Question

Determine whether each of these sets is finite, countably infinite, or uncountable. For those that are countably in- finite, exhibit a one-to-one correspondence between the set of positive integers and that set. a) the negative integers b) the even integers c) the integers less than 100 d) the real numbers between 0 and 1 2 e) the positive integers less than 1,000,000,000 f) the integers that ar emultiples of 7 .

Answer 1

a) the negative integers set A is countably infinite.

one-to-one correspondence with the set of positive integers:

f: Z+ → A, f(n) = -n

b) the even integers set A is countably infinite.

one-to-one correspondence with the set of positive integers:

f: Z+ → A, f(n) = 2n

c) the integers less than 100 set A is countably infinite.

one-to-one correspondence with the set of positive integers:

f: Z+ → A, f(n) = 100 - n

d) the real numbers between 0 and 12 set A is uncountable.

e) the positive integers less than 1,000,000,000 set A is finite.

f) the integers that are multiples of 7 set A is countably infinite.

one-to-one correspondence with the set of positive integers:

f: Z+ → A, f(n) = 7n

Explanation:

A set is finite when its elements can be listed and this list has an end.

A set is countably infinite when you can exhibit a one-to-one correspondence between the set of positive integers and that set.

A set is uncountable when it is not finite or countably infinite.