Chapter 10: Problem 23

Prove that if \(s \in[m]_{n}\) and \(t \in[k]_{n}\) then \(s \times t \in[m\) \(\times k] n\).

Short Answer

Expert verified

The product of two elements from equivalence classes modulo-n will also be an element of an equivalence class modulo-n. Specifically, if \(s \in[m]_{n}\) and \(t \in[k]_{n}\), then their product \(s \times t\) will belong to the equivalence class modulo-n represented by the product of the representatives of the original two classes, \(m \times k\).

Step by step solution

Introduction of arbitrary elements

Let's introduce two arbitrary elements from the equivalence classes \(s\) and \(t\). These can be represented as \(s = m + nx\) and \(t = k + ny\) where \(x\) and \(y\) are integers.

Multiply the elements

Now we multiply these elements together. \(s \times t = (m + nx) \times (k + ny) = mk + n(mx + ky + nxy)\). We can see the product can be written in the form \(a + nb\) where \(a = mk\) and \(b = mx + ky + nxy\), which are both integers since \(m\), \(k\), \(x\), and \(y\) are integers and \(n\) is a natural number.

Conclude the proof

Since the product \(s \times t = mk + n(mx + ky + nxy)\) is of the form \(a + nb\) where \(a\) and \(b\) are integers, this implies that \(s \times t\) is an element of the equivalence class modulo n, specifically the \(mk modulo-n\) class. Thus, it is proven that if \(s \in[m]_{n}\) and \(t \in[k]_{n}\) then \(s \times t \in[m \times k]_{n}\).

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Computer Science Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Prove that if \(s \in[m]_{n}\) and \(t \in[k]_{n}\) then \(s \times t \in[m\) \(\times k] n\).

Short Answer

Step by step solution

Introduction of arbitrary elements

Multiply the elements

Conclude the proof

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Computer Science Textbooks

Theory of Computation

Computer Systems

Functional Programming

Databases

Computer Programming

Data Structures

Study anywhere. Anytime. Across all devices.