Use the following outline to supply proofs for the statementsin Theorem 1.5.8.((i) If A1,A2, . . . Am are each countable sets, then the unionA1 ∪ A2 ∪ · · · ∪ Am is countable.)(a) First, prove statement (i) for two countable sets, A1 and A2. Example1.5.3 (ii) may be a useful reference. Some technicalities can be avoidedby first replacing A2 with the set B2 = A2\A1 = {x ∈ A2 : x /∈ A1}. Thepoint of this is that the union A1 ∪ B2 is equal to A1 ∪ A2 and the setsA1 and B2 are disjoint. (What happens if B2 is finite?)Now, explain how the more general statement in (i) follows.

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Set Theory

Every field in the present world involves a set. The theory of sets was developed by German mathematician Georg Cantor while working on the “problems of trigonometric series”.