Determine if each of the following sets is denumerable count

Determine if each of the following sets is denumerable, countable, uncountable or countably infinite. Check ALL correct answers. {1, 3, 5, 7, 9} countably infinite. countable. denumerable. uncountable. N denumerable. uncountable. countably infinite. countable. Z uncountable. countably infinite. denumerable. countable. Z times Z countable. countably infinite. denumerable. uncountable.

Solution

Countable and Uncountable Sets:

A set A is said to be finite, if A is empty or there is n N and there is a bijection f : {1,...,n} A. Otherwise the set A is called infinite.

Let X be an in nite set. We say that X is countably in finite if there is a bijection f : X N. If X is finite or countably infinite, we say that X is countable. A set is uncountable or uncountably in nite if it is not countable.

An infinite set is denumerable if it is equivalent to the set of natural numbers.

(a) Given that A = { 1 , 3 , 5 , 7 , 9 } which is finite as well as countable set.

(b) N , The set of all natural numbers is countably infinite

(c) The integers Z form a countable set. A bijection from Z to N is given by f ( k ) = 2 k if k 0 and f( k ) = 2( k ) + 1 if

k < 0. So, f maps 0 , 1 , 2 , 3 ... to 0 , 2 , 4 , 6 ... and f maps 1 , 2 , 3, 4 ... to 1 , 3 , 5 , 7 ...

(d) consider the set Z x Z which is a countable set