Question

Prove that the function \(F: \mathbb{Z} \rightarrow \mathbb{Z}\) defined as \(F(n)=n+6\) is a bijection.

Short answer

The function is bijective since it is both injective and surjective.

Step-by-step solution

Prove Function is Injective

A function is injective (one-to-one) if distinct elements in the domain map to distinct elements in the codomain. So assume that for two elements \(m, n \in \mathbb{Z}\), \(F(m) = F(n)\). This implies \(m + 6 = n + 6\). By subtracting 6 from both sides, solving gives \(m = n\). Thus, \(F\) is injective.

Prove Function is Surjective

A function is surjective (onto) if every element in the codomain is the image of at least one element in the domain. Let \(y\) be an arbitrary element in \(\mathbb{Z}\). We want to find \(x\) in the domain such that \(F(x) = y\). Notice that \(x = y - 6\) satisfies \(F(x) = x + 6 = (y - 6) + 6 = y\). Since \(x\) is an integer whenever \(y\) is an integer, \(F\) is surjective.

Conclusion of Bijection

Since the function \(F\) is both injective and surjective, it is bijective by the definition of a bijection (one-to-one and onto mapping). Therefore, \(F(n) = n + 6\) is a bijection from \(\mathbb{Z}\) to \(\mathbb{Z}\).

Key concepts

Injective Function

An injective function, also known as a one-to-one function, is a pivotal concept in understanding bijections. In an injective mapping, different elements in the domain correspond to different elements in the codomain. This is crucial because it assures that no two distinct domain values share the same image post-mapping.

For instance, consider our given function, defined as \( F(n) = n + 6 \) from integers \( \mathbb{Z} \) to integers \( \mathbb{Z} \). To demonstrate it is injective, assume that \( F(m) = F(n) \) for some integers \( m \) and \( n \). Substitution gives us \( m + 6 = n + 6 \). By simply subtracting 6 from both sides, we discern that \( m = n \).

This result illustrates that distinct input values (\( m \) and \( n \)) cannot map to the same output value, confirming that the function \( F(n) = n+6 \) is injective.

Understanding injectivity helps when considering how functions could be reversed, which is essential in proving a function’s bijection.

Surjective Function

Surjective functions, or onto functions, complete another half of the bijection puzzle. Here, every target element in the codomain is tied to a source element in the domain. This means nothing in the codomain gets left unmatched.

In the case of our function \( F(n) = n + 6 \), to prove surjectivity, we take an arbitrary element \( y \) from the codomain, \( \mathbb{Z} \), and show that there exists an \( x \) within the domain such that \( F(x) = y \).

Consider solving \( y = x + 6 \) for \( x \), we find that \( x = y - 6 \). Since \( y \) is an integer, \( x \), or \( y - 6 \), will also be an integer, meeting the domain's condition of consisting entirely of integers. Therefore, for every integer \( y \), there is a corresponding integer \( x \) creating an image under \( F \), establishing the function as surjective.

This property ensures whole codomain coverage, essential for a function to be bijective.

Integer Mapping

Mapping integers is a foundational exercise in understanding functions like \( F(n) = n + 6 \). It involves linking elements from one integer set (the domain) to another (the codomain).

The function we examine connects integers to integers, mapping each \( n \) in \( \mathbb{Z} \) to \( n+6 \) in \( \mathbb{Z} \). This consistent addition of 6 highlights a linear shift rather than a distortion or interruption in mapping.

This type of transformation ensures each integer is translated systematically; \( n \) becomes \( n+6 \). Such mappings are powerful in providing clear demonstrations of how different mathematical properties—like injectivity and surjectivity—interact in the function. The shift by 6 guarantees that the range is densely populated with integers, which are the outputs.

Overall, understanding integer mapping is key in visualizing how specific algebraic manipulations result in functions that may or may not possess special properties like bijectiveness.