/*! 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 2 Evaluate each, where \(n\) is an... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 2

Evaluate each, where \(n\) is an integer. $$\lfloor n / 2\rfloor$$

Short Answer

Expert verified
In both cases, when \(n\) is an even integer (\(n = 2k\)) or an odd integer (\(n = 2k+1\)), the value of the expression \(\lfloor \frac{n}{2} \rfloor\) is equal to \(k\).

Step by step solution

01

Case 1: n is even

If \(n\) is even, then we can write it as \(n = 2k\), where \(k\) is an integer. Now, let's evaluate the given expression: \(\lfloor \frac{n}{2} \rfloor = \lfloor \frac{2k}{2} \rfloor = \lfloor k \rfloor\) Since \(k\) is an integer, the floor function will have no effect, and the result will be: \(\lfloor k \rfloor = k\)
02

Case 2: n is odd

If \(n\) is odd, then we can write it as \(n = 2k+1\), where \(k\) is an integer. Now, let's evaluate the given expression: \(\lfloor \frac{n}{2} \rfloor = \lfloor \frac{2k+1}{2} \rfloor = \lfloor k + \frac{1}{2} \rfloor\) Since \(k\) is an integer, the value of \(k+\frac{1}{2}\) will be a half-integer value. As the floor function will return the largest integer less than or equal to the value inside, the result will be: \(\lfloor k + \frac{1}{2} \rfloor = k\) So, in both cases, the value of the expression \(\lfloor \frac{n}{2} \rfloor\) is equal to \(k\) for integer values of \(n\).

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.

Integer Division
Integer division is a mathematical operation where you divide one integer by another and then round down to the nearest whole number. In simpler terms, instead of providing a result with a decimal or fractional part, integer division delivers the quotient without any remainder.
  • For example, when you divide 7 by 3 in integer division, the result is 2, since 7 divided by 3 is approximately 2.33, but integer division ignores anything after the decimal point.
  • The floor function, indicated by \(\lfloor x \rfloor\), helps in integer division by returning the greatest integer less than or equal to \(x\). So even if the division yields a value like 2.5, the floor function makes it fall back to 2.
Understanding integer division is critical in computing and mathematics, as it turns continuous actions into discrete ones, so everything is a whole number, much like counting distinct objects.
Even and Odd Numbers
In mathematics, even and odd numbers form the basic foundation for modular arithmetic, which involves integers. Recognizing whether a number is even or odd is straightforward and crucial for working with integer division.
  • An even number can be divided by 2 without leaving a remainder. It has the form \(2k\), where \(k\) is an integer. Examples include: 4, 6, 8, etc.
  • An odd number, in contrast, leaves a remainder of 1 when divided by 2, expressed as \(2k + 1\). Examples include: 3, 5, 7, etc.
When you consider the floor function with these numbers:
  • For even numbers, dividing by 2 results in an integer, so the floor function doesn鈥檛 change the value. Thus, \(\lfloor \frac{2k}{2} \rfloor = k\).
  • For odd numbers, dividing by 2 gives a fraction \(k + \frac{1}{2}\). The floor function then rounds down to the nearest whole number, \(k\).
Discrete Mathematics
Discrete mathematics is a branch of mathematics focusing on distinct and separate values rather than continuous. It plays a vital role in mathematics and computer science by helping us model real-world problems with distinct entities, usually integers.
  • It covers a wide range of topics such as graph theory, combinatorics, logic, and integer arithmetic. Through these disciplines, discrete math aids in forming the theoretical foundation of computer science.
  • The concept of even and odd numbers, along with integer division, helps in computer algorithms and data structure, often defining how a program handles data.
By understanding how the floor function interacts with integers through integer division, discrete mathematics simplifies complex problems, translating them into concrete results that fit smoothly into programming languages used in computation and the digital world.

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

Prove. The cartesian product of two countable sets is countable.

Prove each, where \(x \in \mathbb{R}\) and \(n \in \mathbf{Z}.\) \(\left\lfloor\frac{n}{2}\right\rfloor=\frac{n-1}{2}\) if \(n\) is odd.

Evaluate each sum, where \(\delta_{i j}\) is defined as follows. $$ \delta_{i j}=\left\\{\begin{array}{ll}{1} & {\text { if } i=j} \\ {0} & {\text { otherwise }}\end{array}\right. $$ \(\left[\delta_{j} \text { is called Kronecker's delta, after the German mathematician Leopold }\right.\) Kronecker \((1823-1891) . ]\) $$ \sum_{i=1}^{3} \sum_{j=1}^{i}(j+3) $$

Evaluate each sum. $$\sum_{i=-1}^{4} 3$$

Determine if the functions are bijective. If they are not bijective, explain why. \(g: \Sigma^{*} \rightarrow \Sigma^{*}\) defined by \(g(w)=a w a,\) where \(\Sigma=\\{a, b, c\\}.\)
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation

Applied 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.