Question

Prove that a disjoint union of any finite set and any countably infinite set is countably infinite.

Proof: Suppose A is any finite set, B is any countably infinite set, and A and B are disjoint. By definition of disjoint,

A ∩ B = ∅

Then h is one-to-one because f and g are one-to one and A ∩ B = 0. Further, h is onto because f and g are onto and given any element x in

A ∪ B, x is in A or x is in B.

In case x is in A, then, since f is onto, there is an integer r in {1, 2, 3, ..., m} such that f(r) = x. Since r is in {1, 2, 3, ..., m}, r ≤ m, and so h(r) =_______.

In case x is in B, then, since g is onto, there is an integer s in ℤ⁺ such that g(s) = x. Let t = s + m. Then s = t − m. Also t __ m + 1, and thus h(t) = g(t − m) = g(s) =____.

Therefore, h is a one-to-one correspondence from ℤ⁺ to A ∪ B, and so A ∪ B is countably infinite by definition of countably infinite.

Answer 1

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