Question

Determine whether each of these sets is nite, countably in nite, or uncount- able. For those that are countably in nite, exhibit a one-to-one correspondence between the set of positive integers and that set. For those that are nite or uncountable, explain your reasoning. a. integers that are divisible by 7 or divisible by 10?

Answer 1

Integers that are divisible by 7 are countably infinite

The set are {7, 14, 21, 28, 35, 42, 49,......

Integers that are divisible by 10 are countably infinite

The set are {10, 20, 30, 40, 50, 60, 70,........

Also for the negative integers divisible by 7 or 10{.......,-49, -42, -35, -28, -21, -14, -7} or {........,-70, -60, -50, -40, -30, -20, -10} are countably infinite

Explanation:

See the set of integers divisible by 7 or 10,

{7, 14, 21, 28, 35, 42, 49,........} Or

{10, 20, 30, 40, 50, 60, 70,.......}

Can be map one-to-one to {1,2,3,4,5,6,7,...... The set of natural numbers or positive Integers.

So also, it is applicable in negative integers divisible by 7 or 10.

Therefore, they are countably infinite

A set is uncountable if it contains so many elements that they cannot be put in one-to-one correspondence with the set of natural numbers(Positive Integers). In other words, there is no way that one can count off all elements in the set in such a way that, even though the counting will take forever, you will get to any particular element in a finite amount of time.

WHILE

A set is countable or countably finite if it contains so many elements that they CAN be put in one-to-one correspondence with the set of natural numbers(Positive Integers)