Question

Give an example of two uncountable sets A and B such that A−B is

a) finite.

b) countably infinite

c) uncountable

Short answer

(a) The uncountable sets A and B are same for difference of these set to be finite.

(b) The uncountable sets A and B are set of real numbers and set of whole numbers such that difference of these two is uncountably infinite.

(c) The uncountable sets A is set of positive real numbers greater than 0 and B are set of set of positive real numbers less than 5 such that difference of these two is uncountable.

Step-by-step solution

Determination of uncountable sets for difference to be finite(a)

The subset of countably finite is countable and subset of countably infinite is countable infinite with one to one correspondence. The one to one correspondence is the relation of every element of one set with every element of other set.

Suppose uncountable sets A and B are such that elements of both sets are same and identical then difference of these sets will be a null set which is a example of finite set.

Therefore, the uncountable sets A and B are same for difference of these set to be finite.

Determination of uncountable sets for difference to be countably infinite(b)

Suppose uncountable sets A and B are such that A is set of real numbers R and set B is set of whole numbers W, then the set A-B will be a set containing only negative integers. The number of negative integers is countably infinite.

Therefore, the uncountable sets A and B are set of real numbers and set of whole numbers such that difference of these two is uncountably infinite.

Determination of uncountable sets for difference to be uncountable(c)

Suppose uncountable sets A and B are such that A is set of positive real numbers greater than 0 and set B is set of positive real numbers less than 5 then the set A-B will be a set containing only positive real numbers greater than 5 which will be uncountable.

Therefore, the uncountable sets A is set of positive real numbers greater than 0 and B are set of set of positive real numbers less than 5 such that difference of these two is uncountable.