/*! 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 53 Show that if \(x\) is a real num... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 53

Show that if \(x\) is a real number and \(n\) is an integer, then a) \(x < n\) if and only if \(\lfloor x\rfloor < n .\) b) \(n < x\) if and only if \(n < \lceil x\rceil\)

Short Answer

Expert verified
a) \(x < n\) if and only if \(\lfloor x\rfloor < n\). b) \(n < x\) if and only if \(n < \lceil x\rceil\).

Step by step solution

01

Understand Floor and Ceiling Functions

The floor function, denoted by \(\floor{x}\), is the greatest integer less than or equal to \(x\). The ceiling function, denoted by \(\ceil{x}\), is the smallest integer greater than or equal to \(x\).
02

Restate the Given Information

We need to show two things: a) \(x < n\) if and only if \(\lfloor x\rfloor < n\). b) \(n < x\) if and only if \(n < \lceil x\rceil\).
03

Prove Part (a)

To prove \(x < n\) if and only if \(\lfloor x\rfloor < n\): Suppose \(x < n\). Since \(\lfloor x\rfloor \) is the greatest integer less than or equal to \(x\), it must be that \(\floor{x} \leq x < n\), thus \(\lfloor x\rfloor < n\). Conversely, suppose \(\lfloor x\rfloor < n\). Since \(\lfloor x\rfloor\) is the greatest integer less than or equal to \(x\), it follows that \(\lfloor x\rfloor \leq x < n\). Therefore, \(x < n\).
04

Prove Part (b)

To prove \(n < x\) if and only if \(n < \lceil x\rceil\): Suppose \(n < x\). Since \(\lceil x\rceil\) is the smallest integer greater than or equal to \(x\), it must be that \(n < x \leq \lceil x\rceil\). Therefore, \(n < \lceil x\rceil\). Conversely, suppose \(n < \lceil x\rceil\). Since \(\lceil x\rceil\) is the smallest integer greater than or equal to \(x\), it follows that \(\floor{x} < \lceil x\rceil \leq x\), hence \(n < x\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Floor function
The floor function is a critical concept in discrete mathematics, denoted as \(\floor{x}\). It represents the greatest integer that is **less than or equal to** a given real number, \(x\). Essentially, the floor function rounds down to the nearest integer. For instance, \(\floor{3.7} = 3\) and \(\floor{-2.3} = -3\).Understanding the floor function is vital for dealing with inequalities involving real numbers and integers. In terms of the exercise provided, to show \(x < n\) if and only if \(\floor{x} < n\), we leverage the property that \(\floor{x}\) is always less than or equal to \(x\). Thus, if \(x < n\), it directly implies \(\floor{x} < n\). Conversely, if \(\floor{x} < n\), since \(\floor{x} \leq x\), it follows that \(x < n\) remains true.
Ceiling function
The ceiling function, written as \(\floor{x}\), represents the smallest integer **greater than or equal to** a given real number \(x\). This function rounds up to the nearest integer. For example, \(\floor{2.1} = 3\) and \(\floor{-1.9} = -1\).This function comes into play when proving inequalities like \(n < x\) if and only if \(n < \floor{x}\). Given \(n < x\), the inherent property of the ceiling function tells us that \(\floor{x}\) is either equal to or greater than \(x\). This assures us that \(n < \floor{x}\) holds true. Conversely, if \(n < \floor{x}\), then clearly \(n < x\), since \(\floor{x}\) is the smallest integer not less than \(x\).
Inequalities
Inequalities are fundamental in understanding relationships between numbers and applying the floor and ceiling functions. They help us describe ranges and conditions under which certain statements hold true.In the outlined problem, inequalities are utilized to connect real numbers with integers through floor and ceiling functions. In the steps provided, we see assertions like \(x < n\) corresponding to \(\floor{x} < n\) which leverage the understanding of inequalities. By correctly applying these concepts, students can solve and prove these relationships effectively.It鈥檚 important to visualize these inequalities when using floor and ceiling functions. Consider how changing values of \(x\) affect the floor or ceiling of \(x\), ensuring the inequalities hold strong. This deeper grasp of inequalities helps students master discrete mathematics problems.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

What subsets of a finite universal set do these bit strings represent? a) the string with all zeros b) the string with all ones

a) Show that the system of simultaneous linear equations $$ \begin{array}{l}{a_{11} x_{1}+a_{12} x_{2}+\cdots+a_{1 n} x_{n}=b_{1}} \\\ {a_{21} x_{1}+a_{22} x_{2}+\cdots+a_{2 n} x_{n}=b_{2}} \\ {\vdots} \\ {a_{n 1} x_{1}+a_{n 2} x_{2}+\cdots+a_{n n} x_{n}=b_{n}}\end{array} $$ in the variables \(x_{1}, x_{2}, \ldots, x_{n}\) can be expressed as \(\mathbf{A X}=\mathbf{B},\) where \(\mathbf{A}=\left[a_{i j}\right], \mathbf{X}\) is an \(n \times 1\) matrix with \(x_{i}\) the entry in its \(i\) th row, and \(\mathbf{B}\) is an \(n \times 1\) matrix with \(b_{i}\) the entry in its ith row. b) Show that if the matrix \(\mathbf{A}=\left[a_{i j}\right]\) is invertible (as defined in the preamble to Exercise 18\()\) , then the solution of the system in part (a) can be found using the equation \(\mathbf{X}=\mathbf{A}^{-1} \mathbf{B}\) .

What do we know about the sizes of the matrices \(\mathbf{A}\) and \(\mathbf{B}\) if both of the products \(\mathbf{A B}\) and \(\mathbf{B} \mathbf{A}\) are defined?

a) Show that if a set \(S\) has cardinality \(m,\) where \(m\) is a positive integer, then there is a one-to-one correspondence between \(S\) and the set \(\\{1,2, \ldots, m\\} .\) b) Show that if \(S\) and \(T\) are two sets each with \(m\) elements, where \(m\) is a positive integer, then there is a one-to-one correspondence between \(S\) and \(T\) .

Let \(A\) be an \(m \times n\) matrix and let 0 be the \(m \times n\) matrix that has all entries equal to zero. Show that \(A=0+A=\) \(\mathbf{A}+0 .\)
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.