Bijection from Integers to Multiples of Two

Describe a bijection $heta$ from the set mathbb{Z} of integers to its proper subset $E$, such that $E$ is the set of multiples of two of each integer. Show that the function $heta(x) = 2x$ is bijective.

In this problem, you are tasked with exploring the concept of bijections within the realm of set theory and functions. Specifically, you must establish that the function theta, mapping integers to their multiples of two, forms a bijection between the set of integers and a proper subset consisting of these multiples. The core idea here is to rigorously demonstrate two major properties of bijective functions: injectivity and surjectivity.

First, examine the injectivity of the function. A function is injective, or one-to-one, if different inputs map to different outputs. For the function theta defined as $2x$, you will need to show that if theta(x1) equals theta(x2), then x1 must equate to x2, ensuring no two distinct integers map to the same output. This reflects the foundational understanding of how multiplication by two preserves uniqueness across inputs.

Next, the problem demands proof of surjectivity, or onto. A function is surjective if every element in the target set $E$ (multiples of two) corresponds to some integer mapped from the domain. You need to establish that for every integer $y$ in the subset $E$, there exists an integer $x$ in the domain of integers such that theta(x) yields $y$. Effectively, this demonstrates the full coverage of the subset $E$ by the function theta, thus proving that it is indeed surjective. Through mastering these concepts, foundational understanding of bijections in mathematics is expanded, shedding light on broader implications in mathematical theory and functions.