subsets of countable sets are countable

The definition of countable sets would not serve us well if it did not conform with our intuition about countable sets. So let us prove that countability is in a sense hereditary.

Theorem 1.

Every subset of a countable set is itself countableMathworldPlanetmath.


Let BA and A countable with f:AK, K a bijective function as in the definition of countable sets.

Let us consider f|B, the function f restricted to B, i.e. f|B:Bf(B). Then f|B is trivially onto, but also one-to-one (f was one-to-one!). So we have a bijective function from B onto f(B)K, which the proof. ∎

Title subsets of countable sets are countable
Canonical name SubsetsOfCountableSetsAreCountable
Date of creation 2013-03-22 15:45:56
Last modified on 2013-03-22 15:45:56
Owner beke (12826)
Last modified by beke (12826)
Numerical id 7
Author beke (12826)
Entry type Corollary
Classification msc 03E10