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. (a) {1, 3, 5, 7, 9} A. countably infinite. B. countable. C. denumerable. D. uncountable. (b) N A. denumerable. B. uncountable. C. countably infinite. D. countable. (c) Z A. uncountable. B. countably infinite. C. denumerable. D. countable. (d) Z times Z A. countable. B. countably infinite. C. denumerable. D. uncountable.

Solution

Cardinal Number is the number of elements in a set.

If a set has a cardinal number less than the cardinal number of a set of all Natural numbers than it is COUNTABLE otherwise uncountable.

Cartesian Product A x A of a set A is countable if A is countable.

Also, Z is Countable. Any finite set is countable.

Using above theoritical Statements we can say that....

(a) {1,3,5,7,9} is countable. As it is a finite set. It cannot be countable infinite as it is finite. Also not denumerable as it is not surjective. It is not uncountable for sure.

(b) N is countable. It is countably infinite as it is infinite. Also denumerable as it is bijective to N. It is not uncountable for sure as it is already countable.

(c) Z is countable. As there exist a bijective function from Z -> N. It is countably infinite. It is denumerable is surjective.

(a) Z x Z is countable. It is countably infinite. It is not denumerable as it is not surjective to N. It is not uncountable for sure as already countable

Hence,

(a) B

(b) A C D

(c) B C D

(d) A B