Question

We know that the union of two countable sets is countable. If A is an uncountable set and B is a countable set, must A l B be uncountable? Justify your answer. (AIB represents A - B)

Answer 1

The union of an uncountable set A and a countable set B can result in either a countable or an uncountable set, depending on the specific sets involved.

Step-by-step explanation:

No, the union of an uncountable set A and a countable set B does not necessarily result in an uncountable set. In fact, it can be both countable and uncountable, depending on the specific sets involved. The size of a set is determined by its cardinality, and if the union of two sets leads to a set with the same cardinality as one of the original sets, then the resulting set will also be countable. However, if the resulting set has a larger cardinality than either of the original sets, then it will be uncountable.

For example, let's consider the set A = {1, 2, 3,...}, which is an uncountably infinite set, and the set B = {a,b, c,...}, which is a countably infinite set. If we take the union of A and B, we get A ∪ B = {1,2,3,...,a,b,c,...}. In this case, the resulting set is still countable because it has the same cardinality as set A. However, if we take the union of A and another uncountable set C, the resulting set may be uncountable.