4.3.1. Theorem on Union of Countable Sets

Now we will prove some general results about the cardinality of sets both with respect to finite sets and infinite sets. So the first theorem is that if you have 2 sets A and B and if they are countable then their union is also countable.

This section introduces a theorem which states that the union of two countable sets is also countable. A set is defined as countable if there are either a finite number of elements or the same number of elements as the positive integers. In simpler terms, we can count the elements either through enumeration or a bijection with natural numbers. The theorem applies to any two countable sets without making assumptions about the nature of their elements, whether finite or infinite.

Imagine you have two baskets, one filled with apples and one with oranges, both having a finite number of fruits. If you were to combine these baskets into one larger basket, the total number of fruits in the new basket is still countable. If you had infinite baskets of apples and oranges, you can still count the total fruits in a larger sense. This analogy provides a tangible understanding of how merging two countable sets maintains their countable nature.

First of all there might be a possibility that A and B are not disjoint but to keep our proof simple without loss of generality we assume that A and B are disjoint. The proof can be simply adapted for the case when A and B are not disjoint.

In this explanation, the theorem is approached by first assuming that the two countable sets A and B are disjoint. This means there are no common elements between them. The reasoning behind this assumption is that it simplifies the proof without losing the generality of the theorem since the same argument can be applied even if sets overlap. Therefore, we focus on the cases where sets do not share elements to establish clarity in the proof process.

Think of a library with two sections: Fiction and Non-Fiction. If we treat each section as a disjoint set (no overlapping books), it is easier to count the total books. If there were some books that belonged to both sections, we would still be able to count all unique titles by adjusting our method but starting with the assumption of no overlap makes our counting simpler.

So we can have various cases depending upon whether A, and B are countably finite or countably infinite. Case 1 when both A and B are finite that means say if the cardinality of A is m and the cardinality of B is n then it is easy to see that the cardinality of union of A and B will be m + n which is a finite number and hence A U B is also countable.

The first case discusses what happens when both sets A and B are finite. If A has m elements and B has n elements, their union (A U B) will simply consist of m + n elements. This sum is finite and hence demonstrates that the union of two finite sets is indeed countable. This serves as a foundation for understanding more complex cases involving infinite sets.

Consider two boxes: one containing 3 red balls and the other containing 5 blue balls. The total number of balls when combined into one box is 3 + 5 = 8, which is a finite number. We can easily count the total number of balls in the combined box, underscoring the idea that the union of two finite groups results in a total that we can still count.

Case 2 is when exactly 1 of the set A and B is finite whereas the other set is countably infinite. Now again we can have 2 possible cases depending upon which of the 2 sets is countably infinite.

In this scenario, one of the sets is finite while the other is countably infinite. Regardless of which is which, since the infinite set contains an unending number of elements, the union (A U B) will also be countably infinite. The logic here is that adding any finite number of elements to an infinite set does not change its infinite nature. Thus, the union remains countable.

Imagine you have a box with infinitely many marbles (representing the infinite countable set). If you add a single marble to this box from another collection, you still have infinitely many marbles. The addition of a finite number doesn't alter the overall infinity; hence, it remains countable in a practical manner.

The third case is when both A and B set are infinite and countable, because I am assuming my A and B sets are countable and if A and B are both infinite that means both the cardinality of A as well as the cardinality of B is א .

This case involves two sets where both are countably infinite. When we take the union of them and through a specified enumeration sequence, we can list the elements ensuring none are missed. The method typically involves alternating between elements from sets A and B, allowing for a comprehensive count without skipping any elements. This establishes that A U B remains countably infinite.

Think of two infinite queues of people waiting for a concert ticket. One line represents all those in line for general admission tickets, while the other line is for VIP tickets. By alternating who gets the next ticket based on which line they came from, you are able to serve everyone in both lines completely without missing anyone, demonstrating how their union is still manageable.

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