Problem 6 - Applied Discrete Structures

Question

Inverse images. If \(f\) is any function from \(A\) into \(B,\) we can describe the inverse image as a function from \(B\) into \(\mathcal{P}(A)\), which is also commonly denoted \(f^{-1}\). If \(b \in B, f^{-1}(b)=\\{a \in A \mid f(a)=b\\}\). If \(f\) does have an inverse, the inverse image of \(b\) is \(\left\\{f^{-1}(b)\right\\}\). (a) Let \(g: \mathbb{R} \rightarrow \mathbb{R}\) be defined by \(g(x)=x^{2} .\) What are \(g^{-1}(4), g^{-1}(0)\) and \(g^{-1}(-1) ?\) (b) If \(r: \mathbb{R} \rightarrow \mathbb{Z},\) where \(r(x)=\lceil x\rceil,\) what is \(r^{-1}(1) ?\)

Step-by-Step Solution

Answer

(a) \(g^{-1}(4) = \{-2, 2\}\), \(g^{-1}(0) = \{0\}\), \(g^{-1}(-1) = \emptyset\). (b) \(r^{-1}(1) = (0, 1]\).

Step 1: Understand the Function

The problem provides a function \(g: \mathbb{R} \rightarrow \mathbb{R}\) defined by \(g(x) = x^2\). Our goal is to find the inverse image for specific values. Remember, \(g^{-1}(b) = \{ x \in \mathbb{R} \mid g(x) = b \}\).

Step 2: Find Inverse Image g^{-1}(4)

To find \(g^{-1}(4)\), solve the equation \(x^2 = 4\). This equation has two solutions: \(x = 2\) and \(x = -2\). Thus, \(g^{-1}(4) = \{ -2, 2 \}\).

Step 3: Find Inverse Image g^{-1}(0)

To find \(g^{-1}(0)\), solve the equation \(x^2 = 0\). This equation has a single solution: \(x = 0\). Therefore, \(g^{-1}(0) = \{ 0 \}\).

Step 4: Find Inverse Image g^{-1}(-1)

To find \(g^{-1}(-1)\), solve the equation \(x^2 = -1\). Since no real number squared gives a negative number, there are no solutions. Thus, \(g^{-1}(-1) = \emptyset \).

Step 5: Understand Second Function

The second function \(r: \mathbb{R} \rightarrow \mathbb{Z}\) is defined by \(r(x) = \lceil x \rceil \). The inverse image \(r^{-1}(1)\) is the set of all real numbers that \(r\) maps to 1.

Step 6: Find Inverse Image r^{-1}(1)

\(r(x) = 1\) implies that for real number \(x\), \(\lceil x \rceil = 1\). This means \(0 < x \leq 1\). Therefore, \(r^{-1}(1) = (0, 1]\).

Key Concepts

Functions

In mathematics, a function is a special kind of relationship between sets of things. Specifically, each input in one set (called the domain) is associated with exactly one output in another set (called the codomain). This process of mapping each input to an output is what defines a function. Some important properties of functions include:

• A function can only have one output for each input.
• Functions can often be represented by equations. For example, the function \( g(x) = x^2 \) takes a real number \( x \) from the domain and maps it to its square in the codomain.
• Functions are versatile and can be visualized as graphs, with inputs on the x-axis and outputs on the y-axis.

In our exercise, the function \( g: \mathbb{R} \rightarrow \mathbb{R} \) maps real numbers to real numbers by squaring them. When looking for inverse images, understanding the nature of this function is crucial.

Real Numbers

Real numbers are the set of all possible numbers along the number line. This set includes integers, fractions, and irrational numbers like \(\sqrt{2}\) or \(\pi\). Real numbers are characterized by their ability to be represented as either terminating or non-terminating decimals.Real numbers have a few important features:

• They can be positive, negative, or zero.
• They form a continuum without gaps, meaning between any two real numbers, there exist infinitely many others.
• In functions, they serve as a common domain and codomain, as seen in our \( g(x) = x^2 \) function.

For example, when evaluating \( g^{-1}(4) \), you explore which real numbers squared give you 4. Understanding real numbers helps solve such problems effectively.

Inverse Functions

An inverse function reverses the operation of an original function. Given a function \( f \), its inverse (if it exists) maps each element of the codomain back to the corresponding element in the domain.What to know about inverse functions:

• Not all functions have an inverse. For a function to have an inverse, it must be bijective (one-to-one and onto).
• In our exercise, we look at the inverse image \( g^{-1} \), which is not technically an inverse function but shares the idea of reversing the map.
• Inverse functions are symbolized as \( f^{-1} \) and operate as the opposite of \( f \).

In our scenario, finding \( g^{-1}(b) \) is about determining which inputs from the domain map to the output \( b \) in the codomain, like finding real numbers \( x \) such that \( x^2 = b \).

Set Theory

Set theory is the mathematical study of collections of objects, known as sets. A set is a well-defined group of distinct objects, regarded as an object itself.Key aspects of set theory include:

• Elements: objects within a set, denoted usually inside curly brackets { }.
• Operations: such as union, intersection, and difference of sets.
• Special sets: such as the empty set \( \emptyset \) (a set with no elements).

In this exercise, we use set notation to express inverse images. For instance, we use \( \{ -2, 2 \} \) to denote the set of numbers that are the pre-images under \( g(x) = x^2 \) for \( g(4) \). Understanding how set theory applies helps clarify how inverse images and functions interact within a mathematical framework.