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An injective function, also known as a one-to-one function, ensures that every element in the domain is mapped to a distinct and unique element in the codomain. No two different inputs produce the same output. This property of being injective is crucial when establishing a function as a bijection.

In the case of our exercise, to demonstrate that the function \(f(n) = 2n + 13\) is injective, we assume that \(f(a) = f(b)\) for any natural numbers \(a\) and \(b\). By manipulating the equation, we see:\[(2a + 13) = (2b + 13)\]

This simplifies to \(2a = 2b\), which further simplifies to \(a = b\). Since the assumption that two distinct inputs could yield the same output leads us back to stating the inputs are equal, the function is confirmed to be injective. This means each natural number \(n\) maps to a unique odd integer greater than 13, guaranteeing no repetition or overlap in mapping.

A surjective function, or onto function, is characterized by its ability to map every element of the codomain to at least one element of the domain. Essentially, every element of the codomain has a pre-image within the domain. Surjectivity is pivotal for proving a function is a bijection, as it confirms complete coverage of the target set.

To verify the surjectivity of the function \(f(n) = 2n + 13\) for mapping from natural numbers to odd integers greater than 13, consider this: for each odd integer \(s\) in the set \(S\), we must find a natural number \(n\) such that \(f(n) = s\).

Given any \(s\), we can rearrange the function to find \(n\): \(n = (s - 13) / 2\). Since the output \(s\) belongs to the set of odd numbers greater than 13, this ensures that \(n\) remains a valid natural number. Therefore, each odd integer greater than 13 has a corresponding natural number \(n\), proving the function is surjective.

Natural numbers, often denoted by \(\mathbb{N}\), consist of the positive integers starting from 1, extending indefinitely: 1, 2, 3, 4, and so on. They are the foundation of counting and basic arithmetic operations. Natural numbers are simple, sequential, and intuitive, making them a fundamental set in mathematics.

In our function \(f(n) = 2n + 13\), \(n\) represents a natural number. Each natural number serves as a mechanism to generate odd integers greater than 13. This transformation is possible since each increment of \(n\) by 1 results in an increase by 2 in the output set \(S\), aligning perfectly with the integers in \(S\) being odd numbers spaced by 2 units each.

• Starts from 1: Begins with the first positive number
• Sequential and infinite: Continues indefinitely without skipping
• Used for simple counting: Acts as a basic counting set

This intuitive understanding helps when grasping how our function seamlessly pairs \(\mathbb{N}\) with the set of odd numbers greater than 13, showcasing both concepts through a clear mathematical relationship.