Question

(a) Prove that if \(X\) and \(Y\) are countable sets, so are \(X \cup Y, X \cap Y, X-Y\), and \(X \times Y\). (Caution: Countable means either finite or countably infinite, so there may be separate cases to consider.) (b) If \(X\) and \(Y\) are countably infinite, which of the following sets must be countably infinite: \(X \cup Y, X \cap Y, X-Y,\) and \(X \times Y ?\)

Short answer

(a) All sets are countable. (b) \(X \cup Y\) and \(X \times Y\) are countably infinite, \(X \cap Y\) and \(X - Y\) can be finite or countably infinite.

Step-by-step solution

Definitions and Cases for Countable Sets

A set is countable if it is either finite or countably infinite, meaning there is a bijection with the natural numbers. Consider sets \(X\) and \(Y\), each of which is countable.

Proving \(X \cup Y\) is Countable

If \(X\) and \(Y\) are both finite, \(X \cup Y\) is also finite, hence countable. If either or both are countably infinite, their union, \(X \cup Y\), can be shown to be countably infinite by constructing a bijection using the fact that the countable union of countable sets is countable (using interlacing or the pairing function).

Proving \(X \cap Y\) is Countable

The intersection \(X \cap Y\) is a subset of both \(X\) and \(Y\). A subset of a countable set is either finite or countably infinite; hence \(X \cap Y\) is countable (finite or countably infinite).

Proving \(X - Y\) is Countable

The set difference \(X - Y\) is a subset of \(X\), which is countable. A subset of a countable set is countable, so \(X - Y\) is countable (either finite or countably infinite).

Proving \(X \times Y\) is Countable

For \(X \times Y\), if both \(X\) and \(Y\) are finite, \(X \times Y\) is finite. If both are countably infinite, use the Cantor pairing function or diagonal argument to show \(X \times Y\) is countably infinite, as a countable product of countable sets is countable.

Analyzing Countably Infinite Cases

When both \(X\) and \(Y\) are countably infinite, \(X \cup Y\) can be shown to be countably infinite by interleaving elements or using bijections. \(X \cap Y\) could be finite or countably infinite, depending on the overlap of elements. \(X - Y\) remains countably infinite. \(X \times Y\) is countably infinite, proven by the pairing function or diagonal argument.

Key concepts

Countably Infinite Sets

Understanding what countably infinite sets are is essential in mathematics, particularly in set theory. A set is deemed countable if we can list its elements in a sequence similar to that of natural numbers. In other words, we can establish a one-to-one correspondence (bijection) between the elements of the set and the natural numbers.
Furthermore, a countably infinite set implies that though the set is infinite, its elements can be arranged in a sequence, just like natural numbers (e.g., the set of integers). Countable does not necessarily mean all sets are infinite. A finite set is also countable because you can count its elements, even if the counting eventually stops. The distinction between finite and infinite (when there’s a bijection with natural numbers) sets is at the heart of many set-related proofs.

Bijection

A bijection is a fundamental concept in understanding countable sets. It refers to a function that perfectly pairs every element of one set with a unique element of another set. No two elements in the first set are mapped to the same element in the second. Hence, both to-and-from directions of the function achieve this unique pairing.
Bijections help demonstrate if a set is countably infinite. If we can create a bijection between a set's elements and natural numbers, we confirm the set is countably infinite. For example, considering two sets, if every element in the first set corresponds to a unique element in the second set without omission, they are bijective.

Union and Intersection of Sets

The concepts of union and intersection of sets play a crucial role in determining if combined or overlapping sets maintain countability.

• Union of Sets (X \cup Y"): The union combines all elements from two sets, eliminating duplicates. If sets \(X\) and \(Y\) are countable, \(X \cup Y\) is countable because even in cases where one or both are countably infinite, constructing a bijection (through interleaving or pairing functions) shows it can be sequenced like natural numbers.
• Intersection of Sets (X \cap Y"): The intersection only includes elements that appear in both sets. This is always a subset of the original sets. A subset of a countable set is countable too. Consequently, \(X \cap Y\) stays countable, whether finite or countably infinite.

Cantor Pairing Function

The Cantor pairing function is an innovative mathematical tool that helps show how the Cartesian product of countable sets is still countable. The Cartesian product \(X \times Y\) involves forming pairs consisting of one element from \(X\) and another from \(Y\).
When both \(X\) and \(Y\) are countably infinite, the Cartesian product will also be countably infinite. The Cantor pairing function explicitly constructs a bijection between these pairs and natural numbers. It uses a formula to uniquely encode each pair of numbers into a single natural number, preserving countability. This concept underpins many proofs concerning the countability of products of infinite sets.