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Chapter 12: Problem 28

\((r \mid s) r\) and \(r t \mid s t\)

Short Answer

Expert verified
In summary, given the relations \((r \mid s)r\) and \(rt \mid st\), we can express \(s = ar\) and substitute this into the second relation, obtaining \(st = art = a(rt)\). This demonstrates that there exists an integer \(a\) such that \(st = a(rt)\), which is the divisibility relation \((r \mid s)t\). Therefore, the given relations imply \((r \mid s)t\).

Step by step solution

01

Define the Given Relations

We are given two relations: 1. \((r \mid s)r\): this means that there exists an integer \(a\) such that \(s = ar\). 2. \(rt \mid st\): this means that there exists an integer \(b\) such that \(st = brt\).
02

Express s in terms of r

From the first relation, we can express \(s\) as follows: \[ s = ar \]
03

Substitute s in the second relation

Now, let's substitute the expression of \(s\) from Step 2 into the second relation: \[ st = art = brt \]
04

Simplify the Expression

Divide both sides of the equation by \(rt\), so we get: \[ a = b \]
05

Express (r | s)t

Now, we need to express \((r \mid s)t\). Since \(a = b\), we can substitute \(a\) for \(b\) in the second relation: \[ st = art = a(rt) \] From Step 2, we know that \(s = ar\). So, multiply both sides by \(t\): \[ st = (ar)t = a(rt) \] This expression shows that \((r \mid s)t\), as there exists an integer \(a\) such that \(st = a(rt)\). Hence, the given divisibility relations imply \((r \mid s)t\).

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Key Concepts

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

Mathematical Proof
When tackling complex mathematical problems, the aim is to establish certainty and clarity about the correctness of conjectures. This is achieved through the use of mathematical proofs, which are systematic and logical arguments that validate whether a statement is true or false. A proof must be rigorous, meaning that each step follows naturally and necessarily from the preceding ones, using established mathematical principles and theorems.
Integer Division
Integer division is a fundamental concept in mathematics that deals with the division of whole numbers, producing a quotient and sometimes a remainder. In the language of divisibility, we say that an integer 'a' divides another integer 'b' if there is an integer 'c' such that 'b = ac'. This relationship is denoted as 'a | b' and read as 'a divides b'. In the context of the exercise, it's essential to understand that if 'r | s', then there exists an integer 'a' such that 's = ar'. Multiplying both sides of a divisible relationship by the same integer does not change the nature of the divisibility.
Number Theory
Number theory is a branch of mathematics focused on the study of integers and integer-valued functions. One of the central themes in number theory is the concept of divisibility and understanding various properties of prime and composite numbers. In our exercise, we apply number theory principles to explore the relationship between divisibility and multiplication, which are foundational in solving problems related to factors, multiples, and the structure of numbers.
Discrete Mathematics
Within the realm of discrete mathematics, which revolves around the study of mathematical structures that are distinct and separable (as opposed to continuous), divisibility relations play a critical role. Problems that fall under discrete mathematics often involve finite or countably infinite sets, such as integers, graphs, and statements in logic. The given problem from our exercise draws upon discrete mathematical concepts by utilizing logical reasoning to demonstrate divisibility relationships, illustrating the compound use of integers and their properties.

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Most popular questions from this chapter

In 22-24 indicate whether the given strings belong to the language defined by the given regular expression. Briefly justify your answers. 22\. Expression: \((b \mid \epsilon) a(a \mid b)^{*} a(b \mid \epsilon)\), strings: \(a a a b a, b a a b b\)

The language consisting of all strings of \(a\) 's and \(b\) 's in which the third character from the end is a \(b\).

\(\left(x \mid y z^{*}\right)^{*}\left(y x \mid(y z)^{*} z\right)\)

In \(I\) and 2 let \(\Sigma=\\{x, y\\}\) be an alphabet. 1\. a. Let \(L_{1}\) be the language consisting of all strings over \(\Sigma\) that are palindromes and have length \(\leq 4\). List the elements of \(L_{1}\) between braces. b. Let \(L_{2}\) be the language consisting of all strings over \(\Sigma\) that begin with an \(x\) and have length \(\leq 3\). List the elements of \(L_{2}\).

Write a regular expression to define the set of strings of 0 's and 1 's with an even number of 0 's and even number of 1 's.
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