/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 1 Let \(R_{5}=\\{0,1,2,3,4\\} .\) ... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 1

Let \(R_{5}=\\{0,1,2,3,4\\} .\) Define \(f: R_{5} \rightarrow R_{5}\) by \(f(x)=x^{2}+4(\bmod 5)\), and define \(g: R_{5} \rightarrow R_{5}\) by \(g(x)=(x+1)(x+4)(\bmod 5)\). (a) Calculate \(f(0), f(1), f(2), f(3),\) and \(f(4)\). (b) Calculate \(g(0), g(1), g(2), g(3),\) and \(g(4)\). (c) Is the function \(f\) equal to the function \(g\) ? Explain.

Short Answer

Expert verified
The functions f and g are equal because their values match for each element in \(R_5\): f(0) = g(0) = 4, f(1) = g(1) = 0, f(2) = g(2) = 3, f(3) = g(3) = 3, and f(4) = g(4) = 0.

Step by step solution

01

(a) Calculate the values of f(x) for each element in \(R_5\)

For each x in \(R_5\), we calculate f(x) using the function definition: \(f(x)=x^2+4 (\mod 5)\) 1. f(0) = \(0^2+4 (\mod 5) = 4\) 2. f(1) = \(1^2+4 (\mod 5) = 5 (\mod 5) = 0\) 3. f(2) = \(2^2+4 (\mod 5) = 8 (\mod 5) = 3\) 4. f(3) = \(3^2+4 (\mod 5) = 13 (\mod 5) = 3\) 5. f(4) = \(4^2+4 (\mod 5) = 20 (\mod 5) = 0\) The values for f(x) are: f(0) = 4, f(1) = 0, f(2) = 3, f(3) = 3, f(4) = 0.
02

(b) Calculate the values of g(x) for each element in \(R_5\)

For each x in \(R_5\), we calculate g(x) using the function definition: \(g(x) = (x + 1)(x + 4) (\mod 5)\) 1. g(0) = \((0+1)(0+4) (\mod 5) = 4 (\mod 5) = 4\) 2. g(1) = \((1+1)(1+4) (\mod 5) = 10 (\mod 5) = 0\) 3. g(2) = \((2+1)(2+4) (\mod 5) = 18 (\mod 5) = 3\) 4. g(3) = \((3+1)(3+4) (\mod 5) = 28 (\mod 5) = 3\) 5. g(4) = \((4+1)(4+4) (\mod 5) = 40 (\mod 5) = 0\) The values for g(x) are: g(0) = 4, g(1) = 0, g(2) = 3, g(3) = 3, g(4) = 0.
03

(c) Determine if f and g are equal functions

To determine if f and g are equal functions, we need to compare the values of f(x) and g(x) for each element in \(R_5\). We can observe that: - f(0) = 4 and g(0) = 4 - f(1) = 0 and g(1) = 0 - f(2) = 3 and g(2) = 3 - f(3) = 3 and g(3) = 3 - f(4) = 0 and g(4) = 0 All the values of f(x) match the corresponding values of g(x) for each element in \(R_5\). Therefore, the functions f and g are equal.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Function Equality
Two functions are considered equal if they produce the same output for every input in their domain. To check for equality, we need to evaluate both functions at each element within the domain and compare the results.

In the given exercise, functions \(f\) and \(g\) are defined over the set \(R_5\), which contains the elements \{0, 1, 2, 3, 4\}. We calculated the outputs of both functions at each element and compared them.
  • For \(f(0)\) and \(g(0)\), both outputs are 4.
  • For \(f(1)\) and \(g(1)\), both outputs are 0.
  • For \(f(2)\) and \(g(2)\), both outputs are 3.
  • For \(f(3)\) and \(g(3)\), both outputs are 3.
  • For \(f(4)\) and \(g(4)\), both outputs are 0.
This comparison of outputs confirms that functions \(f\) and \(g\) are indeed equal, as they produce indistinguishable results across the domain \(R_5\). Thus, in terms of function equality, even if the expressions look different, their equivalency is proved through equal mappings of inputs to outputs.
Function Evaluation
Function evaluation involves calculating the output of a function for a given input. In this exercise, we evaluated two different mathematical expressions \(f(x)\) and \(g(x)\) over a set of elements. Function \(f\) was \(f(x) = x^2 + 4 \, (\bmod \, 5)\), and function \(g\) was \(g(x) = (x + 1)(x + 4) \, (\bmod \, 5)\).

Evaluating a function in modular arithmetic means you perform the operations and take the result modulo the given number (in this case, 5). Let's break it down with examples:
  • For evaluating \(f(3)\), compute \(3^2 + 4 = 13\), then \(13 \, (\bmod \, 5) = 3\).
  • For evaluating \(g(3)\), compute \((3+1)(3+4) = 28\), then \(28 \, (\bmod \, 5) = 3\).
This shows that to evaluate a function, you substitute the input into the function's formula, execute the arithmetic, and simplify using the modular operation. This process helps determine the functional value for distinct inputs and provides insights into how functions behave under specific modulo constraints.
Finite Fields
A finite field, also known as a Galois field, is a set containing a finite number of elements where you can perform addition, subtraction, multiplication, and division (except by zero ). This forms the basis for modular arithmetic.

In this exercise, the set \(R_5 = \{0, 1, 2, 3, 4\}\) forms a finite field under the modulus 5. Here are a few key concepts about finite fields relevant to our exercise:
  • Closure: Any operation (addition, subtraction, multiplication, division excluding zero-division) between two elements in \(R_5\) results in another element within \(R_5\).
  • Associativity and Commutativity: Addition and multiplication are both associative and commutative within the field.
  • Distributive Law: Multiplication over addition holds, which means \((a+b) \cdot c = a\cdot c + b\cdot c\).
  • Existence of Identical Elements: Element 0 and 1 serve as the additive and multiplicative identities, respectively.
Working within finite fields, as demonstrated in this exercise, allows us to understand function behaviors in modular systems and explore properties like equal outputs for different operations that would otherwise be distinct in regular arithmetic.

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Most popular questions from this chapter

(a) Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be defined by \(f(x)=x^{2} .\) Explain why the inverse of \(f\) is not a function. (b) Let \(\mathbb{R}^{*}=\\{t \in \mathbb{R} \mid t \geq 0\\} .\) Define \(g: \mathbb{R}^{*} \rightarrow \mathbb{R}^{*}\) by \(g(x)=x^{2} .\) Ex- plain why this squaring function (with a restricted domain and codomain) is a bijection. (c) Explain how to define the square root function as the inverse of the function in Exercise (8b). (d) True or false: \((\sqrt{x})^{2}=x\) for all \(x \in \mathbb{R}\) such that \(x \geq 0\) (e) True or false: \(\sqrt{x^{2}}=x\) for all \(x \in \mathbb{R}\)

The identity function on a set \(S,\) denoted by \(I_{S},\) is defined as follows: \(I_{S:} S \rightarrow S\) by \(I_{S}(x)=x\) for each \(x \in S .\) Let \(f: A \rightarrow B\) " (a) For each \(x \in A\), determine \(\left(f \circ I_{A}\right)(x)\) and use this to prove that \(f\). \(I_{A}=f\) (b) Prove that \(I_{B} \circ f=f\).

Let \(A\) be a nonempty set and let \(f: A \rightarrow A\). For each \(n \in \mathbb{N}\), define a function \(f^{n}: A \rightarrow A\) recursively as follows: \(f^{1}=f\) and for each \(n \in \mathbb{N}\) \(f^{n+1}=f \circ f^{n} .\) For example, \(f^{2}=f \circ f^{1}=f \circ f\) and \(f^{3}=f \circ f^{2}=\) \(f \circ(f \circ f)\) (a) Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) by \(f(x)=x+1\) for each \(x \in \mathbb{R}\). For each \(n \in \mathbb{N}\) and for each \(x \in \mathbb{R}\), determine a formula for \(f^{n}(x)\) and use induction to prove that your formula is correct. (b) Let \(a, b \in \mathbb{R}\) and let \(f: \mathbb{R} \rightarrow \mathbb{R}\) by \(f(x)=a x+b\) for each \(x \in \mathbb{R}\). For each \(n \in \mathbb{N}\) and for each \(x \in \mathbb{R},\) determine a formula for \(f^{n}(x)\) and use induction to prove that your formula is correct. (c) Now let \(A\) be a nonempty set and let \(f: A \rightarrow A\). Use induction to prove that for each \(n \in \mathbb{N}, f^{n+1}=f^{n} \circ f .\) (Note: You will need to use the result in Exercise (5).)

Exploring Composite Functions. Let \(A, B,\) and \(C\) be nonempty sets and let \(f: A \rightarrow B\) and \(g: B \rightarrow C\). For this activity, it may be useful to draw your arrow diagrams in a triangular arrangement as follows: It might be helpful to consider examples where the sets are small. Try constructing examples where the set \(A\) has 2 elements, the set \(B\) has 3 elements, and the set \(C\) has 2 elements. (a) Is it possible to construct an example where \(g \circ f\) is an injection, \(f\) is an injection, but \(g\) is not an injection? Either construct such an example or explain why it is not possible. (b) Is it possible to construct an example where \(g \circ f\) is an injection, \(g\) is an injection, but \(f\) is not an injection? Either construct such an example or explain why it is not possible. (c) Is it possible to construct an example where \(g \circ f\) is a surjection, \(f\) is a surjection, but \(g\) is not a surjection? Either construct such an example or explain why it is not possible. (d) Is it possible to construct an example where \(g \circ f\) is surjection, \(g\) is a surjection, but \(f\) is not a surjection? Either construct such an example or explain why it is not possible.

Creating Functions with Finite Domains. Let \(A=\\{a, b, c, d\\}, B=\) \(\\{a, b, c\\},\) and \(C=\\{s, t, u, v\\} .\) In each of the following exercises, draw an arrow diagram to represent your function when it is appropriate. (a) Create a function \(f: A \rightarrow C\) whose range is the set \(C\) or explain why it is not possible to construct such a function. (b) Create a function \(f: A \rightarrow C\) whose range is the set \(\\{u, v\\}\) or explain why it is not possible to construct such a function. (c) Create a function \(f: B \rightarrow C\) whose range is the set \(C\) or explain why it is not possible to construct such a function. (d) Create a function \(f: A \rightarrow C\) whose range is the set \(\\{u\\}\) or explain why it is not possible to construct such a function. (e) If possible, create a function \(f: A \rightarrow C\) that satisfies the following condition: For all \(x, y \in A,\) if \(x \neq y,\) then \(f(x) \neq f(y)\) If it is not possible to create such a function, explain why. (f) If possible, create a function \(f: A \rightarrow\\{s, l, u\\}\) that satisfies the following condition: For all \(x, y \in A,\) if \(x \neq y,\) then \(f(x) \neq f(y) .\) If it is not possible to create such a function, explain why.
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