Injection from a Finite Set to Integers

Describe an injection from a set $X = \{x_1, x_2, \ldots, x_n\}$ with $n$ elements to the set $\mathbb{Z}$ of integers, where $\theta(x_i) = i$.

An injection, also known as an injective function or one-to-one function, is a type of mapping between two sets where each element of the first set is mapped to a unique element of the second set. In this problem, we explore the concept of constructing such an injection from a finite set X to the set of integers, Z. The specific mapping, theta, is defined such that each element x_i of X is mapped to an integer i. This guarantees that no two different elements in X map to the same integer, highlighting the one-to-one nature of the mapping.

Understanding injections is fundamental in set theory and functions, and is a stepping stone to more advanced concepts such as bijections and surjections. Injections play a critical role in determining the cardinality and structure of sets, allowing us to make comparisons and establish relationships between different mathematical entities.

This problem also emphasizes the importance of ordered mappings and how they maintain the distinctness of elements in their domain and codomain, a key property in many mathematical proofs and applications. The mapping technique used here can be applied broadly in algebra and discrete mathematics, including combinatorics and algorithm design, where the uniqueness of solutions and structures often hinges on constructively defining injective relationships.