Let’s refresh the definitions we will need to solve this exercise.
Definition: “Let A and B be sets. The difference of A and B, denoted by A − B, is the set containing those elements that are in A but not in B. The difference of A and B is also called the complement of B with respect to A.”
Definition: “A set that is either finite or has the same cardinality as the set of positive integers is called countable. A set that is not countable is called uncountable. When an infinite set S is countable, we denote the cardinality of S by א0 (where א is aleph, the first letter of the Hebrew alphabet). We write |S| = א0 and say that S has cardinality “aleph null.”
Both definitions were taken from the textbook Discrete Mathematics and its Applications by Rosen.
a) finite
A=B=ℝ
A-B=𝝓
Therefore, A-B is finite.
b) countably infinite
A=ℝ, B=ℝ – ℤ
A-B = ℤ
Therefore, A-B is countable infinite.
c) uncountable
A is the set of real numbers between 0 and 1.
B is the set of real numbers between 2 and 3.
A-B equals the set of real numbers between 0 and 1.
Therefore, A-B is uncountable.
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