Question

Let \(M=\\{1,2, \ldots, 10] .\) Define a relation \(R\) on elements \(x, y \in M\) such that \((x, y) \in\) \(R\) if and only if there is a positive integer \(k\) such that \(x=k y\). Find the elements of \(R\).

Short answer

The set of ordered pairs in the relation \(R\) is: \(R = \{(1,1), (2,1), (3,1), ..., (10,10)\}\).

Step-by-step solution

Understand the Problem

We need to find the set of ordered pairs \((x, y)\) where \(x\) and \(y\) are elements of \(M = \{1, 2, \ldots, 10\}\), such that there exists a positive integer \(k\) with \(x = k \cdot y\). This means \(x\) must be a multiple of \(y\).

Analyze Possible Pairs

For each element \(y\) in \(M\), determine all possible \(x\) such that \(x = k \cdot y\) for some positive integer \(k\). This means we examine multiples of each possible \(y\) within the set \(M\).

Evaluate for Each y in M

Consider each \(y\) from 1 to 10:- For \(y=1\): Multiples are \(1, 2, 3, \ldots, 10\). - For \(y=2\): Multiples are \(2, 4, 6, 8, 10\). - For \(y=3\): Multiples are \(3, 6, 9\). - For \(y=4\): Multiples are \(4, 8\).- For \(y=5\): Multiples are \(5, 10\).- For \(y=6\): Multiple is \(6\).- For \(y=7\): Multiple is \(7\).- For \(y=8\): Multiple is \(8\).- For \(y=9\): Multiple is \(9\).- For \(y=10\): Multiple is \(10\).

Compile the Relation Set R

Combine the results from each \(y\) to compile the set \(R\). \[ R = \{(1,1), (2,1), (3,1), (4,1), (5,1), (6,1), (7,1), (8,1), (9,1), (10,1), (2,2), (4,2), (6,2), (8,2), (10,2), (3,3), (6,3), (9,3), (4,4), (8,4), (5,5), (10,5), (6,6), (7,7), (8,8), (9,9), (10,10)\} \]

Key concepts

Sets and Relations

In mathematics, a **set** is a collection of distinct objects, considered as an object in its own right. Sets are fundamental objects in mathematics. In this context, we are dealing with the set \( M = \{1, 2, \ldots, 10\} \). A **relation** on a set is a rule that assigns a relationship between elements of that set. Here, we define a relation \( R \) on set \( M \).A relation \( R \) is a set of ordered pairs \((x, y)\), where \( x \) and \( y \) are elements of \( M \). For the given exercise, \((x, y)\) is in relation \( R \) if there exists a positive integer \( k \) such that \( x = k \cdot y \). This means \( x \) must be a multiple of \( y \).Understanding such relations helps in examining how elements within a set can be interconnected based on specific rules. Relations are a key component in analyzing properties of sets and functions in discrete mathematics.

Multiples in Mathematics

**Multiples** play a critical role in understanding relations in mathematics. A multiple of a number \( y \) is any number that can be expressed as \( y \times k \) where \( k \) is a positive integer. To determine if two numbers are in the relation \( R \) from the set \( M \), we check if one number is a multiple of the other.Let's illustrate this with an example: consider the number 3 within the set \( M \). The multiples of 3 within this set are obtained by multiplying 3 with the integers 1, 2, and 3, giving us 3, 6, and 9, respectively. Hence, the ordered pairs for multiples of 3 include \((3,3)\), \((6,3)\), and \((9,3)\). Understanding multiples allows us to construct ordered pairs like these, which together make up the relation \( R \).Overall, grasping the concept of multiples assists in delineating structured relationships between numbers, a frequent task in discrete mathematics and number theory.

Discrete Mathematics

**Discrete Mathematics** is a branch of mathematics dealing with discrete elements that utilize algebra and arithmetic. It is the mathematical language of computer science, underpinning algorithms, data structures, and other computer-based concepts.In the context of this exercise, discrete mathematics focuses on the study of sets and their relations. Exploring relations like \( R \) requires us to list all possible ordered pairs where a condition (being a multiple) is satisfied. This systematic approach is key in discrete mathematics, helping mathematicians and computer scientists describe and resolve complex problems. When you explore such relations, you uncover patterns and connections within data, leading to efficient solutions in computation. Through mastering these techniques, students are better prepared for real-world problem-solving within the technological landscape.