Question

For each of the following functions, prove that the function is \(1-1\) or find an appropriate pair of points to show that the function is not \(1-1:\) (a) \(F: \mathbb{Z} \rightarrow \mathbb{Z}\) $$F(n)=\left\\{\begin{array}{ll}n^{2} & \text { for } n \geq 0 \\ -n^{2} & \text { for } n \leq 0\end{array}\right.$$ (b) \(F: \mathbb{R} \rightarrow \mathbb{R}\) $$F(x)=\left\\{\begin{array}{ll}x+1 & \text { for } x \in \mathbb{Q} \\ 2 x & \text { for } x \notin \mathbb{Q}\end{array}\right.$$ (c) \(F: \mathbb{R} \rightarrow \mathbb{R}\) $$F(x)=\left\\{\begin{array}{ll}3 x+2 & \text { for } x \in \mathbb{Q} \\ x^{3} & \text { for } x \notin \mathbb{Q}\end{array}\right.$$ (d) \(F: Z \rightarrow \mathbb{Z}\) $$F(n)=\left\\{\begin{array}{ll}n+1 & \text { for } n \text { odd } \\ n^{3} & \text { for } n \text { even }\end{array}\right.$$

Short answer

Functions (a), (c), and (d) are not one-to-one; function (b) is one-to-one.

Step-by-step solution

Analyze Function (a)

Function (a) is defined as \( F(n) = n^2 \) for \( n \geq 0 \) and \( F(n) = -n^2 \) for \( n \leq 0 \). To check if this is a one-to-one function, consider \( F(n_1) = F(n_2) \). If \( n_1 \) and \( n_2 \) are both non-negative, it implies \( n_1^2 = n_2^2 \) which means they could have opposite signs. Similarly, for negative values, \( F(n_1) \) and \( F(n_2) \) might still be equal if they have the same magnitude, hence \( n_1 \) and \( n_2 \) might not be equal. Therefore, \( F \) is not one-to-one. For instance, substitute \( n_1 = 1 \) and \( n_2 = -1 \), we find \( F(1) = 1 \) and \( F(-1) = -1 \); however, \( F(0) = 0 \) and conflicts like \( F(-1) = 1 \) can occur. Hence, \( F(n) \) is not one-to-one.

Analyze Function (b)

Function (b) is defined as \( F(x) = x + 1 \) for rational \( x \) and \( F(x) = 2x \) for irrational \( x \). Check \( F(x_1) = F(x_2) \), requiring both rational or both irrational numbers, or one rational and one irrational forming equal outputs, which is impossible. Thus, consistency for \( F(x_1) = F(x_2) \) holds only trivially when \( x_1 = x_2 \), indicating \( F \) is one-to-one.

Analyze Function (c)

Here, \( F(x) = 3x + 2 \) for rational \( x \) and \( F(x) = x^3 \) for irrational \( x \). Demonstrate inconsistency: consider \( x_1 = 1 \) (rational) and \( x_2 = 1.26^{1/3} \) (irrational), we get \( F(1)=5 \) and \( F(x_2)=2 \), but \( F(x) \) results vary radically without equivalence for differing rationals and irrationals leading to the same \( F(x) \). The function is not one-to-one due to this overlap.

Analyze Function (d)

For \( F(n) = n + 1 \) when \( n \) is odd and \( F(n) = n^3 \) when \( n \) is even, check \( F(n_1)=F(n_2) \). Considering odd \( n \) mapped to even odd increasing linearity or cubes for even, leads to clash. For example, consider odd \( n=3,5 \) or even \( 2,4 \) values yielding conflicts like \( F(3)=4 \) and \( F(-3)=28 \) (cubic even changes). Hence, outputs vary precluding singular mapping to irreducible equal values, indicting F is not one-to-one.

Key concepts

Functions and Relations

Functions are a fundamental concept in mathematics that provide a way to model relationships between elements of one set, called the domain, and elements of another set, called the range. A function, often denoted by letters such as \( F \) or \( f \), is defined precisely when each element from the domain is paired with exactly one element from the range. However, two different elements in the domain might correspond to the same element in the range. This concept is central when discussing functions' injectiveness or being one-to-one, where unique outputs are expected for distinct inputs.
Relations are broader, encompassing any association between elements of two sets without the strict restriction of uniqueness found in functions. In the exercises provided, the focus lies in determining when a function outputs a unique result for each input, which requires checking if the function is indeed one-to-one, as seen in the analysis processes for various functions provided in different examples.

Rational and Irrational Numbers

Rational and irrational numbers are two fundamental classifications within real numbers. A rational number is any number that can be expressed as the quotient of two integers, where the denominator is non-zero. Common examples include fractions like \( \frac{1}{2} \) and negative fractions like \( -\frac{5}{3} \). Everything that can be terminated or repeating decimal can also be transformed into a fraction, being thus rational.
Irrational numbers, conversely, cannot be expressed as such a simple fraction. Their decimal expansions are non-repeating and non-terminating. Prominent examples include \( \pi \), the base of the natural logarithms \( e \), and the square root of non-perfect squares like \( \sqrt{2} \).
In the analyzed functions, differentiation between rational and irrational inputs impacts injectiveness. For example, function (b) differentiates actions based on the rationality of \( x \); hence, how these numbers are treated is crucial in deciding if a function remains one-to-one.

Mathematics Proofs

Mathematical proofs are rigorous arguments used to show the truth of a mathematical statement beyond all doubt. They play a critical role in formal mathematics, ensuring that each theorem and result is verified logically.
When verifying if a function is injective, mathematical proofs employ direct proofs or sometimes proofs by contradiction. In the given exercises, such proofs involve assuming apparent equal outputs from different inputs and showing or disproving their possibility. For instance, to show that function (b) is injective, one assumes \( F(x_1) = F(x_2) \) and checks rational/irrational patterns, verifying that distinct domain inputs cannot coincide at the same function value unless they are identical.

Discrete Mathematics

Discrete mathematics focuses on study areas that do not require the notion of continuity. It emphasizes countable objects and is foundational in computer science, cryptography, and combinatorics.
In relation to functions and proofs, discrete mathematics concerns itself with properties like bijections, injections, and surjections—all forms of function relations that define specific traits about them. The exercises dealing with integer mappings and rational versus irrational numbers use discrete definitions to form clear boundaries among function behaviors, such as injectiveness.
Through several sample functions, we see discrete arrangements in juxtaposing odd/even integers or rational/irrational numbers, each revealing injectiveness properties dependent on distinct mappings in their domain and range as discretely defined.