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Chapter 5: Problem 2

For each of the following functions \(f: \mathbf{Z} \rightarrow \mathbf{Z}\), determine whether the function is one-toone and whether it is onto. If the function is not onto, determine the range \(f(\mathbf{Z})\). a) \(f(x)=x+7\) b) \(f(x)=2 x-3\) c) \(f(x)=-x+5\) d) \(f(x)=x^{2}\) e) \(f(x)=x^{2}+x\) f) \(f(x)=x^{3}\)

Short Answer

Expert verified
Function a) is one-to-one and onto with range all integers. Function b) is one-to-one, not onto, range: even integers 卤3. Function c) is one-to-one, not onto, range: integers 鈮�5. Function d) is not one-to-one, onto for nonnegative integers, range: nonnegative integers. Function e) is not one-to-one, onto for integers 鈮� -1/4, range: integers 鈮� -1/4. Function f) is one-to-one and onto with range all integers.

Step by step solution

01

Function \(f(x)=x+7\)

This function is one-to-one and onto. Any two different integers will result in two different outputs, making it one-to-one. Every integer in the codomain can be reached by subtracting 7 from that integer in the domain, hence it is onto. The range is all integers.
02

Function \(f(x)=2x-3\)

This function is one-to-one but it is not onto. For any two different integers, the output will also be different, making it one-to-one. However, the function cannot produce all integers in the codomain. Specifically, it cannot produce odd integers. Therefore, the range is the set of all even integers plus or minus 3.
03

Function \(f(x)=-x+5\)

This function is one-to-one, but not onto. A mapping of different integers will yield different results, making it one-to-one. However, the function is limited in the integers it can produce, specifically it can only output integers less or equal to 5. Therefore, the range is the set of integers less or equal to 5.
04

Function \(f(x)=x^{2}\)

This function is not one-to-one, but it is onto when the codomain is limited to nonnegative integers. For two different integers, the output might be the same (e.g. \(f(-2)=f(2)=4\)), so the function is not one-to-one. However, any nonnegative integer can be produced by taking the square root. Thus, the range is the set of all nonnegative integers.
05

Function \(f(x)=x^{2}+x\)

Like the previous function, this function is not one-to-one and is onto when the codomain is limited to integers greater or equal to -1/4. For two different integers, there might be same output (e.g. \(f(-1)=f(0)=0\)), making it not one-to-one. The difference with the previous example is that one needs to solve a quadratic equation to determine possible outputs. Thus, the range is the set of all integers greater or equal to -1/4.
06

Function \(f(x)=x^{3}\)

This function is both one-to-one and onto. It is similar to the first example, but with a cubic function. Any two different integers produce different results since the cubic function is monotonically increasing for all integers. Therefore, this function is both one-to-one and onto. The range is all integers.

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Key Concepts

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

One-to-One Functions
In the realm of discrete mathematics, one-to-one functions play a significant role. Sometimes referred to as injective functions, they have a distinctive quality. A function is considered one-to-one if distinct inputs map to distinct outputs. This means that for every unique input value in the domain, there is a unique output value in the codomain. An easy way to verify if a function is one-to-one is to check if different values of the domain always lead to different values of the range.

Let's take, for example, the function \( f(x) = x + 7 \). It's one-to-one because adding 7 to any two different integers will always result in different outcomes. Conversely, the function \( f(x) = x^2 \) is not one-to-one. For instance, both \( f(2) = 4 \) and \( f(-2) = 4 \), which means different inputs yield the same output.
  • Injective functions ensure distinct mappings from domain to codomain.
  • Verification involves checking if distinct inputs lead to distinct outputs.
Onto Functions
Another key player in discrete mathematics is the onto function, also known as a surjective function. An onto function perfectly covers every element of the codomain. In simpler terms, every possible output in the codomain has at least one corresponding input from the domain.

Consider the function \( f(x) = x + 7 \), which is onto. Every integer can be expressed by subtracting 7 from another integer, ensuring all integers are covered. On the other hand, \( f(x) = 2x - 3 \) is not onto when the codomain is all integers because it cannot produce odd integers.
  • Surjective functions cover the entire codomain.
  • Determining onto nature involves checking if all codomain elements are achieved by the function.
Integer Functions
In mathematics, integer functions are those that take integer inputs and yield integer outputs. They are framed within the context of \( f: \mathbf{Z} \rightarrow \mathbf{Z} \), denoting functions from the set of integers to the set of integers.

Discussing integer functions involves understanding various operations like addition, subtraction, and multiplication applied to integers. For instance, the function \( f(x) = 2x - 3 \) takes an integer \( x \), multiplies it by 2, and subtracts 3, inevitably resulting in another integer.
  • Integer functions operate by mapping integers to integers.
  • They help in analyzing discrete processes and data structures.
  • Often used in computer science and mathematical reasoning problems.
Function Range
The range of a function refers to the set of all possible output values which the function can produce. Calculating the range of a function involves determining what outputs result from plugging all possible inputs into the equation.

For the function \( f(x) = x^2 \), the range is the set of all nonnegative integers due to the square operation. On the other hand, the range of \( f(x) = -x + 5 \) is integers less than or equal to 5. Exploring the range helps in understanding the limits and behavior of the function under certain conditions.
  • Range represents all potential outputs of a function.
  • Understanding range informs on function behavior and limitations.

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Most popular questions from this chapter

For \(f, g: \mathbf{Z}^{+} \rightarrow \mathbf{R}\) we define \(f+g: \mathbf{Z}^{+} \rightarrow \mathbf{R}\) by \((f+g)(n)=f(n)+g(n)\), for \(n \in \mathbf{Z}^{+}\). [Note: The plus sign in \(f+g\) is for the addition of the functions \(f\) and \(g\), where the plus sign in \(f(n)+g(n)\) is for the addition of the real numbers \(f(n)\) and \(g(n)\).] a) Let \(f_{1}, g_{1}: \mathbf{Z}^{+} \rightarrow \mathbf{R}\) with \(f \in O\left(f_{1}\right)\) and \(g \in O\left(g_{1}\right)\). If \(f_{1}(n) \geq 0, g_{1}(n) \geq 0\), for all \(n \in \mathbf{Z}^{+}\), prove that \((f+g) \in O\left(f_{1}+g_{1}\right)\). b) If the conditions \(f_{1}(n) \geq 0, g_{1}(n) \geq 0\), for all \(n \in \mathbf{Z}^{+}\), are not satisfied, as in part (a), provide a counterexample to show that \(f \in O\left(f_{i}\right)\). \(g \in O\left(g_{1}\right) \Rightarrow(f+g) \in O\left(f_{1}+g_{1}\right) .\)

For distinct primes \(p, q\) let \(A=\left\\{p^{m} q^{n} \mid 0 \leq m \leq 31,0 \leq n \leq 37\right\\}\). a) What is \(|A|\) ? b) If \(f: A \times A \rightarrow A\) is the closed binary operation defined by \(f(a, b)=\operatorname{gcd}(a, b)\), does \(f\) have an identity element?

Let \(A_{i}, 1 \leq i \leq 5\), be the domains for a table \(D \subseteq X_{i=1}^{5} A_{i}\), where \(A_{1}=\\{1,2\\}\) (used to identify the daily vitamin capsule produced by two pharmaceutical companies), \(A_{2}=\) \(\\{\mathrm{A}, \mathrm{D}, \mathrm{E}\\}\), and \(A_{3}=A_{4}=A_{5}=\mathbf{Z}^{+}\). The table \(D\) is given as Table \(5.8\). Table \(5.8\) \begin{tabular}{|c|c|c|c|c|} \hline Vitamin Capsule & Vitamin Present in Capsule & Amount of Vitamin in Capsule in IU" & Dosage: Capsules/Day & No. of Capsules per Bottle \\ \hline 1 & A & 10,000 & 1 & 100 \\ 1 & D & 400 & 1 & 100 \\ 1 & E & 30 & 1 & 100 \\ 2 & A & 4,000 & 1 & 250 \\ 2 & D & 400 & 1 & 250 \\ 2 & E & 15 & 1 & 250 \\ \hline \end{tabular} \- IU = international units a) What is the degree of the table? b) What is the projection of \(D\) on \(A_{1} \times A_{2}\) ? on \(A_{3} \times A_{4} \times A_{5}\) ? c) This table has no primary key. (See Exercise 15.) We can, however, define a composite primary key as the cross product of a minimal number of domains of the table, whose components, taken collectively, uniquely identify each list of \(D\). Determine some composite primary keys for this table.

For \(A=\\{a, b, c\\}\), let \(f: A \times A \rightarrow A\) be the closed binary operation given in Table \(5.6\). Give an example to show that \(f\) is not associative. Table \(5.6\) \begin{tabular}{|l|lll|} \hline\(f\) & \(a\) & \(b\) & \(c\) \\ \hline\(a\) & \(b\) & \(a\) & \(c\) \\ \(b\) & \(a\) & \(c\) & \(b\) \\ \(c\) & \(c\) & \(b\) & \(a\) \\ \hline \end{tabular}

The following Pascal program segment implements an algorithm for determining the maximum value in an array \(A[1], A[2], A[3], \ldots, A[n]\) of integers. The array and the value of \(n(\geq 2)\) are supplied earlier in the program; the integer variables \(i\) and Max are declared at the start of the program. (Here the entries in the array need not be distinct) Begin \(\mathrm{Max}:=\mathrm{A}[1]\) For \(1:=2\) to \(n\) do If \(A[1]>M a x\) then Max : = A[1] End; a) If the worst-case complexity function \(f(n)\) for this segment is determined by the number of times the comparison \(A[i]>\mathrm{Max}\) is executed, find the appropriate "big-Oh" form for \(f\). b) What can we say about the best-case and average-case complexities for this implementation?
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