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Chapter 4: Problem 23

Show that if \(n\) and \(k\) are positive integers, then \(\lceil n / k\rceil=\) \(\lfloor(n-1) / k\rfloor+ 1\)

Short Answer

Expert verified
True, \(\lceil \frac{n}{k} \rceil = \lfloor \frac{n-1}{k} \rfloor + 1\).

Step by step solution

01

- Understand Ceiling Function

The ceiling function \(\lceil x \rceil\) of a real number \(x\) is the smallest integer greater than or equal to \(x\).
02

- Understand Floor Function

The floor function \(\lfloor x \rfloor\) of a real number \(x\) is the largest integer less than or equal to \(x\).
03

- Change Variable

Let \(y = \frac{n}{k}\). We need to show that \(\lceil y \rceil = \lfloor (y-\frac{1}{k}) \rfloor + 1\).
04

- Apply Floor Function on \(y-\frac{1}{k}\)

Consider \(y-\frac{1}{k} = \frac{n}{k} - \frac{1}{k} = \frac{n-1}{k}\). Applying the floor function, we obtain \(\lfloor \frac{n-1}{k} \rfloor\).
05

- Add 1

According to the requirement, add 1 to the result from Step 4: \(\lfloor \frac{n-1}{k} \rfloor + 1\).
06

- Compare with Ceiling Function

Notice that \(\lceil \frac{n}{k} \rceil\) by definition is the smallest integer greater than or equal to \(\frac{n}{k}\). Meanwhile, \(\lfloor \frac{n-1}{k} \rfloor + 1\) adjusts for any fractional part ensuring it aligns with \(\lceil \frac{n}{k} \rceil\). Thus both expressions yield the same result.

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Key Concepts

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

Ceiling Function
The ceiling function, denoted as \(\backslash lceil x \rceil\), returns the smallest integer greater than or equal to a given real number \(x\). For example, \(\backslash lceil 2.3 \rceil = 3\) and \(\backslash lceil 5 \rceil = 5\). It's useful when you need to 'round up' a number to its nearest whole integer.
Floor Function
The floor function, denoted as \(\backslash lfloor x \rfloor\), gives you the largest integer less than or equal to a given real number \(x\). For instance, \(\backslash lfloor 4.7 \rfloor = 4\) and \(\backslash lfloor 3 \rfloor = 3\). This operation 'rounds down' a number to its nearest whole integer. In mathematical expressions, the floor function assesses the integer part of a number.
Proof in Discrete Mathematics
Proofs in discrete mathematics often involve demonstrating the equality of two mathematical expressions or showing that a condition holds for all elements in a specific set. In the provided exercise, we are proving the relationship between the ceiling and floor functions. We began by setting a variable \(\backslash y = \backslash frac{n}{k}\). Using definitions, we showed that \(\backslash lceil \backslash frac{n}{k} \rceil = \lfloor \backslash frac{n-1}{k} \rfloor + 1\), by breaking down the problem and applying the properties of ceiling and floor functions. These methods are foundational tools in learning how to logically, and rigorously, prove mathematical statements.

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Most popular questions from this chapter

Prove that a parity check bit can detect an error in a string if and only if the string contains an odd number of errors.

The Vigen猫re cipher is a block cipher, with a key that is a string of letters with numerical equivalents \(k_{1} k_{2} \ldots k_{m},\) where \(k_{i} \in \mathbf{Z}_{26}\) for \(i=1,2, \ldots, m .\) Suppose that the numerical equivalents of the letters of a plaintext block are \(p_{1} p_{2} \ldots p_{m} .\) The corresponding numerical ciphertext block is \(\left(p_{1}+k_{1}\right)\) mod 26 \(\left(p_{2}+k_{2}\right) \bmod 26 \ldots\left(p_{m}+k_{m}\right)\) mod \(26 .\) Finally, we translate back to letters. For example, suppose that the key string is RED, with numerical equivalents \(1743 .\) Then, the plaintext ORANGE, with numerical equivalents \(141700130604,\) is encrypted by first splitting it into two blocks 141700 and 13 \(0604 .\) Then, in each block we shift the first letter by 17 , the second by \(4,\) and the third by \(3 .\) We obtain 52103 and 0410 \(07 .\) The cipherext is FVDEKH. The ciphertext OIKYWVHBX was produced by encrypting a plaintext message using the Vigen猫re cipher with key HOT. What is the plaintext message?

Show that every positive integer can be represented uniquely as the sum of distinct powers of 2 . (Hint: Consider binary expansions of integers.)

Describe the steps that Alice and Bob follow when they use the Diffie-Hellman key exchange protocol to generate a shared key. Assume that they use the prime \(p=23\) and take \(a=5,\) which is a primitive root of \(23,\) and that Alice selects \(k_{1}=8\) and Bob selects \(k_{2}=5 .\) (You may want to use some computational aid.)

We describe a basic key exchange protocol using private key cryptography upon which more sophisticated protocols for key exchange are based. Encryption within the protocol is done using a private key cryptosystem (such as AES) that is considered secure. The protocol involves three parties, Alice and Bob, who wish to exchange a key, and a trusted third party Cathy. Assume that Alice has a secret key \(k\) alice that only she and Cathy know, and Bob has a secret key \(k_{\text { Bob }}\) which only he and Cathy know. The protocol has three steps: (i) Alice sends the trusted third party Cathy the message "request a shared key with Bob" encrypted using Alice's key \(k_{\text { Alice }}\) . (ii) Cathy sends back to Alice a key \(k_{\text { Alice Bob }},\) which she generates, encrypted using the key \(k_{\text { Alice }},\) followed by this same key \(k_{\text { Alice, Bob }},\) encrypted using Bob's key, \(k_{\text { Bob }}\) (iii) Alice sends to Bob the key \(k_{\text { Alice }, \mathrm{Bob}}\) encrypted using \(k_{\text { Bob }},\) known only to Bob and to Cathy. Explain why this protocol allows Alice and Bob to share the secret key \(k_{\text { Alice }, \text { Bob }},\) known only to them and to Cathy.
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