Answered: Prove that a finite union of countable…

Prove that a finite union of countable sets is countable. Therefore, if A is uncountable and B ⊂ A is countable, then A \ B is uncountable.

College Algebra

1st Edition

ISBN:9781938168383

Author:Jay Abramson

Publisher:Jay Abramson

Chapter9: Sequences, Probability And Counting Theory

Section9.5: Counting Principles

Problem 38SE: Suppose a set A has 2,048 subsets. How many distinct objects are contained in A?

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Question

Prove that a finite union of countable sets is countable. Therefore, if A is uncountable and B ⊂ A is countable, then A \ B is uncountable.

Expert Solution

Step 1

First of all we have to prove that,

The union of countable family of countable sets is countable.

PROOF:

Without loss of generality, we can denote a countable family of sets by $A_{1}, A_{2}, A_{3} \dots \dots \dots \dots$

Suppose $a_{i 1}, a_{i 2}, a_{i 3}, \dots \dots \dots$ is an enumeration for $A_{i}$ . Then,

$Y_{i = 1}^{\infty} A_{i} = {a_{i j} : i \in ℕ, j \in ℕ}$

To the element $a_{i j}$ we assign a natural number $n$ so that there corresponds at most $n - 1$ distinct elements of A.

Therefore, A is countable by Countable Lemma.

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