Mapping Even Numbers to Natural Numbers

Show that all even numbers are countable by mapping them onto the set of natural numbers.

In this problem, the objective is to demonstrate the countability of even numbers by establishing a bijection with the set of natural numbers. Countability is a core concept in real analysis and set theory, focusing on the ability to list elements of a set in a one-to-one correspondence with the natural numbers. The task at hand reveals that both infinite sets, the set of even numbers and the set of natural numbers, share this property despite their differing appearances at first glance.

To approach this problem, consider the fundamental definition of even numbers: any integer that can be expressed as 2 times another integer. Constructing a mapping, or a function, that pairs each natural number to a unique even number involves a strategic selection of such pairs. A simple and elegant method is to associate each natural number n with the even number 2n. This creates a function from the natural numbers to the even numbers that is both injective (no two natural numbers map to the same even number) and surjective (every even number has a natural number that maps to it), thus forming a bijection.

Understanding why this mapping works requires grasping the notions of injectivity and surjectivity: injectivity ensures no duplication in the mapping process, while surjectivity confirms completeness. Such problem solving enhances comprehension of not just the mechanics of countability, but of broader mathematical communication about the structure and comparison of infinite sets. Exploring these mappings also sets the stage for higher order concepts such as the countability of rational numbers and the uncountability of real numbers.