive integers
• rational numbers $𝑞$, with $0<𝑞<1$

$\left\{\frac{1}{2},\frac{1}{3},\frac{2}{3},\frac{1}{4},\frac{2}{4},\frac{3}{4},\frac{1}{5},\frac{2}{5},\dots \right\}$

2. Uncountable (Continuous)

• Intervals ($\left[0,1\right]$)
• Reals
• Plane