Countable

Definition 65   A set $S$ is countable if either $S=\emptyset$ or there is an epimorphism $f:\ensuremath{\mathbb{N}}\to S$.

Theorem 154   A set $S$ is countable if and only if there is a monomorphism $m:S\to \ensuremath{\mathbb{N}}$.

(4 Points )

Proposition 155   The finite cardinal set $[n]$ is countable for any $n\in \ensuremath{\mathbb{N}}$.

(2 Points )

Proposition 156   is countable.

(2 Points )

Proposition 157   If $A$ and $B$ are countable then so are $A\times B$, $A+B$, and $A\cup B$.

(6 Points )

Proposition 158   If $\Lambda$ is a countable set and $A_\lambda$ is countable for each $\lambda\in \Lambda$ then $${\bigcup_{\lambda\in \Lambda}A_\lambda}$$ is countable. (Any countable union of countable sets is countable.)

(4 Points )

Definition 66   A number $r$ is called algebraic if it is the root of an equation with integer coefficients. A real number which is not algebraic is called transcendental.

Proposition 159   There are countably many algebraic real numbers.

(3 Points )

Total for section: 71.