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Chapter 1: Problem 9

Prove in detail that if \(S\) and \(T\) are denumerable, then \(S \cup T\) is denumerable.

Short Answer

Expert verified
In summary, \(S \cup T\) is denumerable because it represents the union of two denumerable sets which could be mapped one-to-one to the set of natural numbers.

Step by step solution

01

Define Terms

Start by defining the terms. A set \(S\) is called denumerable if there exists a bijective function from \(S\) to the set of natural numbers (or some subset of natural numbers). This means you can set up a one-to-one correspondence between the elements in \(S\) and the elements in the set of natural numbers. Similarly, a set \(T\) is denumerable if there exists such a bijective function for \(T\). The union of the two sets \(S \cup T\) is a set containing all the elements that are in \(S\), in \(T\), or in both \(S\) and \(T\).
02

Form Subsets

Denumerable sets are countable. This means that every element in \(S\) and \(T\) can be associated with a unique natural number. Even if the combined set \(S \cup T\) has 'more' elements, there is no limit on natural numbers which can be associated with the elements in \(S \cup T\). Split \(S \cup T\) into two disjoint subsets - one subset that comprises the elements only found in \(S\) but not \(T\), and another that includes the elements only found in \(T\) but not \(S\). Both these subsets are denumerable since the elements of each can be associated with natural numbers in a one-to-one manner.
03

Combine Subsets

The union \(S \cup T\) can be thought as the union of two denumerable (and hence countable) sets obtained in step 2. Therefore, to demonstrate \(S \cup T\) is denumerable, one must establish a bijection between \(S \cup T\) and a set of natural numbers. Since both subsets are denumerable, one can create two lists of natural numbers, corresponding to the two subsets. These lists can then be merged into one list by alternating elements from each list. The resulting list is still countable (since we're drawing from natural numbers, which are infinite), meaning \(S \cup T\) is denumerable.

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