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?

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

Step-by-step 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)